11 Elasto-optic, Electro-opticandNonlinearOptical Properties

11.1 ELASTO-OPTIC EFFECT

11.1.1 Group-IV Semiconductor Alloy

Data relating to elasto-optic behavior plays an important role not only in the design ofelasto-optic devices, such as light modulators, deflectors and switches, but also in theanalysis of lattice-mismatched semiconductor heterostructures. Photoelasticity is depen-dent on wavelength [1–4]. The photoelastic coefficient in 6H-SiC has been measured byGeidur [5]. Since the fundamental absorption edge of 6H-SiC is an indirect band gap, itscontribution to the photoelasticity is negligibly small. Neither theoretical nor experimentalstudies relating to photoelasticity have been carried out on any group-IV semiconductoralloy.

11.1.2 III–V Semiconductor Alloy

A method for calculating the photoelastic coefficients in III–V semiconductors has beendescribed byAdachi and Oe [6]. This modelwas applied to AlxGa1�xAs andGaxIn1�xPyAs1�y/InP. The photoelastic coefficientape for AlxGa1�xAs underX||[100] stress has been reproducedin Figure 11.1. The black and gray circles correspond to the experimental data for GaAs (x¼ 0)and Al0.5Ga0.5As, respectively. The definition of ape can be found in Adachi [1].

More recently, Kikkarin et al. [7] have calculated the spectral dependence of the photo-elastic constants, p11, p12 and p44, for AlxGa1�xAs with x up to 0.3. They also discussed theacousto-optic interaction in AlxGa1�xAs/GaAs waveguides.

11.1.3 II–VI Semiconductor Alloy

The photoelastic constants pij can be determined from the Brillouin scattering cross-sections [1]. Ziobrowski et al. [8] used Brillouin scattering on BexZn1�xSe and obtained pijat l¼ 488 nm (0� x� 0.25). Adachi and Hamaguchi [9] carried out resonant Brillouinscattering measurements on ZnxCd1�xTe and obtained the spectral dependence of p44 forx¼ 0.65 and 0.80. The data suggest that the sign for p44 in ZnxCd1�xTe is negative in the regionremote from theE0 edge and becomes positivewhen thewavelength approaches the band edge,as in ZnTe and CdTe [4].

Properties of Semiconductor Alloys: Group-IV, III–V and II–VI Semiconductors Sadao Adachi© 2009 John Wiley & Sons, Ltd. ISBN: 978-0-470-74369-0

11.2 LINEAR ELECTRO-OPTIC CONSTANT

11.2.1 Group-IV Semiconductor Alloy

Principally, the diamond lattice shows no linear electro-optic effect [1]. The linear electro-opticconstant r41 for 3C-SiC has been reported to be�2.7� 0.5 pm/Vat l¼ 632.8 nm [10]. There isan ordered alloy phase in SixGe1�x (Section 1.3.2) and this phase will exhibit a linear electro-optic effect [11], but no data are available at present.

11.2.2 III–V Semiconductor Alloy

The linear electro-optic effect in AlxGa1�xAs has been studied by Glick et al. [12]. Theymeasured the electro-optic phase difference in an Al0.17Ga0.83As waveguide and determinedr41 at l¼ 1.1523mm, as shown in Table 11.1.

Bach et al. [13] havemeasured r41 for GaxIn1�xPyAs1�y/InP (y¼ 0.20 and 0.34). The valuesthey obtained are summarized in Table 11.1. Figure 11.2 also shows the theoretical r41dispersion obtained by Adachi and Oe [14], together with the data produced by Bach et al.(y¼ 0.34) [13]. It can be seen that the theoretical r41 curve shows strong dispersion near E0

(l�1.1mm). This curve was obtained from an electric-field-induced modulation of theelectronic energy-band structure.

11.2.3 II–VI Semiconductor Alloy

The linear electro-optic constant r41 at l¼ 1.5mm of Bridgman-grown Zn0.1Cd0.9Te has beenmeasured by Zappettini et al. [15] (see Table 11.1). The implementation of a Zn0.1Cd0.9Te-based electro-optic switch which reaches �30 dB of extinction ratio and sub millisecondresponse time has also been presented by the same authors.

