Anisotropic optical matrix elements in [hhk]-oriented quantum wires

MATEIIALS SCIENCE & ENGINEERING E LSE V IE R

B

Materials Scienceand Engineering B35 (1995) 288-294

Anisotropic optical matrix elements in [hhk]-oriented quantum wires A. Atsushi Yamaguchi*, Akira Usui Fundamental Research Laboratories, NEC Corporation, 34 Miyukigaoka, Tsukuba, lbaraki 305, Japan

Abstract

Optical polarization properties in quantum wires (QWIs) are theoretically investigated as functions of wire crystallographic directions taking the valence band anisotropy into account. Optical matrix elements and gain spectra are calculated for GaAs cylindrical QWIs with infinite barriers. It is shown that the optical matrix element for light polarized to the wire direction shows weak dependence on the wire crystallographic direction. In contrast, the valence band anisotropy causes strong dependence on the wire direction for light polarized to the perpendicular directions, and large in-plane optical anisotropy appears for [110]- and [ll2]-oriented QWls. It is considered, from the calculated results of the gain spectra, that a [111]-QWI laser shows the lowest threshold current and that the [1, - 1,0]-QWls on a (110) substrate are the most suitable for polarization controlled vertical cavity surface emitting lasers. These results indicate that the structural optimization from the viewpoint of the crystallographic direction is important for optical devices using QWls. Keywords: Quantum wire; Optical matrix element

1. Introduction Semiconductor quantum wire (QWI) structures are promising for optical devices such as lasers with low threshold current [1,2]. The optical polarization properties are important to determine the characteristics of such devices. The valence band structure in ordinary semiconductors such as GaAs is not spherical, but has a cubic symmetry [3]. This valence band anisotropy effect modifies the optical polarization properties in semiconductor quantum structures; also, the properties are mainly determined by the structural anisotropy. It is known that optical anisotropy in quantum wells varies with the substrate orientation [4-7]. Similarly, in QWI structures, optical matrix elements depend on the crystallographic wire direction and optical anisotropy can appear in the plane perpendicular to the wire direction, even if the cross-sectional shape of the wire is symmetrical, as in a circle [8]. These effects should cause the threshold current of QWI lasers to depend on the wire crystallographic direction. The QWI structures are also expected to be applicable to polarization controlled vertical cavity surface emitting lasers (VCSELs), because the optical matrix elements of the lowest en* Corresponding author.

ergy transition for light polarized to the wire direction are larger than those for the light polarized to the perpendicular direction. The crystallographic direction of the wire and the substrate orientation, which determine the polarization properties of the QWI, should also affect the characteristics of this device. The wire crystallographic direction dependence of the optical properties in QWls has been studied by several authors [9,10]. Citrin and Chang [9] calculated the optical matrix elements in [hk0]-oriented QWIs. Vurgaftman et al. [10] compared the characteristics of [001]-QWI and [111]-QWI lasers. However, there are no studies on the wire direction dependence of the optical anisotropy in QWIs which orient to a more general [hhk] direction. There are also no studies which compare the characteristics of optical devices with [ll0]QWls usually fabricated experimentally [11] to those with QWls oriented to the other directions. Therefore, a more systematic study is needed for understanding the effect of the valence band anisotropy in QWIs. In this paper, we calculate the optical matrix elements in arbitrary [hhk]-oriented QWls for light polarized to any direction in order to investigate the effect of the valence band anisotropy. The calculations are performed under envelope function approximation in cylindrical GaAs QWls with infinite barriers. The calElsevier Science S.A. SSDI 0921-5107(95)01339-3

A. Atsushi Yamaguchi, A. Usui / Materials Science and Engineering B35 (1995) 288-294

culated results show that the optical matrix elements for light polarized to the directions perpendicular to the wire strongly depend on the wire crystallographic direction, while those for light polarized to the wire direction have a weak dependence. Furthermore, we also calculate the gain spectra in [hhk]-QWls to investigate the wire direction dependence of the characteristics of the QWI lasers. The results reveal that the structural design from the viewpoint of crystallographic direction is important for QWI optical devices. The structural optimization of low threshold QWI lasers and polarization controlled QWI-VCSELs are presented as examples.

