- Email: [email protected]
GaN multiple quantum wells
ARTICLE IN PRESS
Microelectronics Journal 38 (2007) 80–86 www.elsevier.com/locate/mejo
Spin–orbit splitting dependent resonant third-order nonlinear optical susceptibility in InGaN/GaN multiple quantum wells Youqing Yu, Fei Gao, Guiguang Xiong Department of Physics, Wuhan University, Luojia Hill, Wuchang, Wuhan, Hubei Province 430072, China Received 31 July 2006; accepted 25 September 2006 Available online 22 November 2006
Abstract For the transition between valence band and conduction band, the third-order nonlinear optical susceptibility w(3) for degenerated four-wave mixing in InxGa1xN/GaN multiple quantum wells (MQWs) has been calculated. The contributions of spin–orbit split-off energy to the resonant third-order nonlinear optical susceptibility of the modes, whose polarization is vertical to the [0 0 1] direction of the MQWs, are discussed in detail. The correlations between the peaks of w(3), which are due to the transitions from the spin–orbit splitoff energy level to first conduction subband, and the width of the quantum well and the constituents of the semiconductor material are obtained. r 2006 Elsevier Ltd. All rights reserved. PACS: 78.67.He; 42.65.An; 71.70.Ej Keywords: Optical properties of quantum well; Optical susceptibility; Spin-orbit split-off energy
1. Introduction In recent years, the wide band-gap semiconductor materials and devices, with the representation of group III–V compounds, have attracted tremendous interest of people for numerical advantages and special properties [1–4]. Among them, much attention has been paid to nonlinear optical effects of GaN-based multiple quantum wells (MQWs), which mean much for both practicable applications and understanding of fundamental physical principles [1–7]. As one means of coherent dynamics researches, the calculation of third-order nonlinear optical susceptibility is significant. Compared with bulk material, the third-order nonlinear optical susceptibility of MQWs has a great enhancement resulting from the effect of quantum confinement in one or several directions [8]. For the two different transition processes, the interband transitions (the transitions between valence band and conduction band), which involve both electrons and holes, are more important than the intraband transitions Corresponding author. Tel.: +86 27 6875 2481; fax: +86 27 6875 2569.
E-mail address: [email protected] (G. Xiong). 0026-2692/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2006.09.008
(the transitions between subbands in valence or conduction band). Because the energy gap of GaN-based material is very wide and the spin–orbit split-off energy is similar to the energy difference between valence subbands, the contribution of spin–orbit split-off energy cannot be neglected in the calculation of energy band structure and nonlinear optical susceptibility. However, in many published papers, in order to simplify the process, only the contributions of heavy holes and light holes have been included [9–11]. On the other hand, in many experiments of MQWs, incident laser beams are often vertical to the surfaces of samples because of their thickness. In this situation, only the modes, whose polarization is vertical to the [0 0 1] direction of the MQWs, are working. To combine theory with practice, we focus on the third-order nonlinear optical susceptibility of the modes with vertical polarization for the transitions between valence band and conduction band, including the contribution of the spin–orbit split-off energy. In addition, the research of spin in MQWs plays an important role in spin-based electronics and photonics and in quantum information technologies [12]. So we got the correlations between the peaks of w(3), which are due to the transitions from the
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
spin–orbit split-off energy level to first conduction subband, and the width of the quantum well (QW), and the constituents of the semiconductor material. The result is of real significance for optical experiments and referential value to study the competitions between different modes in MQWs. 2. Model and theory InGaN/GaN MQWs and the optical field propagating in the structure are shown in Fig. 1. The MQWs grow along the m-direction. The optical fields propagate in the k-direction and oscillate in the e-direction and thus the m-, k-, and e-directions form a coordinate system. In addition, the Descartes coordinate system is defined by the crystallographic axes of the semiconductor material. Respectively, the axes x, y, z are coincident with the [1 0 0], [0 1 0], [0 0 1] directions of the InGaN crystal. So there are two different coordinate systems: the (m, k, e) system from the optical field and the (x, y, z) system from the material crystal. Comparing these two coordinate systems, the x- and y-axes rotate the angle of y from the k and e-direction, respectively, while the direction of m is coincident with the z-axis. Because the energy gap of InGaN is smaller than that of GaN, potential wells are formed. The potential barrier between two neighboring wells is wide enough so that the wave functions in the wells cannot overlap and the MQWs can be treated as a single QW. The direction in which the MQWs grow along is defined as the z-axis. The QW forms a series of discrete quantized energy levels in the direction of z. Under the selection rules, from the valence band to the conduction band, only the transitions occurring between two energy levels with same quantum numbers can exist. At room temperature, the electrons distribute mainly in the bottom of the conduction band and the holes mainly in the top of the valence band. So only the energy levels with the main quantum number n equal to one need to be taken into account. Thus there is a two-fold degenerated state with opposite spins in the conduction band. Considering the impacts of the spin–orbit split-off energy, the situation of
the valence band is somewhat complicated. According to the value of q2 E=q2 k, there are three two-fold degenerated states in the valence band. The first is heavy holes state (HH1), which denotes that the values of q2 E=q2 k and corresponding effective masses are larger. The second is light holes state (LH1), which denotes the values of q2 E=q2 k and corresponding effective masses are smaller. The third is spin–orbit split-off holes state (SO1), which is generated by the interaction between the magnetic torque from the spin of valence electrons and the magnetic field from the orbits of other valence electrons. The conversion matrix between the (m, k, e) coordinate system and the (x, y, z) coordinate system is defined as 0 1 cos y sin y 0 B C (1) T ¼ @ sin y cos y 0 A. 0
0
1
The density-matrix theory can be chosen to calculate the nonlinear optical susceptibility. For the four-wave mixing (FWM) between the valence band and the conduction band, with two incident laser beams of different frequencies, the third-order nonlinear optical susceptibility of the modes, with the polarization vertical to the [0 0 1] direction of the InGaN/GaN MQWs, can be expressed as [13] e e 3 wð3Þ ðo; o0 o0 þ oÞ ¼ L 2_ 8 > > Z < X X dk2 A Acc0 vv0 0 vv0 þ cc ð2pÞ2 > > c¼e"#;c0 ¼e"# : c¼e"#;c00 ¼e"# v¼hh"#;v ¼hh"# v¼lh"#;v0 ¼lh"# X X þ Acc0 vv0 þ Bcvv0 c¼e"#;c0 ¼e"# v¼so"#;v0 ¼so"#
X
þ
c¼e"# v¼hh"#;v0 ¼lh"#
c¼e"# v¼hh"#;v0 ¼so"#
c¼e"# v¼lh"#;v0 ¼so"#
Acc0 vv0
θ
θ InGaN GaN InGaN GaN
Bcvv0 , > > ;
k
6 6 1 2 6 ¼ T 11 6 6 o mcv 4
χ
þxvc xcv0 yv0 c0 yc0 v þ xvc ycv0 xv0 c0 yc0 v Þ o o0 mv0 v
rð0Þ rð0Þ rð0Þ rð0Þ vv c0 c0 v0 v0 c0 c0 0 o mc 0 v 0 o mc 0 v
!