Figure 11.1 Photoelastic coefficientape for AlxGa1�xAswith increments in x of 0.2. The black and graycircles correspond to the experimental data for GaAs (x¼ 0) and Al0.5Ga0.5As, respectively. [Reprintedwith permission from S. Adachi and K. Oe, J. Appl. Phys. 54, 6620 (1983). Copyright (1983), AmericanInstitute of Physics]

358 PROPERTIES OF SEMICONDUCTOR ALLOYS

The electro-optic constants rij of w-CdSxSe1�x have been measured by several authors[16,17]. The results obtained byBaldassarre et al. [16] are r13¼�1.5 pm/Vand r33¼ 2.4 pm/Vat l¼ 632.8 nm for x¼ 0.75. More recently, Bini and Razzetti [17] reported the unclampedelectro-optic constant

rTc ¼ rT13�none

� �3

rT33 ð11:1Þ

to be 5.81� 0.03 pm/V for x¼ 0.5, where no and ne are, respectively, the ordinary andextraordinary refractive indices and rTij is the unclamped electro-optic tensor component [14].These authors also obtained rTc ¼ 6:45� 0:03 pm=V for w-CdS and 7.23� 0.04 pm/V for

Figure 11.2 Theoretical r41 and R11�R12 curves along with the experimental data of Bach et al. [13] forGaxIn1�xPyAs1�y/InPwith y¼ 0.34. [Reprinted with permission from S. Adachi andK. Oe, J. Appl. Phys.56, 1499 (1984). Copyright (1984), American Institute of Physics]

Table 11.1 Linear electro-optic constant r41 for some III–V and II–VI semiconductor alloys.S¼ clamped value

System Alloy x (y) r41 (pm/V) l (mm) Comments

III–V AlxGa1�xAs 0 �1.80a 10.6 S0.17 �1.43b 1.1523

GaxIn1�xPyAs1�y/InP 0.20 (y) �1.34c 1.320.34 (y) �1.43c 1.250.34 (y) �1.44c 1.321.00 (y) �1.68a 1.50 S

II–VI ZnxCd1�xTe 0 4.1a 10.6 S (|r41|)0.1 4.3d 1.5 S (|r41|)1.0 3.9a 10.6 S (|r41|)

aS. Adachi, Properties of Group-IV, III–V and II–VI Semiconductors. John Wiley & Sons, Ltd, Chichester, 2005bM. Glick et al., J. Appl. Phys. 63, 5877 (1988)cH. G. Bach et al., Appl. Phys. Lett. 42, 692 (1983)dA. Zappettini et al., J. Electron. Mater. 30, 743 (2001)

ELASTO-OPTIC, ELECTRO-OPTIC AND NONLINEAR OPTICAL PROPERTIES 359

w-CdSe. A smaller alloy value than those of the endpoint binaries is considered to be due to themicroscopic strains present in the alloy.

11.3 QUADRATIC ELECTRO-OPTIC CONSTANT

11.3.1 Group-IV Semiconductor Alloy

No detailed data are available for group-IV semiconductor alloys.

11.3.2 III–V Semiconductor Alloy

Bach et al. [13] have studied the quadratic electro-optic effect in GaxIn1�xPyAs1�y/InPdouble-heterostructure waveguides. They obtained the quadratic electro-optic constantsR11�R12¼�0.8� 10�20m2/V2 at l¼ 1.32 mm for y¼ 0.20 and R11�R12¼� 5.8� 10�20

(�3.1� 10�20) m2/V2 at l¼ 1.25 (1.32) mm for y¼ 0.35. These results (y¼ 0.35) are shownin Figure 11.2, together with the theoreticalR11�R12 values calculated byAdachi andOe [18].It can be seen that the quadratic electro-optic constant R11�R12 shows very strong dispersioncompared with the linear electro-optic constant (r41).

11.3.3 II–VI Semiconductor Alloy

No detailed data are available for II–VI semiconductor alloys.

11.4 FRANZ–KELDYSH EFFECT

11.4.1 Group-IV Semiconductor Alloy

Although the Franz–Keldysh effect in the indirect absorption edge of SiC has been dis-cussed [19], no detailed data are available for group-IV semiconductor alloys.

11.4.2 III–V Semiconductor Alloy

To our knowledge, there have been no publications relating to the direct measurement ofthe absorption edge shift in AlxGa1�xAs under an applied electric field. The Franz–Keldysheffect in GaxIn1�xPyAs1�y/InP has been studied experimentally by several authors [13,20–22].The change in the optical absorption coefficient Da observed in InP/GaInPAs/InP p-i-n doubleheterostructures [22] is found to be comparable with that obtained from the quantum-confinedStark effect (�10�3 cm�1).

11.4.3 II–VI Semiconductor Alloy

Samuel et al. [23] measured the absorption coefficients near the fundamental absorption edgeof ZnxCd1�xSe and CdSxSe1�x. They found that most of the results can be explained by theDow–Redfield theory of the internal Franz–Keldysh effect. An enhancement of the two-photon

360 PROPERTIES OF SEMICONDUCTOR ALLOYS

absorption coefficient has also been observed to occur in the space charge region of theCdxHg1�xTe p-n junction, which can be attributed to the Franz–Keldysh effect induced by thebuilt-in electric field [24].

11.5 NONLINEAR OPTICAL CONSTANT

11.5.1 Group-IV semiconductor alloy

The independent non-vanishing tensor elements of the nonlinear optical constants forsemiconductors of certain symmetry classes are given in [1]. Principally, the diamond latticedoes not show a second-order nonlinear optical effect [1]. No detailed data are available on thethird-order nonlinear optical constants for group-IV semiconductor alloys.