2. Calculation M e t h o d s

We consider cylindrical GaAs QWls which orient to arbitrary [hhk] directions. The barrier height of the surrounding material is taken to be infinite and the radius of the wire R is taken to be 5 nm. The calculations of the valence band structures of the [hhk]-QWIs are essential for this study. The Hamiltonian we treat here for valence band states is a 4 x 4 Luttinger-Kohn Hamiltonian for kp perturbation. The Luttinger parameters for GaAs are taken to be ?1 =6.79, ?2 1.924, ?3 = 2.681 [12]. The difference between ?2 and ?s corresponds to the valence band anisotropy of the bulk crystal. Some studies on valence band structures in QWIs use a spherical approximation, in which the valence band anisotropy is completely neglected by assuming ?2 = ?3 in the, Luttinger parameters [13,14]. We also calculate under this approximation by taking both values of ?2 and i/3 to be (72 + ?3)/2 in order to investigate the validity of the approximation. The coordinate system of the Hamiltonian is transformed from the ordinal crystal coordinate system into a coordinate system (x,y,z) shown in Fig. 1 more suitable for the calculations, where the :,'-axis is along the wire direction ([hhk] direction), and x- and y-axes are respectively along the [1, - 1,0] and [k,k, - 2h] directions, which are perpendicular to the wire direction. Eigenenergies and wavefunctions are calculated by diagonalization of the Hamiltonian matrix represented by the base wavefunctions. The base wavefunctions used here are written as follows, using Bessel functions that satisfy the boundary condition at r = R =

1

3

NL, Ji. (kL, r ) exp (iL~b) exp (ikzz)l ~ ,Jz )

ilL,,

Here kz is a wavenumber along the z direction and Jz is the z-component of the angular momentum in the Bloch function. NL, is a normalization factor of the wavefunction. The eigenenergies and wavefunctions at any points in the Brillouin-zone are calculated by the diagonalizations for the corresponding values of kz. The squared optical matrix element for the transition between the conduction and valence band states takes the general form

IMI== I.AI =

(3)

where p is the momentum operator and A is the polarization vector of the linearly polarized optical wave, (c] and Iv) are the wavefunctions of the conduction and valence states respectively. The calculation of conduction band states is quite simple compared with that for the valence band states described above. The conduction eigenenergies and wavefunctions in cylindrical QWIs are straightforwardly calculated by the one-band effective mass theory [15]. The conduction wavefunctions are written as 1

1

1

-~JL(kL,r ) exp (iLO) exp (ikzz)l 5' --- 2 1

(4)

where Arc is a normalization factor. The energy of the state is given by h2

E~(kz) = ~ m . ( k L + k~)

(5)

The effective mass of a conduction electron (m2) is taken to be 0.067m0 for GaAs and the band nonparabolicity is neglected. Now that we get the conduc-

Y[

X [1,-1,0]

( 1)

Here (r,O,z) is a cylindrical coordinate of the (x,y,z) coordinate system. JL is an Lth order Bessel function and kL, is expressed using ilL, (nth zero point of Lth order Bessel function) by

kL, = R

289

(2)

Z [h,h,k] Fig. 1. Coordinate system used in the calculations: z-axis is along the wire direction ([hhk] direction); x- and y-axes are respectively along the [ 1 , - 1,0] and [k,k,- 2h]] directions, which are perpendicular to the wire direction.