T 211 xvc xcv0 xv0 c0 xc0 v þ T 212 ðxvc ycv0 yv0 c0 xc0 v Emitting
þ
InGaN Fig. 1. The structure of the InGaN/GaN MQWs and the two different coordinate systems.
ð2Þ
T 211 xvc xcv0 xv0 c0 xc0 v þ T 212 ðxvc ycv0 yv0 c0 xc0 v
H E
9 > > =
with
Z(m) y
X
Bcvv0 þ
2 e
81
þxvc xcv0 yv0 c0 yc0 v þ xvc ycv0 xv0 c0 yc0 v Þ o o0 mcc0 rð0Þ rð0Þ c0 c0 v0 v0 o0 mc0 v0
rð0Þ rð0Þ v0 v0 cc o mcv0
!
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
82
T 211 xvc xcv0 xv0 c0 xc0 v þ T 212 ðxvc ycv0 yv0 c0 xc0 v þ
< c|
þxvc xcv0 yv0 c0 yc0 v þ xvc ycv0 xv0 c0 yc0 v Þ mv0 v rð0Þ rð0Þ vv c0 c0 o 0 mc 0 v
rð0Þ c0 c0
rð0Þ v0 v0 o mc0 v0
!
ω1
ω2
ω3
ω
T 211 xvc xcv0 xv0 c0 xc0 v þ T 212 ðxvc ycv0 yv0 c0 xc0 v þ
þxvc xcv0 yv0 c0 yc0 v þ xvc ycv0 xv0 c0 yc0 v Þ mcc0 ð0Þ ð0Þ rð0Þ rð0Þ cc ru0 u0 cc ru0 u0 o0 mcv0 o0 mc0 v0
< v|
!# ,
ð3Þ
Fig. 2. Schematic diagram of the transition of type I, where c ¼ e"# , v ¼ hh"# , lh"# , so"# , and o1 ¼ o2 ¼ o3 ¼ o.
and Bcvv0 ¼
1 T2 o mcv 11
ð0Þ ð0Þ rð0Þ rð0Þ cc rv0 v0 cc rvv o0 mcv0 o mcv
!
T 211 xvc xcv0 xv0 c xcv þ T 211 ðxvc ycv0 yv0 c xcv
ω1
þxvc xcv0 yv0 c ycv þ xvc ycv0 yv0 c ycv o o 0 m v0 v
1 þ T2 o mcv0 11
ð0Þ ð0Þ rð0Þ rð0Þ cc rv0 v0 cc rvv o0 mcv o mcv0
ω2
þxv0 c xcv yvc ycv0 þ xv0 c ycv yvc ycv0 Þ o o0 mvv0
,
E c ðkÞ E v ðkÞ igcv , _
< v' |
ð4Þ
where L, e and _ are the width of the QW, the electronic charge and the Planck’s constant, respectively. c, c0 and v, v0 denote the spin-up and spin-down states of the electrons and holes, respectively. rc0 and rv0 are the quasi-Fermi energy levels of the electrons in the conduction band and the holes in the valence band, respectively. mcv(k) is defined as follows: mcv ðkÞ ¼
ω
!
T 211 xv0 c xcv xvc xcv0 þ T 211 ðxv0 c ycv yvc xcv0
ω3
(5)
where Ec(k) and Ev(k) are the energies of the electrons and holes, respectively, and gcv is the relaxation of the energy levels. Analyzing Eqs. (2)–(4), there are two different kinds of transitional processes. The first one is shown as Fig. 2. The upper energy level of the transition corresponds to the twofold degenerated state with opposite spins in the conduction band and the lower energy level corresponds to the random one of the three two-fold degenerated states with opposite spins in the valence band. The second kind of transitional process is shown in Fig. 3, which is different from the first one. The upper energy level is still the two-fold degenerated state with opposite spins in the conduction band while the lower energy levels are not the same. The two lower energy levels correspond to random two states of the three two-fold
Fig. 3. Schematic diagram of the transition of type II, where c ¼ e"# , v, v0 ¼ hh"# , lh"# , so"# , vav0 , o2 ¼ o3 , and o1 ¼ o.
degenerated states of the heavy holes, light holes and spin–orbit split-off holes in the valence band. Since the relaxation of the energy levels has been included in gcv, gcc, and gvv, the frequency of incident laser can be chosen at the resonant frequency. When the formula _o ¼ _o0 ¼ E c ðkÞ E v ðkÞ
(6)
is satisfied, Acc0 vv0 reaches extreme value and the corresponding physical process is the degenerated four-wave mixing (DFWM) of three resonances. On the other hand, when the formulae _o ¼ E c ðkÞ E v ðkÞ,
(7)
and _o0 ¼ E c ðkÞ E v0 ðkÞ,
(8)
are satisfied at the same time, Bcvv0 reaches extreme value and the corresponding physical process is the FWM. Without losing general properties, we choose special parameters to simplify the expression of w(3). First of all, we set the angle y of the transition matrix between the former two different coordinate systems as zero. That is to say, the (m, k, e) system from the optical field coincides with the (x, y, z) system from the material crystal. So the conversion
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
matrix can be simplified to 0 1 1 0 0 B C T ¼ @ 0 1 0 A. 0 0 1
and 1 , f v ðkÞ ¼ exp ðE v ðkÞ F v Þ=kB T þ 1 (9)
Furthermore, we assume that the two incident laser beams have the same frequency ðo0 ¼ oÞ. On this condition, Acc0 vv0 is far larger than Bcvv0 and the main physical process is the DFWM with three resonances. Thus the third-order nonlinear optical susceptibility of the modes with the polarization vertical to the [0 0 1] direction of the MQWs can be simplified to the expression Z e e 3 dk2 wð3Þ ðo; o o þ oÞ ¼ L 2_ ð2pÞ2 8 > > > > > > > < X xvc xcv0 xv0 c0 xc0 v > o mcv > > c; c0 ¼ e "# > > > > : v; v0 ¼ hh "#; lh "#; so "# 39 > > > 6 7> 6 7= 6 7 6 ð0Þ ð0Þ 7 . ð0Þ ð0Þ r 0 0 r 0 0 rcc r 0 0 5> > 4 1 c v c v v v > þ m om 0 > ; om0 0 0 cv cc cv 2
1 m0 vv
r
ð0Þ ð0Þ r c0 c0 v 0 v0 om 0 0 cv
r
83
ð0Þ ð0Þ rvv c0 c0 omc0 v
(13)
where kB is the Boltzmann’s constant, T denotes the temperature, and Fc and Fv are the quasi-Fermi energy levels of the electrons and holes, respectively. If the densities of the carriers are given, the quasi-Fermi energy level of the electrons and holes can be calculated from Eqs. (11)–(13). Thus the zero-order diagonal elements rð0Þ cc and rð0Þ vv of the density matrix can be obtained. Up to now, the only thing required for the calculation of the third-order nonlinear optical susceptibility is the energy structure of the QW. To get it, the k~ ~ p method and the effective-mass theory, namely the real masses of the electrons are replaced by the effective masses of those near the bottom of the conduction band and the real masses of the holes are replaced by the effective masses of those near the top of the valence band [16,17], are mainly used. With this approximation, the wave function of the electrons and holes near the G point in the first Brillouin zone has the form X c;v Cc;v ðrÞ ¼ U c;v (14) n ðrÞF n ðrÞ, n
