Extreme optical anisotropy in strained (110) quantum wells

Superlattices

and Microstructures,

355

Vol. 12, No. 3, 1992

EXTREME OPTICAL ANISOTROPY IN STRAINED (110) QUANTUM WELLS Y.Kajikawaa and M.Hatab

‘Optoelectronics Technology Research Laboratory, 5-5 Tohkodni, Tsukuba, Ibaraki 300-26, Japan bSemiconductor Research Center, Sanyo Electric Co., Lrd., 1-18-13 Hashiridani, Hirakuta, Osaka 573, Jopan (Received 4 August 1992)

The effects of strain on optical anisotropy in (110) GaxInl_,As quantum wells (QW’s) are compared within a multiband effective-mass approximation between two cases: (1) when a uniaxial compressive stress is externally applied normal to the QW plane, and (2) when an in-plane internal biaxial tensile strain is induced due to a lattice mismatch. In both cases it is shown that the lowest interband transition can be tuned by the strain so that it is forbidden for light linear-polarized along one of the principal axes in the QW plane. However, the in-plane polarization direction for which the transition is forbidden in the two cases is perpendicular to each other, i.e., [OOl] for case (1) and [TlO] for case (2).

1.

Introduction

It has recently been shown both theoretically1V2 and experimentally3*4 that (110) quantum wells (QW’s) show optical anisotropy with respect to a rotation of the polarization vector in the (110) well plane. Though the extent of the anisotropy is not so large (no more than 8 9%) in the absence of strain, there is a possibility that it is enhanced to be remarkable by strain effects. To date, however, most of the studies concerning the strain effects on QW’s were those on (001) QW’s. Here, we examine the effects of strain on the optical anisotropy in (110) QW’s for two cases: (1) when a uniaxial compressive stress is applied externally normal to the QW plane and (2) when an in-plane internal biaxial tensile strain is induced due to a lattice mismatch between the well layer and the substrate. In case of (001) QW’s, a uniaxial compressive stress normal to the QW plane has the same effect as a biaxial tensile strain in the QW plane regarding the relative energy position between the first heavy- and light-hole level: Both shift these two levels so that they approach each other; at a critical strain, the two levels cross. On the other hand, in the case of (110) QW’s, the crossing is replaced by an

0749-6036/92/070355

+ 04 ao8.00/0

anticrossing due to the lower symmetry. The lowest two levels in (110) QW’s have well-mixed characters between the heavy- and light-hole type within the anticrossing region. The magnitude of matrix element as well as its polarization dependence is expected to vary according to the extent of mixture. In the following we first derive analytical expressions for the matrix elements in (110) strained QW’s within an effective-mass approximation; we then compare the calculated results of the matrix elements between the above-mentioned two cases.

2. Effective-Mass Approximation for the Calculation of Matrix Elements in (110) QW’s The 4x4 Luttinger Hamiltonian for holes in a (110) QW in the IJ, M> basis for J=3/2 is given in Refs. 5 and 6. By setting k,=k,=O, the Luttinger Hamiltonian at the Brillouion zone center is written as’

H=

li2k 2 -71 2m0

I

0 1992 Academic Press Limited

356

Superlattices and Microstructures, EU=E

E

(1)

21 =

AE = where 7, = (l/2) y2 + (312) y3, 7, = (v%2) (r3 - r2). I is the 4x4 unit matrix, and the yis are Luttinger parameters. The coordinate axes are taken as x//[TlO], y//[OOl], and z//[ 1 lo]. Here, we adopt an infinitely high barrier model, in which k, can take only discrete values of nn/L, where L is the well width. The Pikus-Bir Hamiltonian, i.e., the strain Hamiltonian in the V, M> basis, for holes in a (110) QW can be obtained by substitution in the Luttinger Hamiltonian using the following correspondence relations:7

kikj c)

E g

fl2

(i, j = X, Y, z), 2m, tf -Dd,

(2)

yy =

E ,, -

c,,

Vol. 72, No. 3, 1992

EII’

AE , E ij = 0 (i+j).