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tion and valence band states, the optical matrix elements for light polarized to any direction can be calculated by Eq. (3). The gain spectrum at 300 K can be calculated from the optical matrix elements and the dispersion relations of the conduction and valence bands as follows. If we take Eph as the photon energy, the material gain spectrum g(Eph) is given by 2 e22h2 Z 2 f c¢~'dkz g(Epn) = p nomoEocR Eph ~ a d - ~

× ]M~(kz,A)]2[fc(Ec~(kz)) -fv(Evp(kz))] x

~'in (Eph - E=4(k~))2 + (h/2"in) 2

(6)

where ~ and fl are the sub-band indices of the conduction and valence band. Ec~(kz) (E,.a(kz)) is the dispersion relation of ~(fl) sub-band of the conduction (valence) band, and E~p(kz) is the energy difference between the ~ sub-band of the conduction band and the sub-band of the valence band at point kz. The band gap of GaAs at 300 K is taken to be 1.424 eV. Tin and no are respectively the intraband relaxation time and refractive index. Here we take qn = 1 ps and no = 3.6.f. andf~ are respectively the electron and hole quasi-Fermi functions at 300 K. The quasi-Fermi energy of the conduction and valence bands are determined from carrier density n by the following equations. n~

c

1===

v

Here D,(E) and D~(E) are density of states of the conduction and valence bands. Gain spectra g(Eph) are calculated in arbitrary [hhk]-QWIs as functions of the polarization vector and carrier densities.

3. Results and discussion

Fig. 2 shows the valence band structures in QWIs which have various crystallographic orientations. The valence band structure calculated under the spherical approximation is also shown in Fig. 2(e). The valence band structures in QWIs strongly depend on the wire directions, as shown in this figure. For example, for the lowest sub-band, confinement energies are maximum in the [001]-QWI and minimum in the [lll]-QWI. The dispersion curve in the [1 l l]-QWI is nearly parabolic and the average density of states in the range of energies close to the band-edge is significantly lower, as already pointed by Vurgaftman et al. [10], compared with QWIs which orient to the other directions. It is also noticed that the feature of sub-band anti-crossing due to state mixing depends on the wire direction. These wire direction dependences of the valence band structures are the origin of the wire direction depen-

dence of the optical properties discussed below. The main optical transitions in the near-band-edge region are C I - V 1 and C1-V3 transitions, because the C1 V2 transition is not allowed at the F point, where Cn-(Vn) denotes the nth lowest energy conduction (valence) sub-band. The optical matrix elements of the C 1 - V1 and C 1 - V3 transitions at the F point for light polarized to the x, y and z directions are shown in Fig. 3 as functions of the wire direction. The horizontal axis 0 represents the angle between the wire direction and the [001] direction. The value of 0 changes from 0 ° to 90 ° during the change of direction from the [001] to the [110] through to the [111] direction. The matrix elements under the spherical approximation are shown by the broken lines in Fig. 3. For the C1-V1 transition, the matrix element for light polarized to the wire direction (z direction) is four times larger than those for light polarized to the directions perpendicular to the wire (x, y directions) under the spherical approximation, as already known [16], and no optical anisotropy appears in the x y plane, because the cross-section of the wire is a circle. The solid lines in Fig. 3 show calculated results with consideration of the valence band anisotropy. For light polarized to the z direction, the dependence on the wire direction is weak and the spherical approximation can be considered valid. The situation is different for x and y polarizations. The matrix elements strongly depend on the wire direction. The valence band anisotropy causes in-plane optical anisotropy in QWIs other than [001]-QWIs (0 = 0 °) and [lll]-QWIs (0~55°). Large anisotropy appears in [ll0]-QWIs (0 = 90 °) and the matrix element for light polarized to the y ([001]) direction is about twice as large as that for the x ( [ 1 , - 1,0]) direction. Large anisotropy also appears in O ~ 35 ° (close to [ll2]-oriented QWIs). In this case, the matrix element for polarization to the x ( [ 1 , - 1,0]) direction is larger than that for the y ([1,1,-1]) direction. For the C1-V3 transition, the matrix elements for x and y polarization are large, and that for z polarization is zero under the spherical approximation, as shown by the broken lines in Fig. 3(b). The calculated results with consideration of the valence band anisotropy (solid lines) show that the matrix elements for z polarization show weak dependence on wire directions and that those for x and y polarizations strongly depend on the wire direction, as seen for the C1-V1 transition. It is interesting in physical meaning that the probability of C I - V3 transition for z polarization is not zero in QWIs other than [001]- and [lll]-QWIs. This breaking of the selection rule is also caused by the valence band anisotropy effect, x - y in-plane optical anisotropy appears in QWIs other than [001]- and [lll]-oriented ones. The largest anisotropy also appears in [110]-QWIs, as was the case for the C1-V1 transition. The matrix element for x polarization is larger than that for y polarization, which