ð10Þ
In the expression of the summation, the suffixes c and c0 denote the two-fold degenerated state with opposite spins in the conduction band, v and v0 denote one of the three two-fold degenerated states with opposite spins in the valence band. ð0Þ The zero-order diagonal elements rð0Þ cc and rvv denote the quasi-Fermi energy level of the electrons and holes when the whole system is in thermal equilibrium at the temperature of 0 K, without outside interaction. If the energy of incident photons equals or exceeds the energy gap of InGaN, the transitions will generate from the valence band to the conduction band and there will be a lot of couples of electrons and holes. For the existence of the carriers, the former Fermi energy level is not appropriate any more and the corresponding quasi-Fermi level should be adopted to describe the electrons in the conduction band and the holes in the valence band [14,15]. For the semiconductor material, the areal densities of the electrons and holes are assumed to be equal and the relationship is X Z dk2 nc ¼ nv ¼ f c ðkÞ ð2pÞ2 c X Z dk2 1 f v ðkÞ . ð11Þ ¼ 2 ð2pÞ v Then, the Fermi distribution functions of electrons and holes are 1 f c ðkÞ ¼ exp ðE c ðkÞ F c Þ=kB T þ 1 , (12)
where the suffix c denotes the electrons in the conduction band and the suffix v denotes the holes in the valence band. Un(r) is the Bloch function and Fn(r) is the slowly-varying envelope function. For the electrons in the conduction band, U i ðrÞ ¼ jCi, and jCi is the ground state wave function of electrons. For the holes in valence band, U n ðrÞ is corresponding to six band-edge wave functions jV n i (n ¼ 1, 2, 3, y, 6). In this way, the effective mass equations of the electrons and holes can be obtained. Because the QW forms discrete quantized energy levels in the direction of z and the electrons and holes are only confined in the zdirection correspondingly, the slowly varying envelope functions of the electrons and holes can be dealt with the method of separation of variables. Setting the energy at the top of the valence band as zero, the wave function and energy at the G point for the electrons at the bottom of the conduction band can be written as 1 FC n ðkx ; ky ; zÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp½iðk x x þ k y yÞ Lx Ly rffiffiffiffi 2 npz , sin L L EC ¼ Eg þ
_2 ðk2x þ k2y Þ _2 n2 p2 þ 2, 2me 2me L
ð15Þ
(16)
where n is the main quantum number and equal to 1, 2, 3, etc. Lx and Ly are the constants of normalization and denote the length of QW in the direction of x and y, respectively. Eg is the energy gap of the semiconductor material. me is the effective mass of the electrons. On the other hand, for the holes in the valence band, the Hamiltonian can be expressed by the 6 6 Luttinger–Kohn
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
matrix [16,17] and the envelope wave function of the holes can be expanded by the wave functions of infinite potential pits as rffiffiffiffi M 2X n mpz V . (17) F n ðzÞ ¼ C sin L m¼1 m L By this way, the effective mass equation of holes can be rewritten as an eigenvalue problem of a 6M 6M matrix. The number M is the order of the plane wave expansion in Eq. (17). Thus, by solving the eigenvalue problem, the energy structure of holes near the top of the valence band can be obtained.
0.8 0.6 Re (χ(3)) (10-9 esu)
84
0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 3.10
3.12
3. Results and discussion
3.14 3.16 Frequency (ev)
3.18
3.20
Fig. 5. The real part of w(3).
Table 1 The parameters of GaN and InGaN used in the paper Parameters Length of primitive cell a0 (A˚) Forbidden gap Eg (ev) Effective mass me Luttinger parameters g1 g2 g3
GaN
InN
In0.05Ga0.95N
4.50
4.98
4.50
3.299
1.94
3.08855
0.15
0.12
0.1485
2.67 0.75 1.10
3.72 1.26 1.63
2.7225 0.7755 1.1265
|χ(3)| (10-9 esu)
0.1
0.01
3.16 3.18 Frequency (eV)
Fig. 4. The modulus of w(3).
3.20
0.01
3.14
3.16 Frequency (ev)
3.18
3.20
Fig. 6. The imaginary part of w(3).
1
3.14
0.1
3.12
3 2
3.12
1 Im (χ(3)) (10-9 esu)
The parameters of GaN and InGaN at room temperature used in this paper are given in Table 1 [18]. In order to understand the dispersion and gain or loss of the optical field in the MQWs, setting the width of the wells as 7.0 nm and the concentration of In as 0.05, the figures of the modulus, real and imaginary part of w(3) are obtained in Figs. 4–6, respectively.
3.22
From Fig. 5, the real part of w(3) has the peak-valley trends near the resonant frequencies, which indicates that the dispersion of the InGaN/GaN MQWs we used is normal. From Fig. 6, the imaginary part of w(3) has the shape of upward peaks, which indicates that the semiconductor material we used has nonlinear absorption to the incident laser. In addition, it can be seen that there are two obvious peaks in the figures of the modulus, real and imaginary part of w(3). The two obvious peaks denote the transitions from the energy levels of heavy and light holes in the valence band (HH1 and LH1) to the first conduction subband respectively. In fact, there should be three peaks in all the three figures. The reason for the disappearance of the third peak is that there is a great difference of values between the third peak and the former two. This difference is due to the probability of the transition from the energy level of different holes in the valence band to the conduction subband, which can be determined by the wave functions obtained from the method outlined above.