2(C,, + 2C,2) + c,, + 2c,,

(6)

ofI’

where the C’s are stiffness constants. Therefore, the strain Hamiltonian can be written in the same manner as in eq. (4), but with o=Dd(3e,,5 = (l/60,

AE), + l/W,‘)

A.E,

(7)

q=&(D,‘-D,)AE. In both cases the total Hamiltonian H,=H+H, at the zone center is block diagonal. There are therefore two energy levels for each n; each of the energy levels is doubly degenerate. The two energy levels for each n can be obtained by diagonalizing the total Hamiltonian with k,=nx/L; they are written aslo*ll

where the D’s are deformation potentials. Under uniaxial compression X along [l lo], the strain components can be written as6

Evln = y1 E, - 0 - (an2 + Pn2)1nT (8) Ev2n= y1 ~~~ 0 + (an2 + Pn2P.

E

21

= - (l/2) (St, + S,, + S&)X,

Exx = - (l/2) (St, + St2 - S,,/2)X? EYY=

-

S,$X,

E

ij

=

(3)

can be written as linear combinations m.spectively:lo*l ’

0 (i #j),

where the S’s are compliance strain Hamiltonian becomes

where q,,=(E2/2m,$(nx/L)2, a,, = y, E,, - 5 and fi,, = q. The eigenstates corresponding to ~,t,, and E,T~ r, %n

constants.

Therefore, the

Ivl,(f)>

of the IV,M> states,

= cos en l3/2, +3/2> - sin 8,13/2, r1/2>, (9)

lv2,(f)> = sin en 1312,f3/2> + cos 8,13/2,71/2>,

H,=-oI-

[

:

;

i

;],

(4)

where 8, = tan-’ [( 4 an2 + pn2 - a,,)/P,]. They can also be written as linear combinations of Ix>, ly>, and Iz> states, e.g.,

o = -DDd (S,, + 2S,,) X, Ivl,(+)> = - (2/3)“2 [sin (8, + 7t/3)lxt>

5 = (l/2)

E, +

(X3 E;,

(5)

- i sin

(e, - x/3)lyT> + sin e,lzJ>].

(10)

q = (G/2) (Eu’- Euh where E, = (l/3) D, (S,, - St,) X and E,’ = (l/6) D,’ S,, On the other hand, when an in-plane internal biaxial strain is induced in the well layer due to a lattice mismatch, the strain components become8’9

As can be seen from eq. (lo), the !x> or ly> component vanishes in the Ivl,(+)> state when 8,=frt/3. Since alp,lx> = = -dp,lz> = P, the effective matrix elements can be written as lo*’’ lMJ2 = 21Mb12sin*

(en + x/3),

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Table I. Parameters used in calculations, where a, y’s, D’s and S’s are the lattice constant

(A),Luttinger

deformation potentials (eV), and elastic compliance constants (10-12 bar-‘). Parameters for Ga,In,_S\s

parameters,

are derived by

linear interpolation.

s&f

Y2

Y3

Du

DLL’

Sl 1

s12

&AS

5.6533

2.1

2.9

2.57

3.94

1.15

-0.358

1.657

InAs

6.0583

8.37

9.29

2.7

3.1 I

I.945

-0.6X.5

1.526

InP

5.8697

11

IM I2 = 21Mb12sin2 (8 n - 7t/3)' Y

IM 2I2=

(11)

2lMb I2 sin2 ElIt'

for the c,-vl, transitions, where IMb12=1/3P2 is the matrix element for the bulk. Thus, when B,=fn/3 the matrix elements for the c,-vl n transitions vanish for x or y polarization, corresponding to a vanishing of the Ix> or ly> component in the Ivl,> state. 3.

Results

'\,, [ii 01 \\ ‘\

‘.

‘.._

-.