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Fig. 2. Calculated valence band structures in GaAs cylindrical QWIs with radius of 5 nm oriented to various crystallographicdirections: (a) [001]; (b) [112]; (c) [111]; (d) [110]. The result calculated under the spherical approximation is shown in (e). The original point of energy is the top of the valence band in bulk material. is the opposite of the :result for the C 1 - V1 transition. A large anisotropy also appears in [ll2]-QWIs (0 ~ 35°). It should be: noted that the matrix elements for the x and y polarizations tend to decrease with an increase in 0, and that the value for 0 above 40 ° is about half of the value for 0---0 °.

These results show that the optimization of crystallographic direction of wires and light polarization used in optical devices is important. Especially, this optimization is important for devices which use light polarized to directions perpendicular to the wires. We consider optimization of the Q W I laser structure as an example.

A. Atsushi Yamaguchi, A. Usui / Materials Science and Engineering B35 (1995) 288-294

292

Fig. 4 shows material gain spectra in [001]-QWI lasers for various carrier densities. Since optical anisotropy in the x - y polarization plane does not appear in [001]QWIs, two curves are drawn for the z and xy polarizations in Fig. 4. For small values of carrier density, such as 1 × 105 cm-~ (Fig. 4(a)), gain is negative and absorption takes place in all spectrum regions. The main absorption peaks are the C1- V1 transition for z polarization and the C I - V3 transition for xy polarization. The C1-V3 transition peak is higher than the C1-V1 transition peak because the joint density of states of the C 1 - V 3 transition is larger than that for the C 1 - V 1 transition. As carrier density increases (Fig. 4(b)-4(d)), absorption for the C 1 - V 1 transition decreases and gain takes place at a certain threshold carrier density. The gain due to the C1-V1 transition for z polarization is larger than that for xy polarization due to the anisotropic optical matrix elements mentioned above. Further increase in carrier density makes the gain peak due to the C1-V3 transition grow (Fig. 4(e)), until it becomes higher than the gain peak for the C 1 - V 1 transition, as shown in Fig. 4(f). These features are also seen in QWIs which orient to other directions although gain anisotropy in x and y polarizations appears in QWls which orient to directions other than the [001] and [111]. Now we consider the optimization of QWI lasers as low threshold lasers. This laser should utilize the C1V1 transition of the z polarization because maximum gain occurs in the low carrier density region for this transition, as mentioned above. The optical matrix element for the z polarization is almost independent of the wire direction. However, there are the other factors 0.7 0.6

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I l l l l l t l

-

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0.6

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CI-V1

C1-V3

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0.5

0.

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0.2

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0.1

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20 40 60 80 8 (degree)

I

0 (b)

I

f

20 0

I

40

I

I

60

I

I

80

(degree)

Fig. 3. Calculated optical matrix elements in the QWIs for (a) C1- Vl transition and (b) C1- V3 transition at G point as functions of the wire crystallographic direction. Curves labeled X, Y, and Z correspond to optical matrix elements for light polarized to x, y and z directions. Broken lines show calculated results under the spherical approximation.