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
Under the condition of near resonance, the third-order nonlinear optical susceptibility w(3) of DFWM comes to maximum values when the energy of incident photons equals the difference between the corresponding two energy levels. However, because of the different transition probabilities, there is a great difference between the maximal values in different resonant frequencies, which results in the disappearance of the third peak in Figs. 4 and 5. Analyzing the pictures in detail, the modulus of w(3) comes to the peaks at the photonic energy of 3.151 and 3.158 eV. The values of the two peaks are 2.35 109 and 2.15 109 esu, which correspond to the energy gaps from the subband HH1 of the heavy holes and LH1 of the light holes to the first conduction subband, respectively. At the photonic energy of 3.194 eV, there is a tiny peak in the modulus, which is three orders of magnitude smaller than the former two. This peak corresponds to the energy gap from the valence subband SO1 of the spin–orbit split-off holes to the first conduction subband. Although the value of the peak is very small, the contribution cannot be neglected in the studies of nonlinear optical effects. Fortunately, there are three obvious peaks in Fig. 6 of the imaginary part of w(3), which are contributed by the transitions from the valence subband of the heavy holes, light holes and spin–orbit split-off holes to the first conduction subband respectively. As many previous researches have neglected to discuss the contribution of the spin–orbit split-off energy to the third-order nonlinear optical susceptibility, we focus on the correlations between the third peak of the imaginary part of w(3) vertical to the zdirection and the width of the QW, and the constituents of the semiconductor material. When the concentration of In is kept constant at 0.050, varying the width of the wells from 65 to 90 A˚ with a step of 5 A˚, a set of peaks of the imaginary part of w(3), which
85
are due to the contributions of the transitions from the spin–obit split-off energy level to the first conduction subband, with different well widths are obtained, as shown in Fig. 7. It can be seen from Fig. 7 that the peaks have an infrared divergence with an increase of the well width, which indicates that the energy gap from the spin–orbit split-off energy level to the first conduction subband becomes narrower with an increase of the well width. On the other hand, when the energy of incident laser is equal to the energy gap between the spin–orbit split-off energy level and the first conduction subband, the contribution to the imaginary part of w(3) is enhanced and the enhancement becomes obvious with an increase of the width of the wells, which indicates that wider QW causes more loss to incident laser. When the well width L is fixed at 7 nm, varying the concentration of In from 0.040 to 0.065 in steps of 0.005, another set of peaks of the imaginary part of w(3), which are due to the contributions of the transitions from the spin–obit split-off energy level to the first conduction subband, with different concentrations of In are obtained, as shown in Fig. 8. It is shown from Fig. 8 that with an increase of the concentration of In, the positions of the peaks of the imaginary part of w(3) have red shifts, which indicates that the energy gap between the spin–orbit split-off energy level and the first conduction subband becomes narrower with an increase of the concentration of In, and the values of the peaks vary irregularly. When the concentration of In changes from 0.04 to 0.05, the peak values increase linearly. However, when the concentration of In changes from 0.05 to 0.065, the peak values decrease linearly. This phenomenon is very interesting. The reason, in our opinion, is partly that the width of the forbidden gap of the MQWs lies on the composition of the potential well. When the
0.03 0.01
0.025 L9
0.009
0.02
In0.05 0.008
0.015 L8
Im (χ(3)) (10-9 esu)
Im (χ(3)) (10-9 esu)
L8.5 L7.5
0.01 L7
L6.5
0.005
3.15
3.16
3.17 3.18 3.19 Frequency (ev)
3.20
3.21
Fig. 7. The contribution of spin–orbit split-off energy to the imaginary part of w(3) with different well widths.
0.007
In0.055 In0.065 In0.06
In0.045
In0.04
0.006
0.005
0.004 3.12
3.14
3.16 3.18 3.20 Frequency (ev)
3.22
3.24
Fig. 8. The contribution of spin–orbit split-off energy to the imaginary part of w(3) with different concentrations of In.
ARTICLE IN PRESS 86
Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
material of the potential wells is InGaN, the width of the forbidden gap is determined by the concentration of In. To some extent, the forbidden gap becomes wider with an increase of the concentration of In. But this relationship is not always valid. There should be a value and if the concentration of In is equal to it, the peak value of the imaginary part of w(3) will get to extremum. Therefore, with an increase of the concentration of In, the peak values become larger in the beginning and smaller later. The content of this part needs further study and an experiment is being designed to confirm it in our next work. 4. Conclusion The expression of the third-order nonlinear optical susceptibility w(3) of the modes with the polarization vertical to the [0, 0, 1] direction of the MQWs is discussed. Two different types of transition processes included in FWM are discussed in detail and the expression of w(3) is simplified. Under the condition of three-photon resonance, the third-order nonlinear optical susceptibility of the modes with the vertical polarization has been calculated by the k~ ~ p method and the effective-mass theory. The contribution of the spin–orbit split-off energy to the thirdorder nonlinear optical susceptibility with the variation of the width of QW and the concentration of In, which is neglected in some published literatures, has been discussed. It has been shown from the results that the peaks of the imaginary part of w(3), which are due to the transitions from the spin–orbit split-off energy level to the first conduction subband, have red shifts with an increase of the width of the QW and the concentration of In.
Acknowledgment This work was financially supported by the Natural Science Foundation of China (Grant no. 10534030).
References [1] J. Petalas, S. Logothetidis, S. Boultadakis, M. Alouani, J.M. Wills, Phys. Rev. B. 52 (1995) 8082. [2] H. Schmidt, A. Imamoglu, Opt. Commun. 131 (1996) 333. [3] F. Caposso, K. Mohammed, A.Y. Cho, IEEE J. Quantum Electron. QE-22 (1986) 1853. [4] H. Gao, G. Xiong, X. Feng, Microelectr. J. 37 (2006) 742. [5] J. Chr. Holst, A. Hoffmann, D. Rudloff, F. Bertram, T. Riemann, J. Christen, T. Frey, D.J. As, D. Schikora, K. Lischka, Appl. Phys. Lett. 76 (2000) 2832. [6] C. Wang, G. Xiong, Microelectr. J. 37 (2006) 847. [7] G. Todorovic´, V. Milanovic´, Z. Ikonic´, D. Indjin, Phys. Lett. A 279 (2001) 268. [8] H.C. Lee, A. Hariz, P.D. Dapkus, A. Kost, M. Kawase, Appl. Phys. Lett. 50 (1987) 1182. [9] V.I. Litvinov, J. Appl. Phys. 88 (2000) 5814. [10] X.Y. Liu, J.F. Fa¨lth, T.G. Andersson, P. Holmstro¨m, P. Ja¨nes, U. Ekenberg, L. Thyle´n, J. Cryst. Growth 278 (2005) 397. [11] Yu.M. Sirenko, J.B. Jeon, K.W. Kim, M.A. Littlejohn, M.A. Stroscio, Phys. Rev. B. 53 (1996) 1997. [12] S.A. Wolf, et al., Science 294 (2001) 1488. [13] Y. Takahashi, H. Kawaguchi, IEEE J. Quantum Electron. 36 (2000) 864. [14] F. de Rougement, R. Frey, Phys. Rev. B. 37 (1988) 1237. [15] P.N. Butcher, D. Cotter, The Elements of Nonlinear Optics, Cambridge University Press, Cambridge, 1990. [16] J.M. Luttinger, W. Kohn, Phys. Rev. 97 (1955) 869. [17] J.M. Luttinger, Phys. Rev. 102 (1955) 1030. [18] I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815.
