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¨ Electric ﬁeld effect on the refractive index changes in a Modiﬁed-PoschlTeller quantum well V. Ustoglu Unal a,n, E. Aksahin a, O. Aytekin b a b

Department of Physics, Yeditepe University, Kayısdagı, Atasehir, 34755 Istanbul, Turkey Department of Physics, Adıyaman University, 02040 Adıyaman, Turkey

H I G H L I G H T S c

c

c

We consider an electron in a quantum well modeled by a Modiﬁed ¨ Poschl–Teller potential. We present a numerical calculation of the wave functions and energy levels of the system. The nonlinear optical properties (the refractive index changes) of this system have been investigated.

G R A P H I C A L

A B S T R A C T

Contribution of the second term in the nonlinear refractive index change as a function of photon energy for various applied electric ﬁeld values E, where Z ¼ 6and the well depth is 0:34 eVwith I ¼ 1:0 MW=cm2 .

a r t i c l e i n f o

abstract

Article history: Received 12 June 2012 Received in revised form 10 October 2012 Accepted 23 October 2012 Available online 2 November 2012

The effect of an applied static ﬁeld is studied on the optical properties of a quantum well (QW) ¨ represented by a Modiﬁed Poschl–Teller potential. This potential allows analytical solution of the eigen-values and eigen-functions which in turn makes the numerical calculation of optical properties quite transparent. In this work, we concentrate on the linear and nonlinear refractive index changes. ¨ Comparison of the results using the ﬁnite Modiﬁed-Poschl–Teller (MPT) potential with those in the ¨ literature using the inﬁnite Poschl–Teller (PT) potential shows that the main difference between the two potentials is coming from the depth differences of the two wells. The changes in the refractive indices are bigger than those using inﬁnite PT potential. If one wants a larger change in the total refractive index, one should try to reduce the applied electric ﬁelds and the optical intensities. & 2012 Elsevier B.V. All rights reserved.

1. Introduction Important advances in both epitaxial growth and laser technologies have created an ever growing interest in the linear and nonlinear optical properties of QW [1–19]. This interest is also supported by the communication technology and the need for faster optical switches and communication lines [9–11, 20–22].

n

Corresponding author. Tel.: þ90 216 5780740; fax: þ 90 216 5780672. E-mail address: [email protected] (V.U. Unal).

1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.10.018

The scientiﬁc interest is concentrated on the nonlinearity of the polarization of the medium. The nonlinear optical response depends strongly on the shape of the conﬁning potential experienced by the charge carriers in the medium. The anharmonic character of the potential is crucial. The asymmetry in potential proﬁle may be obtained in real wells by the application of an external ﬁeld or by compositionally grading the QW. There is a growing number of studies in the literature which employs different conﬁning potentials [16]. This is usually based on the better control of the atomic layers of the constituents which makes it possible to design QW and dots with variable shapes.

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V.U. Unal et al. / Physica E 47 (2013) 103–108

The effect of different conﬁning potentials should also be studied from the perspective of optimizing nonlinear optical properties of these low dimensional systems [4]. The one¨ dimensional conﬁning potential is represented by a Poschl–Teller (PT) potential and used in calculating optical properties by Yıldırım and Tomak [23]. They have studied nonlinear optical properties, including the optical absorption, linear and nonlinear changes in the refractive index [24], and the third harmonic generation under an electric ﬁeld [25]. ¨ In the present work, we consider the Modiﬁed-Poschl–Teller (MPT) potential because of its simplicity. This will give us a chance to compare the results. We consider mainly the effect of the applied electric ﬁeld on the linear and nonlinear changes in the refractive index of the medium. The electronic structure of the QW is calculated using the effective-mass approximation [26]. Our results show that the changes in the refractive index depend sensitively on the shape, depth and the width of the potential. The results also depend on the optical intensity of the incident beam. The paper is organized as follows: Section 2 describes the theoretical framework for both the electronic structure and the optical coefﬁcients. Section 3 presents the results and their overall discussion. A brief conclusion is presented in Section 4.

2. Theoretical framework The energy levels En and the envelope wave functions fn(z) are found by solving the one-dimensional effective-mass equation describing the quantization of motion along the growth direction:

_2 d 1 dfn ðzÞ þV ðzÞfn ðzÞ ¼ En fn ðzÞ n dz 2 dz m

ð1Þ

where N¼

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n!ðkn1Þ!ð2k2n1Þ! 1 pﬃﬃﬃﬃ and k ¼ þ Z Z þ 1Þ with n ¼ 0,1,2,. . . 4 p kn3=2 !ð2kn1Þ!