which cause wire direction dependence of the threshold carrier density. These are the joint density of states and the quasi-Fermi energy difference between electrons and holes depending on carrier density. Since the valence sub-band dispersion strongly depends on the wire crystallographic direction as shown in Fig. 2, threshold carrier density can depend on the wire crystallographic direction. Maximum values in the gain spectra for the three polarization directions are shown in Fig. 5 as functions of wire direction at the carrier density of 1.6 x 106 cm 1, which is just above the transparency carrier density. The maximum gain is always higher for z polarization than for x or y polarization, as shown in this figure. This shows that lasing occurs for z polarization as expected. This figure shows that the maximum gain for z polarization strongly depends on the wire crystallographic direction. It is maximum for [111]-oriented QWls and minimum for [001]-QWIs. These results show that the laser threshold carrier density is minimum for [111]-QWI lasers and maximum for [001]QWI lasers. Vurgaftman et al. [10] have already shown that threshold carrier density for [lll]-QWI lasers is smaller than that for [001]-QWI lasers. It is shown, in this paper, that [lll]-QWI lasers give the lowest threshold carrier density of the [hhk]-oriented QWI lasers. This result is mainly caused by the small density of states near the band-edge region of the valence band in [111]-QWI. QWI lasers are also promising as polarization controlled VCSELs, because the QWI structures have large optical anisotropy in the substrate plane, while bulk or quantum well structures have little or no optical anisotropy. Then we consider polarization stability in QWI-VCSEL next. Laser light from QWI-VCSEL is expected to be polarized to a direction parallel to the wire for small carrier densities, as mentioned above. However, as the carrier density increases, the gain of the C1-V3 transition for light polarized to directions perpendicular to the wire increases. Therefore, polarization of the laser light in QWI-VCSELs can switch to the direction perpendicular to the wire at a certain critical carrier density. Large values of the critical carrier density provide high polarization stability in QWIVCSELs. The maximum gain for each polarization direction is shown in Fig. 6 as a function of the wire direction at a carrier density of 2 x 107 cm 1, which is about one order of magnitude larger than the transparency carrier density. The gain of x and y polarization light exceeds that of z polarization in [001]-QWls, while the gain of x and y polarization is still smaller than that for z polarization in [111]- and [ll0]-QWIs at the carrier density. This result is mainly caused by the wire direction dependence of the optical matrix elements for the C1-V3 transition shown in Fig. 3(b). It is shown, from the result, that the polarization stability depends on wire crystallographic direction and that QWI-VC-

A. Atsushi Yamaguchi, d. Usui / Materials Science and Engineering B35 (1995) 288-294 I

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Fig. 4. Calculated material gain spectra in [001]-QWI lasers for various carrier densities: (a) 1 x l0 s cm - '; (b) 1 x 106 cm - '; (c) 1.5 x 106 cm - ~; 1.8 × 106 c m 1; (e) 3 x 10'~ cm - '; (f) 2 x 107 cm - I Solid lines show the gain for z polarization and broken lines show the gain for x and y polarization. (d)

SELs with [111]- or [110]-QWIs give higher polarization stability than those with [001]-QWIs. In the Q W I - V C SELs with [ l l 0 ] - Q W I s on the (001) substrate usually fabricated, the polarization stability for z polarization is determined by the ratio ,of gain for x ([1, - 1,0]) polarization to that o f z polarization. A l t h o u g h the polariza-

tion stability o f the V C S E L s is m u c h higher than the [001]-QWI-VCSELs, it is not the highest in [hhk]-QWIV C S E L s with arbitrary substrate orientations. The ratio described above is the smallest for y polarization in [110]-QWIs as shown in Fig. 6. F r o m this result, the [110]-QWI laser structure in which the direction perpen-

294

A. Atsushi Yamaguchi, A. Usui / Materials Science and Engineering B35 (1995) 288 294 I

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Fig. 5. Maximum values in gain spectra of the QWI lasers with cartier density of 1.6 x l06 c m - i as functions of wire crystallographic direction. Curves labeled X, Y, and Z correspond to the results for light polarized to x, y and z directions.