Microelectronics Journal 38 (2007) 80–86 www.elsevier.com/locate/mejo
Spin–orbit splitting dependent resonant third-order nonlinear optical susceptibility in InGaN/GaN multiple quantum wells Youqing Yu, Fei Gao, Guiguang Xiong Department of Physics, Wuhan University, Luojia Hill, Wuchang, Wuhan, Hubei Province 430072, China Received 31 July 2006; accepted 25 September 2006 Available online 22 November 2006
Abstract For the transition between valence band and conduction band, the third-order nonlinear optical susceptibility w(3) for degenerated four-wave mixing in InxGa1xN/GaN multiple quantum wells (MQWs) has been calculated. The contributions of spin–orbit split-off energy to the resonant third-order nonlinear optical susceptibility of the modes, whose polarization is vertical to the [0 0 1] direction of the MQWs, are discussed in detail. The correlations between the peaks of w(3), which are due to the transitions from the spin–orbit splitoff energy level to first conduction subband, and the width of the quantum well and the constituents of the semiconductor material are obtained. r 2006 Elsevier Ltd. All rights reserved. PACS: 78.67.He; 42.65.An; 71.70.Ej Keywords: Optical properties of quantum well; Optical susceptibility; Spin-orbit split-off energy
1. Introduction In recent years, the wide band-gap semiconductor materials and devices, with the representation of group III–V compounds, have attracted tremendous interest of people for numerical advantages and special properties [1–4]. Among them, much attention has been paid to nonlinear optical effects of GaN-based multiple quantum wells (MQWs), which mean much for both practicable applications and understanding of fundamental physical principles [1–7]. As one means of coherent dynamics researches, the calculation of third-order nonlinear optical susceptibility is significant. Compared with bulk material, the third-order nonlinear optical susceptibility of MQWs has a great enhancement resulting from the effect of quantum confinement in one or several directions [8]. For the two different transition processes, the interband transitions (the transitions between valence band and conduction band), which involve both electrons and holes, are more important than the intraband transitions Corresponding author. Tel.: +86 27 6875 2481; fax: +86 27 6875 2569.
E-mail address: [email protected] (G. Xiong). 0026-2692/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2006.09.008
(the transitions between subbands in valence or conduction band). Because the energy gap of GaN-based material is very wide and the spin–orbit split-off energy is similar to the energy difference between valence subbands, the contribution of spin–orbit split-off energy cannot be neglected in the calculation of energy band structure and nonlinear optical susceptibility. However, in many published papers, in order to simplify the process, only the contributions of heavy holes and light holes have been included [9–11]. On the other hand, in many experiments of MQWs, incident laser beams are often vertical to the surfaces of samples because of their thickness. In this situation, only the modes, whose polarization is vertical to the [0 0 1] direction of the MQWs, are working. To combine theory with practice, we focus on the third-order nonlinear optical susceptibility of the modes with vertical polarization for the transitions between valence band and conduction band, including the contribution of the spin–orbit split-off energy. In addition, the research of spin in MQWs plays an important role in spin-based electronics and photonics and in quantum information technologies [12]. So we got the correlations between the peaks of w(3), which are due to the transitions from the
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
spin–orbit split-off energy level to first conduction subband, and the width of the quantum well (QW), and the constituents of the semiconductor material. The result is of real significance for optical experiments and referential value to study the competitions between different modes in MQWs. 2. Model and theory InGaN/GaN MQWs and the optical field propagating in the structure are shown in Fig. 1. The MQWs grow along the m-direction. The optical fields propagate in the k-direction and oscillate in the e-direction and thus the m-, k-, and e-directions form a coordinate system. In addition, the Descartes coordinate system is defined by the crystallographic axes of the semiconductor material. Respectively, the axes x, y, z are coincident with the [1 0 0], [0 1 0], [0 0 1] directions of the InGaN crystal. So there are two different coordinate systems: the (m, k, e) system from the optical field and the (x, y, z) system from the material crystal. Comparing these two coordinate systems, the x- and y-axes rotate the angle of y from the k and e-direction, respectively, while the direction of m is coincident with the z-axis. Because the energy gap of InGaN is smaller than that of GaN, potential wells are formed. The potential barrier between two neighboring wells is wide enough so that the wave functions in the wells cannot overlap and the MQWs can be treated as a single QW. The direction in which the MQWs grow along is defined as the z-axis. The QW forms a series of discrete quantized energy levels in the direction of z. Under the selection rules, from the valence band to the conduction band, only the transitions occurring between two energy levels with same quantum numbers can exist. At room temperature, the electrons distribute mainly in the bottom of the conduction band and the holes mainly in the top of the valence band. So only the energy levels with the main quantum number n equal to one need to be taken into account. Thus there is a two-fold degenerated state with opposite spins in the conduction band. Considering the impacts of the spin–orbit split-off energy, the situation of
the valence band is somewhat complicated. According to the value of q2 E=q2 k, there are three two-fold degenerated states in the valence band. The first is heavy holes state (HH1), which denotes that the values of q2 E=q2 k and corresponding effective masses are larger. The second is light holes state (LH1), which denotes the values of q2 E=q2 k and corresponding effective masses are smaller. The third is spin–orbit split-off holes state (SO1), which is generated by the interaction between the magnetic torque from the spin of valence electrons and the magnetic field from the orbits of other valence electrons. The conversion matrix between the (m, k, e) coordinate system and the (x, y, z) coordinate system is defined as 0 1 cos y sin y 0 B C (1) T ¼ @ sin y cos y 0 A. 0
0
1
The density-matrix theory can be chosen to calculate the nonlinear optical susceptibility. For the four-wave mixing (FWM) between the valence band and the conduction band, with two incident laser beams of different frequencies, the third-order nonlinear optical susceptibility of the modes, with the polarization vertical to the [0 0 1] direction of the InGaN/GaN MQWs, can be expressed as [13] e e 3 wð3Þ ðo; o0 o0 þ oÞ ¼ L 2_ 8 > > Z < X X dk2 A Acc0 vv0 0 vv0 þ cc ð2pÞ2 > > c¼e"#;c0 ¼e"# : c¼e"#;c00 ¼e"# v¼hh"#;v ¼hh"# v¼lh"#;v0 ¼lh"# X X þ Acc0 vv0 þ Bcvv0 c¼e"#;c0 ¼e"# v¼so"#;v0 ¼so"#
X
þ
c¼e"# v¼hh"#;v0 ¼lh"#
c¼e"# v¼hh"#;v0 ¼so"#
c¼e"# v¼lh"#;v0 ¼so"#
Acc0 vv0
θ
θ InGaN GaN InGaN GaN
Bcvv0 , > > ;
k
6 6 1 2 6 ¼ T 11 6 6 o mcv 4
χ
þxvc xcv0 yv0 c0 yc0 v þ xvc ycv0 xv0 c0 yc0 v Þ o o0 mv0 v
rð0Þ rð0Þ rð0Þ rð0Þ vv c0 c0 v0 v0 c0 c0 0 o mc 0 v 0 o mc 0 v
!