11.5.2 III–V semiconductor alloy

The second-order nonlinear optical susceptibilities for w-AlxGa1�xN with 0� x� 0.666 havebeen measured by Sanford et al. [25]. These authors determined the nonlinear opticalsusceptibility constants d31 and d33 at l¼ 1.064mm as a function of x under the assumptionof d31¼ d15. These results are shown in Figure 11.3(a), together with the endpoint data takenfrom Adachi [1]. The nonlinear susceptibility constant d31 decreases almost linearly withincreasing x, while d33 largely scatters.

The nonlinear optical susceptibility d14 at l¼ 1.064 mm in AlxGa1�xAs has been measuredusing a reflected second harmonicsmethod byOhashi et al. [26]. They reported themagnitudesof d14 relative to GaAs over the whole alloy range 0� x� 1.0. These results are shown inFigure 11.3(b). The GaAs and AlAs d14 values in Adachi [3] are in the ranges 119–170 and32 pm/V, respectively.

0 0.2 0.4 0.6 0.8 1.0-25

-20

-15

-10

-5

0

5

10

15

x

d 31, d

33 (

pm/V

)

(a) w-AlxGa1-xN

d31

d33

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

x

(b) AlxGa1-xAs

|d14

(x)/

d 14(G

aAs)

|

0 0.2 0.4 0.6 0.8 1.00

20

40

60

80

100

120

140

x

(c) ZnxCd1-xTe

d 14 (

pm/V

)

Figure 11.3 Second-order nonlinear optical susceptibility constants dij in (a) w-AlxGa1�xN,(b) AlxGa1�xAs and (c) ZnxCd1�xTe. The experimental data in (a) are taken from Adachi [1] andSanford et al. [25], in (b) from Ohashi et al. [26] and in (c) from Adachi [4] (open circles) and Zappettiniet al. [31] (solid circles)

ELASTO-OPTIC, ELECTRO-OPTIC AND NONLINEAR OPTICAL PROPERTIES 361

The third-order nonlinear susceptibility |x(3)| in GaxIn1�xPyAs1�y/InP (lg� 1.5 mm) hasbeen determined using a forward degenerate four-wave mixing method [27]. The |x(3)| valueobtained was �3.8� 10�3 esu.

The two-photon absorption coefficient b has been discussed both theoretically andexperimentally for non-alloyed group-IV, III–Vand II–VI semiconductors [1]. Krishnamurthyet al. [28] showed that theoretically, b can be increased or decreased by adding In or Al to theGaN compound. Experimentally, Villeneuve et al. [29] obtained the spectral dependence ofb near half the band gap of Al0.18Ga0.82As. The b increased with decreasing wavelengthl (�0.05 cm/GW at l¼ 1.65mm and �1.2 cm/GW at l� 1.49 cm/GW). The three-photonabsorption coefficienta inAl0.18Ga0.82Aswas alsomeasured for photon energies between one-half and one-third the band gap byKang et al. [30]. They found thata decreaseswith decreasingl from a� 0.13 cm3/GW2 at l¼ 1.66mm to a� 0.03 cm3/GW2 at l¼ 1.50mm.

11.5.3 II–VI semiconductor alloy

The second harmonic generation efficiency has been examined in the near-IR region (l¼ 1.5and 1.9 mm) of ZnxCd1�xTewith x¼ 0, 0.1, 0.2 and 1.0 [31]. The d14 data for ZnxCd1�xTe [31],together with the endpoint data taken fromAdachi [4], have been plotted in Figure 11.3(c). Thesolid line represents the linear least-squares fit given by d14¼ 73� 13x pm/V.

The second-order nonlinear constant d14 of CdxHg1�xTe with x¼ 0.277 has been deter-mined to be 350� 40 pm/V [32]. The second harmonic generation has also been measured atl¼ 1.06 mm as a function of x for w-ZnxCd1�xS and w-CdSxSe1�x [33].

The two-photon absorption phenomena have been studied in several II–VI semiconductoralloys [34,35]. The maximum value of the two-photon absorption coefficient b is observed forx¼ 0.4 in c-MgxZn1�xSe (0� x� 0.4) [34]. In c-ZnxCd1�xSe andw-ZnxCd1�xSe (0� x� 1.0),the reduction in the band gap is accompanied by a smooth rise of b [35]. Tunability of the two-photon absorption coefficient has also been demonstrated by adjusting the electrical field in aCdxHg1�xTe photodiode with a cutoff wavelength of 5.2mm [24].

REFERENCES

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362 PROPERTIES OF SEMICONDUCTOR ALLOYS

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ELASTO-OPTIC, ELECTRO-OPTIC AND NONLINEAR OPTICAL PROPERTIES 363