The corresponding energies are given by 2

En ¼

_2 b 2mn

2 1 k nþ 2

ð4Þ

Energies vary with varying b, the parameter controlling the well depth. As expected, the energy difference between levels decreases by decreasing well width. We consider a two-level system under an electric ﬁeld applied in the growth direction (z-direction) of QW. The PT-potential is no longer symmetrical, as shown in Fig. 2. We make the generalization that the Hamiltonian of the system splits into two parts, ! H ¼ H0 þ e9 E 9z

ð5Þ

where H0 is the unperturbed part of the total Hamiltonian, e is the ! absolute value of the electron charge and 9 E 9 is the external electric ﬁeld [29,30]. The second term in Eq. (5) describes the interaction of an electromagnetic wave with the two-level electron system in the dipole approximation. If the inequality, ! ð0Þ e9 E 9z 5 9Eð0Þ 0 E1 9

ð6Þ

is satisﬁed, we determine the wave functions and the corresponding energy levels using the time-independent, non-degenerate perturbation theory [29], such that; 2 ! 2 X 9znj 9

!

2 Dn ¼ En Eð0Þ n ¼ e9 E 9znn þe 9 E 9

where V ðzÞ ¼

V0 2

cosh ðbzÞ

¼

2 _2 b Z Z þ 1 2mn cos h2 ðbzÞ

h

ikn2ð1=2Þ

!X

fn ðzÞ ¼ fð0Þ n ðzÞ þ e9 E 9

þk C n ðtanhðbzÞÞ n

ð3Þ

-14

-8

-2

4

10

16

znj Enð0Þ Eð0Þ j

ð9Þ

and the matrix elements, zij, are evaluated using the unperturbed wave functions. We consider an optical radiation of angular frequency w applied to the system with the polarization along the growth -28

-22

-16

-10

-4 0.00

2

-0.02

-0.09

η=2

-0.14 -0.19 -0.24 -0.29

-0.04 eV

η=4

ð8Þ

znn ¼ 0

22

-0.04

þ :::

ð0Þ

η=6

eV

ð7Þ

with no degeneracy, f0 is expected to be a parity eigen state; hence,

-34

-20

fð0Þ j ðzÞ

jan

ao* -26

þ:::

and ð2Þ

represents the MPT-potential Z 41. The potential is controlled by two parameters Z and b which determines its width and depth, respectively. The parameter dependence of MPT-potential is given in Fig. 1, where ano is effective Bohr radius. The envelope functions can be written in terms of Gegenbauer functions [27,28]

fn ¼ N sech2 ðbzÞ

ð0Þ ð0Þ j a n En Ej

8

14

20

26

32 η=6 η=4 η=2

-0.06 -0.08 -0.10 -0.12 -0.14 ao*

-0.34 Fig. 1. Variation of the well width with the parameter Z, where the depth is ﬁxed to 0.34 eV.

Fig. 2. Variation of the well shape with the effect of applied electric ﬁeld E, for various well widths (various Z), where E ¼ 15 kV/cm and the well depth is 0.34 eV with I¼ 1.0 MW/cm2.

V.U. Unal et al. / Physica E 47 (2013) 103–108

becomes

direction. The incident ﬁeld can be written as X EðtÞ ¼ E wj exp iwj t

nðoÞ ¼

ð10Þ

pﬃﬃﬃ

4pwðwÞ : 2e

e 1 þ Re

j

wð3Þ ðwÞ ¼

rs m10 2 , E10 _oi_G0

Using the expression for the third-order nonlinear susceptibility, one ﬁnds for the nonlinear change in the refractive index;

2pRe wð3Þ ðwÞ Dnð3Þ ðoÞ ¼ : ð16Þ nr

ð11Þ

" 4 2pIrs m10 4 nr cðE10 _oi_G0 Þ ðE10 _oÞ2 þ ð_G0 Þ2 # m m 2 1 00 11 2 ðE10 _oi_G0 ÞðE10 i_G0 Þ m