T 211 xvc xcv0 xv0 c0 xc0 v þ T 212 ðxvc ycv0 yv0 c0 xc0 v Emitting
þ
InGaN Fig. 1. The structure of the InGaN/GaN MQWs and the two different coordinate systems.
ð2Þ
T 211 xvc xcv0 xv0 c0 xc0 v þ T 212 ðxvc ycv0 yv0 c0 xc0 v
H E
9 > > =
with
Z(m) y
X
Bcvv0 þ
2 e
81
þxvc xcv0 yv0 c0 yc0 v þ xvc ycv0 xv0 c0 yc0 v Þ o o0 mcc0 rð0Þ rð0Þ c0 c0 v0 v0 o0 mc0 v0
rð0Þ rð0Þ v0 v0 cc o mcv0
!
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
82
T 211 xvc xcv0 xv0 c0 xc0 v þ T 212 ðxvc ycv0 yv0 c0 xc0 v þ
< c|
þxvc xcv0 yv0 c0 yc0 v þ xvc ycv0 xv0 c0 yc0 v Þ mv0 v rð0Þ rð0Þ vv c0 c0 o 0 mc 0 v
rð0Þ c0 c0
rð0Þ v0 v0 o mc0 v0
!
ω1
ω2
ω3
ω
T 211 xvc xcv0 xv0 c0 xc0 v þ T 212 ðxvc ycv0 yv0 c0 xc0 v þ
þxvc xcv0 yv0 c0 yc0 v þ xvc ycv0 xv0 c0 yc0 v Þ mcc0 ð0Þ ð0Þ rð0Þ rð0Þ cc ru0 u0 cc ru0 u0 o0 mcv0 o0 mc0 v0
< v|
!# ,
ð3Þ
Fig. 2. Schematic diagram of the transition of type I, where c ¼ e"# , v ¼ hh"# , lh"# , so"# , and o1 ¼ o2 ¼ o3 ¼ o.
and Bcvv0 ¼
1 T2 o mcv 11
ð0Þ ð0Þ rð0Þ rð0Þ cc rv0 v0 cc rvv o0 mcv0 o mcv
!
T 211 xvc xcv0 xv0 c xcv þ T 211 ðxvc ycv0 yv0 c xcv
ω1
þxvc xcv0 yv0 c ycv þ xvc ycv0 yv0 c ycv o o 0 m v0 v
1 þ T2 o mcv0 11
ð0Þ ð0Þ rð0Þ rð0Þ cc rv0 v0 cc rvv o0 mcv o mcv0
ω2
þxv0 c xcv yvc ycv0 þ xv0 c ycv yvc ycv0 Þ o o0 mvv0
,
E c ðkÞ E v ðkÞ igcv , _
< v' |
ð4Þ
where L, e and _ are the width of the QW, the electronic charge and the Planck’s constant, respectively. c, c0 and v, v0 denote the spin-up and spin-down states of the electrons and holes, respectively. rc0 and rv0 are the quasi-Fermi energy levels of the electrons in the conduction band and the holes in the valence band, respectively. mcv(k) is defined as follows: mcv ðkÞ ¼
ω
!
T 211 xv0 c xcv xvc xcv0 þ T 211 ðxv0 c ycv yvc xcv0
ω3
(5)
where Ec(k) and Ev(k) are the energies of the electrons and holes, respectively, and gcv is the relaxation of the energy levels. Analyzing Eqs. (2)–(4), there are two different kinds of transitional processes. The first one is shown as Fig. 2. The upper energy level of the transition corresponds to the twofold degenerated state with opposite spins in the conduction band and the lower energy level corresponds to the random one of the three two-fold degenerated states with opposite spins in the valence band. The second kind of transitional process is shown in Fig. 3, which is different from the first one. The upper energy level is still the two-fold degenerated state with opposite spins in the conduction band while the lower energy levels are not the same. The two lower energy levels correspond to random two states of the three two-fold
Fig. 3. Schematic diagram of the transition of type II, where c ¼ e"# , v, v0 ¼ hh"# , lh"# , so"# , vav0 , o2 ¼ o3 , and o1 ¼ o.
degenerated states of the heavy holes, light holes and spin–orbit split-off holes in the valence band. Since the relaxation of the energy levels has been included in gcv, gcc, and gvv, the frequency of incident laser can be chosen at the resonant frequency. When the formula _o ¼ _o0 ¼ E c ðkÞ E v ðkÞ
(6)
is satisfied, Acc0 vv0 reaches extreme value and the corresponding physical process is the degenerated four-wave mixing (DFWM) of three resonances. On the other hand, when the formulae _o ¼ E c ðkÞ E v ðkÞ,
(7)
and _o0 ¼ E c ðkÞ E v0 ðkÞ,
(8)
are satisfied at the same time, Bcvv0 reaches extreme value and the corresponding physical process is the FWM. Without losing general properties, we choose special parameters to simplify the expression of w(3). First of all, we set the angle y of the transition matrix between the former two different coordinate systems as zero. That is to say, the (m, k, e) system from the optical field coincides with the (x, y, z) system from the material crystal. So the conversion
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
matrix can be simplified to 0 1 1 0 0 B C T ¼ @ 0 1 0 A. 0 0 1
and 1 , f v ðkÞ ¼ exp ðE v ðkÞ F v Þ=kB T þ 1 (9)
Furthermore, we assume that the two incident laser beams have the same frequency ðo0 ¼ oÞ. On this condition, Acc0 vv0 is far larger than Bcvv0 and the main physical process is the DFWM with three resonances. Thus the third-order nonlinear optical susceptibility of the modes with the polarization vertical to the [0 0 1] direction of the MQWs can be simplified to the expression Z e e 3 dk2 wð3Þ ðo; o o þ oÞ ¼ L 2_ ð2pÞ2 8 > > > > > > > < X xvc xcv0 xv0 c0 xc0 v > o mcv > > c; c0 ¼ e "# > > > > : v; v0 ¼ hh "#; lh "#; so "# 39 > > > 6 7> 6 7= 6 7 6 ð0Þ ð0Þ 7 . ð0Þ ð0Þ r 0 0 r 0 0 rcc r 0 0 5> > 4 1 c v c v v v > þ m om 0 > ; om0 0 0 cv cc cv 2
1 m0 vv
r
ð0Þ ð0Þ r c0 c0 v 0 v0 om 0 0 cv
r
83
ð0Þ ð0Þ rvv c0 c0 omc0 v
(13)
where kB is the Boltzmann’s constant, T denotes the temperature, and Fc and Fv are the quasi-Fermi energy levels of the electrons and holes, respectively. If the densities of the carriers are given, the quasi-Fermi energy level of the electrons and holes can be calculated from Eqs. (11)–(13). Thus the zero-order diagonal elements rð0Þ cc and rð0Þ vv of the density matrix can be obtained. Up to now, the only thing required for the calculation of the third-order nonlinear optical susceptibility is the energy structure of the QW. To get it, the k~ ~ p method and the effective-mass theory, namely the real masses of the electrons are replaced by the effective masses of those near the bottom of the conduction band and the real masses of the holes are replaced by the effective masses of those near the top of the valence band [16,17], are mainly used. With this approximation, the wave function of the electrons and holes near the G point in the first Brillouin zone has the form X c;v Cc;v ðrÞ ¼ U c;v (14) n ðrÞF n ðrÞ, n