of Calculation

In this section we show that a situation does occur in which the matrix element vanishes for x or y polarization. We present the calculated results of the optical matrix in a (110) elements for the c, -L’1 1 transition Ga,Inl_,As QW grown on an InP substrate with a well width of 100 A. The elastic coefficients and the band parameters used in the calculation are listed in Table I. A linear interpolation was used for the Ga$nl_#s values. Figure 1 shows the effective matrix elements for the lowest transition of cl-\‘1 1 in a (110) Ga0,471n0,53As QW, which is lattice-matched to an InP substrate, as a function of an external uniaxial compression X along [ 1 lo]. The solid and dashed lines indicate the matrix elements for the [OOl] and [TlO] polarizations, respectively. As can be seen in Fig. 1, the matrix element vanishes for the [OOl] polarization at X=20 kbar. Figure 2 shows the matrix elements for the c,-vl 1 transition in a (110) Ga,InI_fis QW, in which an in-plane internal biaxial tensile strain is induced due to a lattice mismatch to an InP substrate, as a function of the Ga composition x. In this case the matrix element vanishes for the [IlO] polarization at x=0.63, as can be seen in Fig. 2. The difference between the two cases above in the polarization direction for which the matrix element vanishes is due to the difference in the sign of On, i.e., the sign of the off-diagonal term (p,) in the total Hamiltonian, at the critical strain. In the uniaxial compression case, the off-

-ol 0

10

20

;0

X (kbar) Fig. 1

Matrix elements of the lowest optical transition in a (110) G%,471n0,53As QW for light linearpolarized in the QW plane as a function of the external [l lo] uniaxial stress.

diagonal term in the strain Hamiltonian. rl, described by eq. (5) is always negative, since E,‘/E,= (l12)D,‘S,,/D,(S,I-S,2)<1 for Ga0,471n0,53As. (This is true for all III-V semiconductors). Therefore, /.$=Y~E~,-~ cannot be negative. Thus, 8, cannot be equal to -x/3, whereas it can be equal lo 71/3. On the other hand, in the in-plane tensile strain case, q described by eq. (7) is posmve since D,‘/D,>l for Ga$nl_x As. Therefore, /3, can be negative for a sufficiently large tensile strain, and 8, can be equal to -n/3 at some critical strain. On the other hand, 8, cannot be equal to rr/3, since 8,,
358

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+ 0

$

Vol. 12, No. 3, 1992

-----------2+--+rl E//

N

and Microstructures,

w

VI

mismatch, while the polarization directions for which the matrix element vanishes are perpendicular to each other in the two cases, i.e., [OOI] in the former case and [TlO] in the latter case. The difference in the direction between the two cases has been explained as a result that E,‘/E,y&>l for Ga,In,_,As. We have shown in both cases that it is possible to get extreme anisotropic optlcal properties in a heterostructure comprising materials which have isotropic optical properties by themselves, by inducing only moderate strain.

--.

‘\\ \

[il

5

01 \ I

References

1. 0'

I

1

I

0.5

I

,,

0.6

6. 7. 8.

Y.Kajikawa, M.Hata, and T.Isu, Japanese Journal of Applied Physics 30, 1944 ( 199 1). G.E.W.Bauer and H.Sakaki, Surface Science 267, 442 (1992). Y.Kajikawa, M.Hata, T.lsu, and Y.Katayama, Surface Science 267,501 (1992). D.Gershoni, I.Brener, G.A.Baraff, S.N.G.Chu, L.N.Pfeiffer, and K.West, GaAs and Related Compounds 1991 (Institute of Physics Conference Series, No.120, Bristol, 1992). p.419. G.E.W.Bauer, Spectroscopy of Semitonductor Microstructures, edited by G.Fasol, A.Fasolino, and P.Lugli (Plenum, New York, 1989), p.381. J.B.Xia, Physical Review B43, 9856 (1991). S.L.Chuang, Physical Review B43,9649 (199 1). J.Hornstra and W.J.Bartels, Journal of Crystal

9.

Growth 44, 513 (1978). E.Anastassakis, Journal

10. 11.

4561 (1990). M.Hata and K.Yagi, unpublished. Y.Kajikawa, to be published in Physical Review B.

I 0.7

2.

Ga composition Fig. 2

Matrix elements of the lowest optical transition in a (110) GaJnl_$s QW which is strained due to a lattice mismatch to an InP substrate, for light linear-polarized in the QW plane as a function of the Ga composition.

3. 4.

5. polarization in the uniaxial stress case, while it vanishes for [TlO] polarization in the in-plane tensile strain case. 4.

Conclusion

We have shown that the matrix element of the lowest transition in (110) Ga,Int_,As QW’s can be completely killed off for one of the in-plane polarizations by applying an external [llO] uniaxial compression, as well as by inducing an in-plane tensile strain through a lattice

of Applied

Physics

68,