ð12Þ

3. Numerical results and discussion

10

The input parameters used in the calculations are

Here, I is the intensity of the incident ﬁeld, rs is the electron density, Eij ¼Ei Ej, nr is the refractive index and mij is the matrix element of the dipole operator D E m ¼ f qz f d 0 ði, j ¼ 0; 1Þ ij

i

j

rs ¼ 3 1016 cm3 , G0 ¼

ð13Þ

where e is the static dielectric constant of the QW material. For sufﬁciently small densities of electrons, the expression

0.25

E1 - E0

eV

eV

0.05 6

mn ¼ 0:067 m0

0.1

E0

4

nr ¼ 3:2,

0.2

E1

0.15

0.05 2

1 s1 , 0:14

where rs is the three dimensional electron density, G0 is the decay rate and m0 is the bare electron mass. These values are chosen so that we can compare our results with the available literature [24]. The electron concentrations depend on the position of the Fermi level. The correct handling of this dependence on the external probes is properly done in earlier investigations [26,31,32].

kn kn

The frequency-dependent refractive index is jpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃk e þ4pwðwÞ nðoÞ ¼ Re

ð14Þ

The change in the refractive index due to the incident ﬁeld can be written as 2pwð1Þ ðwÞ 2pwð1Þ ðwÞ pﬃﬃﬃ ¼ Re Dnð1Þ ðoÞ ¼ Re : ð15Þ nr e

where the summation is over all frequencies. Using the density matrix formalism, one can write the ﬁrst-and the third-order susceptibilities as [26];

wð1Þ ðwÞ ¼

105

8 10 12 14 16

V0 (eV) 0 0.26 0.30 0.34 0.38 0.1

0.15

0.2

0.25

0.3

0.35

0.4

Fig. 3. Energies of ground and excited states as a function of (a) well width and (b) well depth.

20.0

x10-56(Cm)2

16.0

|μ10|²

12.0

|(μ11-μ00)/μ10|²

0.03

0.02

0.01

8.0

4.0

0 2

4 η

6

0

5

10

15

20

Electric Field (kV/cm)

Fig. 4. Variation (a) of the dipole matrix element 9m1092 with well width and (b) of 9(m11 m00)/m1092 with applied electric ﬁeld E.

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V.U. Unal et al. / Physica E 47 (2013) 103–108

In Fig. 3, the energies of the ground state and that of the ﬁrst excited state are plotted as a function of the well width and depth. The difference between the energies of these states E10 decreases with increasing well width (Fig. 3a) but increases with increasing well depth (Fig. 3b). Fig. 4a illustrates the dipole matrix element 9m1092 as a function of well width. The magnitude of the linear change in the refractive index Dn(1) depends on 9m1092 which increases considerably with increasing well width and decreases with increasing well depth. In Fig. 4b, we plot 9(m11 m00)/m1092 term in Eq. (12) for various electric ﬁelds E which is restricted by Eq. (6). The net contribution of the diagonal terms to the nonlinear change in the refractive index Dn(3) increases with the electric ﬁeld. The calculated linear term in the refractive index Dn(1) is displayed in Fig. 5 for various values of Z which characterizes well width of the QW. There are two peaks at E10 7i_G0 , in each curve. The peaks are blue-shifted for decreasing Z at I¼1.0 MW/cm2 and well depth 0.34 eV. This is a result of increasing intersubband energy difference when the well width becomes narrower, increasing the conﬁnement effect. Dn(1) is the largest contribution to the total refractive index change Dn(total). Since the linear change in the refraction index is related to the dipole matrix element 9m1092 which increases with the well width, the peaks in Fig. 5 increase with Z.

n(total) n (¹ ) 0.06

n(³)

η=6

0.04 η=2

n

0.02 0 -0.02 -0.04 -0.06 0.06

0.1

0.14

0.18

hv (eV) Fig. 5. The linear, nonlinear and total refractive index changes as a function of photon energy, for various Z values with I ¼1.0 MW/cm2, where the well depth is 0.34 eV.