ð10Þ
In the expression of the summation, the suffixes c and c0 denote the two-fold degenerated state with opposite spins in the conduction band, v and v0 denote one of the three two-fold degenerated states with opposite spins in the valence band. ð0Þ The zero-order diagonal elements rð0Þ cc and rvv denote the quasi-Fermi energy level of the electrons and holes when the whole system is in thermal equilibrium at the temperature of 0 K, without outside interaction. If the energy of incident photons equals or exceeds the energy gap of InGaN, the transitions will generate from the valence band to the conduction band and there will be a lot of couples of electrons and holes. For the existence of the carriers, the former Fermi energy level is not appropriate any more and the corresponding quasi-Fermi level should be adopted to describe the electrons in the conduction band and the holes in the valence band [14,15]. For the semiconductor material, the areal densities of the electrons and holes are assumed to be equal and the relationship is X Z dk2 nc ¼ nv ¼ f c ðkÞ ð2pÞ2 c X Z dk2 1 f v ðkÞ . ð11Þ ¼ 2 ð2pÞ v Then, the Fermi distribution functions of electrons and holes are 1 f c ðkÞ ¼ exp ðE c ðkÞ F c Þ=kB T þ 1 , (12)
where the suffix c denotes the electrons in the conduction band and the suffix v denotes the holes in the valence band. Un(r) is the Bloch function and Fn(r) is the slowly-varying envelope function. For the electrons in the conduction band, U i ðrÞ ¼ jCi, and jCi is the ground state wave function of electrons. For the holes in valence band, U n ðrÞ is corresponding to six band-edge wave functions jV n i (n ¼ 1, 2, 3, y, 6). In this way, the effective mass equations of the electrons and holes can be obtained. Because the QW forms discrete quantized energy levels in the direction of z and the electrons and holes are only confined in the zdirection correspondingly, the slowly varying envelope functions of the electrons and holes can be dealt with the method of separation of variables. Setting the energy at the top of the valence band as zero, the wave function and energy at the G point for the electrons at the bottom of the conduction band can be written as 1 FC n ðkx ; ky ; zÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp½iðk x x þ k y yÞ Lx Ly rffiffiffiffi 2 npz , sin L L EC ¼ Eg þ
_2 ðk2x þ k2y Þ _2 n2 p2 þ 2, 2me 2me L
ð15Þ
(16)
where n is the main quantum number and equal to 1, 2, 3, etc. Lx and Ly are the constants of normalization and denote the length of QW in the direction of x and y, respectively. Eg is the energy gap of the semiconductor material. me is the effective mass of the electrons. On the other hand, for the holes in the valence band, the Hamiltonian can be expressed by the 6 6 Luttinger–Kohn
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
matrix [16,17] and the envelope wave function of the holes can be expanded by the wave functions of infinite potential pits as rffiffiffiffi M 2X n mpz V . (17) F n ðzÞ ¼ C sin L m¼1 m L By this way, the effective mass equation of holes can be rewritten as an eigenvalue problem of a 6M 6M matrix. The number M is the order of the plane wave expansion in Eq. (17). Thus, by solving the eigenvalue problem, the energy structure of holes near the top of the valence band can be obtained.
0.8 0.6 Re (χ(3)) (10-9 esu)
84
0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 3.10
3.12
3. Results and discussion
3.14 3.16 Frequency (ev)
3.18
3.20
Fig. 5. The real part of w(3).
Table 1 The parameters of GaN and InGaN used in the paper Parameters Length of primitive cell a0 (A˚) Forbidden gap Eg (ev) Effective mass me Luttinger parameters g1 g2 g3
GaN
InN
In0.05Ga0.95N
4.50
4.98
4.50
3.299
1.94
3.08855
0.15
0.12
0.1485
2.67 0.75 1.10
3.72 1.26 1.63
2.7225 0.7755 1.1265
|χ(3)| (10-9 esu)
0.1
0.01
3.16 3.18 Frequency (eV)
Fig. 4. The modulus of w(3).
3.20
0.01
3.14
3.16 Frequency (ev)
3.18
3.20
Fig. 6. The imaginary part of w(3).
1
3.14
0.1
3.12
3 2
3.12
1 Im (χ(3)) (10-9 esu)
The parameters of GaN and InGaN at room temperature used in this paper are given in Table 1 [18]. In order to understand the dispersion and gain or loss of the optical field in the MQWs, setting the width of the wells as 7.0 nm and the concentration of In as 0.05, the figures of the modulus, real and imaginary part of w(3) are obtained in Figs. 4–6, respectively.
3.22
From Fig. 5, the real part of w(3) has the peak-valley trends near the resonant frequencies, which indicates that the dispersion of the InGaN/GaN MQWs we used is normal. From Fig. 6, the imaginary part of w(3) has the shape of upward peaks, which indicates that the semiconductor material we used has nonlinear absorption to the incident laser. In addition, it can be seen that there are two obvious peaks in the figures of the modulus, real and imaginary part of w(3). The two obvious peaks denote the transitions from the energy levels of heavy and light holes in the valence band (HH1 and LH1) to the first conduction subband respectively. In fact, there should be three peaks in all the three figures. The reason for the disappearance of the third peak is that there is a great difference of values between the third peak and the former two. This difference is due to the probability of the transition from the energy level of different holes in the valence band to the conduction subband, which can be determined by the wave functions obtained from the method outlined above.