Fig. 6. Maximum values in gain spectra of the QWI lasers with carrier density of 2 x 107 c m - 1 as functions of wire crystallographic direction. Curves labeled X, Y, and Z correspond to the results for light polarized to x, y and z directions.

dicular to the wire in substrate plane is the y ([001]) direction is considered to show the highest polarization stability. Therefore, it is considered that the QWIVCSEL structure with [1,-1,0]-QWIs on a (110) substrate is the most suitable structure for polarization stabilized surface emitting QWI lasers.

on a (110) substrate, is considered to show the highest polarization stability.

4. Conclusions We calculate the optical matrix elements and gain spectra in arbitrary [hhk]-oriented cylindrical QWls for light polarized to any direction and investigated the effect of valence band anisotropy on optical polarization igroperties. The calculations employed envelope function approximation and took valence band anisotropy into account. The calculated results are shown for GaAs QWls with infinite barriers. It is shown that the optical matrix element for light polarized to the wire direction shows weak dependence on the wire crystallographic direction, that the elements for light polarized to the perpendicular directions show strong dependence on the wire direction, and large in-plane optical anisotropy appears for [ll0]and [ll2]-oriented QWls. It is found, from these resuits, that consideration of the valence band anisotropy is important in the calculation of the optical polarization properties in QWIs, especially for the light polarized to the directions perpendicular to the wires. Furthermore, we consider the structural optimization for low threshold QWI lasers and polarization controlled QWI-VCSELs from the results of the gain spectra. The [lll]-QWI laser is considered to show the lowest threshold current, and the QWI-VCSEL structures, which consist of [ 1 , - 1,0]-QWI lasers

Acknowledgment The authors appreciate the encouragement given by Dr. H. Rangu, Dr. J. Sone and Dr. M. Mizuta. The authors are also very grateful to K. Nishi, S. Ishizaka, H. Sunakawa, Y. Kato, and Dr. C. Sasaoka for their valuable discussions. References [1] Y. Arakawa and H. Sakaki, Appl. Phys. Lett., 40 (1982) 939. [2] M. Asada, Y. Miyamoto and Y. Suemune, Jpn. J. Appl. Phys., 24 (1985) L95. [3] P. Lawaetz, Phys .Rev. B, 4 (1971) 3460. [4] D. Gelshoni, I. Brener, G.A.Baraff, S.N.G. Chu, L.N. Peiffer and K. West, Phys. Rev. B, 44 (1991) 1930. [5] Y. Kajikawa, M. Hata and T. Isu, Jpn .J. Appl. Phys., 30 (1991) 1944. [6] S. Nojima, Jpn. J: AppL Phys., 31 (1992) L765. [7] A.A. Yamaguchi, K. Nishi and A. Usui, Jpn. J. Appl. Phys., 33 (1994) L912. [8] A.A. Yamaguchi and A.Usui, J. Appl. Phys., 78 (1995) 1361. [9] D.S. Citrin and Y.-C. Chang, J. Appl. Phys., 70 (1991) 867. [10] I. Vurgaftman, J.M. Hinckley and J. Singh, IEEE J. Quantum Electron., QE-30 (1994) 75. [11] E. Kapon, D.M. Hwang and R. Bhat, Phys. Rev. Lett., 63 (1989) 430. [12] L.W. Molenkamp, R. Eppellga, G.W. 't Hooft, P. Dawson, C.T. Foxon and K.J. Moore, Phys. Rev. B, 38 (1988) 4314. [13] P.C. Sercel and K.J. Vahala, Phys. Re~. B, 44 (1991) 5681. [14] H. Ando, S. Nojima and H. Kanbe, J..AppL Phys., 74 (1993) 6383. [15] H. Zarem, K.J. Vahara and A. Yariv, 1EEE J. Quantum Electron., QE-24 (1988) 523. [16] P.C. Sercel and K.J. Vahara, Appl. Phys. Lett., 57 (1990) 545.