The nonlinear change in the refractive index Dn(3) is plotted in Fig. 5, as a function of the photon energy for the same values of the intensity and the well depth. Similar to Dn(1), the peaks for Dn(3) are also blue-shifted for decreasing Z values. It is clearly seen that the contribution from Dn(3) to the total change in the refractive index is of opposite sign with that of Dn(1). This is explicitly shown in Fig. 5, where the linear, nonlinear and the total changes in the refractive index are given. Therefore, Dn(3) lowers the total refractive index change. The largest change in the total refractive index is at a photon energy approximately 7 meV off the line center. Fig. 6a, shows the dependence of Dn(1) on the photon energy and the well depth V0. The peaks of Dn(1) are blue-shifted with increasing well depth. This is a result of the dependence of the energy levels on the well depth. The energy difference between levels increases by increasing V0. The drop in the actual peak values is rather small. The dependence of Dn(3) on photon energy and the well depth is shown in Fig. 6b. The peaks are blue-shifted with increasing well depth. The drop in the actual peak values are small, more or less like in the case of Dn(1). In both cases, this drop is related to both the dipole matrix elements in the numerator of Eq. (12) giving w(3)(w), and the energy differences in the denominator. The change in the total refractive index Dn(total) as a function of the photon energy for various intensities with ﬁxed Z ¼6 and the well depth of 0.34 eV is given in Fig. 7. The trend is that the total change in the index of refraction is reduced as the optical intensity is increased. This is a natural result of the increasing negative contribution of nonlinear term which is directly proportional to the intensity. This term has a large variation, and even may exceed the peak value of the linear refractive index change for high optical intensites. The sensitive dependence of the Dn(total) to the depth of the potential well V0 to the photon energy _w and to the optical intensity I makes this system ideal for nonlinear optical material applications. One can control effectively the nonlinear response of the system. There is yet another factor we can change at will which is the applied electric ﬁeld E. Fig. 8 shows the variation of the QW shape with the intensity of the applied electric ﬁeld E. As expected, the main effect of the electric ﬁeld is to make the QW more asymmetrical as E increases. This, of course, increases the nonlinear optical properties. Comparing Figs. 1 and 8, the potential energy proﬁle is tilted by the applied ﬁeld. The asymmetry of the QW becomes stronger as the strength of the applied electric ﬁeld increases and the energy levels E1 and E0 are pushed lower, and also E10 decreases. In this study, the maximum contribution of the second term in Eq. (12) is determined at the incident photon energy 0.089 eV, E ¼15 kV/cm, Z ¼6 and the well depth of 0.34 eV. It is clear that the resulting

Vo=-0.38 eV Vo=-0.34 eV

0.07

0.03

Vo=-0.30 eV

0.05

0.02 0.01

0.01

n (3)

n (1)

0.03

-0.01

-0.01

-0.03

-0.02

-0.05 -0.07 0.06

0

0.08 hv (eV)

0.1

0.12

-0.03 0.06

0.08 0.1 hv (eV)

0.12

Fig. 6. (a) The linear term and (b) the nonlinear term in the refractive index change as a function of photon energy for various well depths, with I ¼ 1.0 MW/cm2.

V.U. Unal et al. / Physica E 47 (2013) 103–108

0.06

107

1.0

I=0 I=1.0 MW/cm² I=2.0 MW/cm²

0.04

I=3.0 MW/cm²

0.0 0.06

x10-5

Δn (total)

0.02 0

0.07

0.08

0.09

0.10

0.11

0.12

-1.0

-0.02

E=15kV/cm

-2.0

-0.04

E=10kV/cm E=5kV/cm

-0.06 0.07

0.08

0.09

0.1

0.11

-3.0

hv (eV)

hν(eV) Fig. 7. The change in the total refractive index as a function of photon energy for various intensities, where Z ¼6 and the well depth is 0.34 eV.

Fig. 9. Contribution of the second term in the nonlinear refractive index change as a function of photon energy for various applied electric ﬁeld values E, where Z ¼ 6 and the well depth is 0.34 eV with I¼ 1.0 MW/cm2.

a 0* -28

-22

-16

-10

-4

8

14

20

1.0

26

E=15 kV/cm E=10 kV/cm E=5 kV/cm E=0

0.0 0.06 x10-5

eV

-0.01

2

0.11

0.16

-1.0 η=3 -2.0

-0.03

η=6

Fig. 8. Variation of the well shape with the effect of applied electric ﬁeld E where Z ¼ 6 and the well depth is 0.34 eV with I¼ 1.0 MW/cm2.