ARTICLE IN PRESS Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
Under the condition of near resonance, the third-order nonlinear optical susceptibility w(3) of DFWM comes to maximum values when the energy of incident photons equals the difference between the corresponding two energy levels. However, because of the different transition probabilities, there is a great difference between the maximal values in different resonant frequencies, which results in the disappearance of the third peak in Figs. 4 and 5. Analyzing the pictures in detail, the modulus of w(3) comes to the peaks at the photonic energy of 3.151 and 3.158 eV. The values of the two peaks are 2.35 109 and 2.15 109 esu, which correspond to the energy gaps from the subband HH1 of the heavy holes and LH1 of the light holes to the first conduction subband, respectively. At the photonic energy of 3.194 eV, there is a tiny peak in the modulus, which is three orders of magnitude smaller than the former two. This peak corresponds to the energy gap from the valence subband SO1 of the spin–orbit split-off holes to the first conduction subband. Although the value of the peak is very small, the contribution cannot be neglected in the studies of nonlinear optical effects. Fortunately, there are three obvious peaks in Fig. 6 of the imaginary part of w(3), which are contributed by the transitions from the valence subband of the heavy holes, light holes and spin–orbit split-off holes to the first conduction subband respectively. As many previous researches have neglected to discuss the contribution of the spin–orbit split-off energy to the third-order nonlinear optical susceptibility, we focus on the correlations between the third peak of the imaginary part of w(3) vertical to the zdirection and the width of the QW, and the constituents of the semiconductor material. When the concentration of In is kept constant at 0.050, varying the width of the wells from 65 to 90 A˚ with a step of 5 A˚, a set of peaks of the imaginary part of w(3), which
85
are due to the contributions of the transitions from the spin–obit split-off energy level to the first conduction subband, with different well widths are obtained, as shown in Fig. 7. It can be seen from Fig. 7 that the peaks have an infrared divergence with an increase of the well width, which indicates that the energy gap from the spin–orbit split-off energy level to the first conduction subband becomes narrower with an increase of the well width. On the other hand, when the energy of incident laser is equal to the energy gap between the spin–orbit split-off energy level and the first conduction subband, the contribution to the imaginary part of w(3) is enhanced and the enhancement becomes obvious with an increase of the width of the wells, which indicates that wider QW causes more loss to incident laser. When the well width L is fixed at 7 nm, varying the concentration of In from 0.040 to 0.065 in steps of 0.005, another set of peaks of the imaginary part of w(3), which are due to the contributions of the transitions from the spin–obit split-off energy level to the first conduction subband, with different concentrations of In are obtained, as shown in Fig. 8. It is shown from Fig. 8 that with an increase of the concentration of In, the positions of the peaks of the imaginary part of w(3) have red shifts, which indicates that the energy gap between the spin–orbit split-off energy level and the first conduction subband becomes narrower with an increase of the concentration of In, and the values of the peaks vary irregularly. When the concentration of In changes from 0.04 to 0.05, the peak values increase linearly. However, when the concentration of In changes from 0.05 to 0.065, the peak values decrease linearly. This phenomenon is very interesting. The reason, in our opinion, is partly that the width of the forbidden gap of the MQWs lies on the composition of the potential well. When the
0.03 0.01
0.025 L9
0.009
0.02
In0.05 0.008
0.015 L8
Im (χ(3)) (10-9 esu)
Im (χ(3)) (10-9 esu)
L8.5 L7.5
0.01 L7
L6.5
0.005
3.15
3.16
3.17 3.18 3.19 Frequency (ev)
3.20
3.21
Fig. 7. The contribution of spin–orbit split-off energy to the imaginary part of w(3) with different well widths.
0.007
In0.055 In0.065 In0.06
In0.045
In0.04
0.006
0.005
0.004 3.12
3.14
3.16 3.18 3.20 Frequency (ev)
3.22
3.24
Fig. 8. The contribution of spin–orbit split-off energy to the imaginary part of w(3) with different concentrations of In.
ARTICLE IN PRESS 86
Y. Yu et al. / Microelectronics Journal 38 (2007) 80–86
material of the potential wells is InGaN, the width of the forbidden gap is determined by the concentration of In. To some extent, the forbidden gap becomes wider with an increase of the concentration of In. But this relationship is not always valid. There should be a value and if the concentration of In is equal to it, the peak value of the imaginary part of w(3) will get to extremum. Therefore, with an increase of the concentration of In, the peak values become larger in the beginning and smaller later. The content of this part needs further study and an experiment is being designed to confirm it in our next work. 4. Conclusion The expression of the third-order nonlinear optical susceptibility w(3) of the modes with the polarization vertical to the [0, 0, 1] direction of the MQWs is discussed. Two different types of transition processes included in FWM are discussed in detail and the expression of w(3) is simplified. Under the condition of three-photon resonance, the third-order nonlinear optical susceptibility of the modes with the vertical polarization has been calculated by the k~ ~ p method and the effective-mass theory. The contribution of the spin–orbit split-off energy to the thirdorder nonlinear optical susceptibility with the variation of the width of QW and the concentration of In, which is neglected in some published literatures, has been discussed. It has been shown from the results that the peaks of the imaginary part of w(3), which are due to the transitions from the spin–orbit split-off energy level to the first conduction subband, have red shifts with an increase of the width of the QW and the concentration of In.
Acknowledgment This work was financially supported by the Natural Science Foundation of China (Grant no. 10534030).
References [1] J. Petalas, S. Logothetidis, S. Boultadakis, M. Alouani, J.M. Wills, Phys. Rev. B. 52 (1995) 8082. [2] H. Schmidt, A. Imamoglu, Opt. Commun. 131 (1996) 333. [3] F. Caposso, K. Mohammed, A.Y. Cho, IEEE J. Quantum Electron. QE-22 (1986) 1853. [4] H. Gao, G. Xiong, X. Feng, Microelectr. J. 37 (2006) 742. [5] J. Chr. Holst, A. Hoffmann, D. Rudloff, F. Bertram, T. Riemann, J. Christen, T. Frey, D.J. As, D. Schikora, K. Lischka, Appl. Phys. Lett. 76 (2000) 2832. [6] C. Wang, G. Xiong, Microelectr. J. 37 (2006) 847. [7] G. Todorovic´, V. Milanovic´, Z. Ikonic´, D. Indjin, Phys. Lett. A 279 (2001) 268. [8] H.C. Lee, A. Hariz, P.D. Dapkus, A. Kost, M. Kawase, Appl. Phys. Lett. 50 (1987) 1182. [9] V.I. Litvinov, J. Appl. Phys. 88 (2000) 5814. [10] X.Y. Liu, J.F. Fa¨lth, T.G. Andersson, P. Holmstro¨m, P. Ja¨nes, U. Ekenberg, L. Thyle´n, J. Cryst. Growth 278 (2005) 397. [11] Yu.M. Sirenko, J.B. Jeon, K.W. Kim, M.A. Littlejohn, M.A. Stroscio, Phys. Rev. B. 53 (1996) 1997. [12] S.A. Wolf, et al., Science 294 (2001) 1488. [13] Y. Takahashi, H. Kawaguchi, IEEE J. Quantum Electron. 36 (2000) 864. [14] F. de Rougement, R. Frey, Phys. Rev. B. 37 (1988) 1237. [15] P.N. Butcher, D. Cotter, The Elements of Nonlinear Optics, Cambridge University Press, Cambridge, 1990. [16] J.M. Luttinger, W. Kohn, Phys. Rev. 97 (1955) 869. [17] J.M. Luttinger, Phys. Rev. 102 (1955) 1030. [18] I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815.