nonlinearity is quite small. This is the reason why we ﬁnd a rather small contribution to the change in the refractive index. The contribution is shown as a function of photon energy for various applied ﬁelds in Fig. 9. As expected, the effect is bigger for stronger electric ﬁelds. Fig. 10 displays the contribution of the nonlinear term to the change in the refractive index as a function of photon energy, for the well widths of Z ¼6, Z ¼3, E¼15 kV/cm, and the well depth of 0.34 eV with I¼1.0 MW/cm2. The peak values are mostly negative and shift to higher energies for wider wells. The peak values also become bigger for stronger electric ﬁelds. This is a result of the fact that the multiplying factor of the dipole matrix elements is an increasing function of the applied electric ﬁeld. This nonlinear term is not dominating the w(3)(w) values because of its small magnitude. In Table 1, Dn(1), Dn(3) and Dn(total) are given for the PT and MPT-potentials with different well widths. The comparison of the results of the MPT and PT-potentials, as given in Table 1, shows that we get more or less the same peak values for Dn(1), Dn(3) and Dn(total) for the chosen values. In this comparison, the MPT-potential is 98 A_ wide and ﬁnite, whereas the PT-potential is 126:5 A_ wide and inﬁnite.

4. Conclusion In the present work, we have studied the changes in linear and nonlinear refractive index for AlGaAs GaAs QW. This study gave

-3.0

hv (eV)

Fig. 10. Contribution of the second term in the nonlinear refractive index change as a function of photon energy for various well widths (various Z) where E ¼ 15 kV/ cm, the well depth is 0.34 eV with I ¼1.0 MW/cm2.

Table 1 Comparison between the (PT)-potential results with Z ¼2 and k ¼ 2.0 in Yıldırım and Tomak [24] and the present results (MPT)-potential with Z ¼ 2, Z ¼ 6 and depth 0.34 eV with I¼ 1.0 MW/cm2. Potential type

Well

PT MPT MPT

126.5 98 126.5

Dn(1) Peak

Dn(3) Peak

Dn(total) Peak

_ width (A) 0.03 (0.171 eV) 0.006 (0.172 eV) 0.025 (0.17 eV) 0.028 (0.165 eV) 0.006 (0.167 eV) 0.024 (0.164 eV) 0.057 (0.084 eV) 0.025 (0.086 eV) 0.042 (0.082 eV)

us a chance to compare our results using the ﬁnite MPT-potential (depth 0.34 eV) with those in the literature [23–25] using inﬁnite PT-potential. The main difference between the two potentials is expected to come from the depth differences of the two wells. We ﬁnd that the conribution from the nonlinear terms is quite small, as found in earlier studies. Our peak values are somewhat smaller than those found by Yıldırım and Tomak [24]. But our values for the changes in the refractive indices are bigger than those in [24]. This is mainly because of the values of the dipole matrix elements resulting from different well depths.

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V.U. Unal et al. / Physica E 47 (2013) 103–108

In summary, we have shown that if one wants a larger change in the total refractive index, one should try to reduce the applied electric ﬁelds and the optical intensities.

Acknowledgments The authors wish to thank Prof. Dr. Mehmet Tomak from Middle East Technical University, Turkey, for providing support during this work, for valuable comments and suggestions. References [1] E. Rosencher, Ph. Bois, Physical Review B 44 (20) (1991) 11315. [2] M.K. Gurnick, T.A. de Temple, IEEE Journal of Quantum Electronics 19 (1983) 791. [3] M.M. Fejer, et al., Physical Review Letters 62 (1989) 1041. [4] S.J.B. Yoo, et al., Applied Physics Letters 58 (1991) 1724. [5] D. Walrod, et al., Applied Physics Letters 59 (1991) 2932. [6] C. Lien, Y. Huang, J. Wong, Journal of Applied Physics 76 (1994) 1008. [7] Y. Huang, C. Lien, Journal of Applied Physics 75 (1994) 3223. [8] A. Sa’ar, et al., Applied Physics Letters 61 (1992) 1263. [9] F. Capasso, K. Mohammed, A.Y. Cho, IEEE Journal of Quantum Electronics QE-22 (1986) 1853. [10] D.A.B. Miller, J.S. Weiner, D.S. Chemla, IEEE Journal of Quantum Electronics 22 (9) (1986) 1816. [11] P.F. Yuh, K.L. Wang, Physical Review B 38 (12) (1988) 8377.

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