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Polarization anisotropies in quantum wells
Surface Science 267 (1992) 442-444 North-Holland
surface science
Polarization anisotropies in quantum wells Gerrit E.W. B a u e r 1 Philips Research Laboratories, 5600 JA Eindhocen, Netherlands
and Hiroyuki Sakaki Research Center for AdL,anced Science and Technology, Unicersity of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153, Japan Received 1 June 1991; accepted for publication 29 June 1991
Three mechanisms are discussed which can cause a dependence of t,~c optical absorption on the direction of the polarization vector in the plane of a quantum well (QW), i.e. the warping distortio~a, strain, and anisotropic scattering potentials. Measured optical anisotropies do not constitute proof of quantum-wire or lateral superlattice formation.
1. Introduction
Perfect quantum wells of tetrahedral semiconductors grown on (001)-oriented substrates belong to the symmetry group D2d (D4h in effective mass theory). If the growth direction is chosen parallel to the z-axis, x and y form a basis of its irrcducible representation E(E,). Therefore, the optical matrix elements for linearily polarized li, ht with wave vector normal to the interfaces do not depend on the field vector. However, when ~;-,e four-fold symmetry is broken intrinsically or by external perturbations, the optical properties become anisotropic with respect to a rotation of the polarization vector in the plane of the interfaces. These optical anisotropies have recently been observed in quantum-wire arrays [1] and grid-inserted quantum wells [2] and were interpreted as evidence of lateral quantum confinement of the carriers. Anisotropies have also beer: measured in strain-patterned quantum wells [3] Part of this work was carried out at the NTT Telecommunications Chair at RCAST, University of Tokyo, ~1-6-i Komaba, Meguro-ku, Tokyo 153, Japan.
and 6-doped GaAs [4]. Here we discuss theoretically the effects of some symmetry breaking perturbations on the optical anisotropies.
2. (110) quantum wells
In quantum wells grown on a (ll0)-oriented substrate the four-fold symmetry axis is broken. In the effective mass approximation the tetragonal distortion is described by the warping term of the Luttinger Hamiltonian, which has been worked out for quantum wells in ref. [5]. The mixing of the heavy-hole and light-hole states by the nondiagonal elements modifies the oscillator strengths anisotropically [1,2]. The order of magnitude of the polarization can be estimated "back-of-the-envelope" by assuming infinite barrier n n t e n t ; n l ~ nncl or'~nciAor;nn t~nl~, ~ho ~+~+,~,- .~ the band edge (k X= k~ = 0). We obtain for the relative modulation of the oscillator strengths for linearily polarized light: [ A l [ / i = [ Ayl/'(2~/) to first order in [Ay[/~, where Ay and ~/ are the difference and the average of the Luttinger parameters )'2 and )'3, respectively. The maximum
()()39-6()28/92/$05.00 !i~ 1992 - Elsevier Science Publishers B.V. and Yamada Science Foundation. All rights reserved
G.E. IV. Bauer, 14. Sakaki / Polar&ation anisotropies in QW's
of intensity of the heavy-hole absorption is achieved when the polarization vector is parallel to the zig-zag bonding chains, which are seen to cause effects similar to quantum wires in the same direction. Polarization spectra are very sensitive to perturbations of the four-fold symmetry. Independent of the well width, the anisotropy amounts to --17 percent for GaAs parameters (T2 = 1.9, T3 = 2.7). Experimental anisotropies in ( l l 0 ) - q u a n t u m wells would be direct evidence of the warping distortion, which continues to elude optical detection in (001)-quantum wells [6].
3. Uniaxial pressure Kash et al. [3] experimentally and theoretically demonstrated optical anisotropies in quantum wells in which the symmetry was broken by lateral strain patterning. Following ref. [3], it is easy to derive the lowest-order effect of a uniaxial pressure P applied to a [001] "hard-wall" quantum well of width d. For the heavy holes we obtain that w h e n P II[ll0]: (I I . ) / [ = 2m o S a a D u , d 2 p / ( 3 h 2 y 2 7 r 2 ) = 0.3 ((J/kbar)(d/lO0 i ) 2 and when PII[100]: (Iit-I ± ) / i = 2moD~(S ~ -
443
4. Anisotropk scattering
The epitaxial growth-controlled fabrication of lateral superlattices and quantum wires on tilted surfaces [7] is an important alternative to lithographic techniques. Recently, anisotropies in the mobilities of a heterostructure with an AlAs grid inserted close to the 2d-electron gas has been interpreted in terms of scattering from AlAs island structures in the inserted sub-monolayer, which are characterized by some degree of shortrange correlations [8]. We have investigated the consequences of such a model for the optical anisotropies [9]. Briefly~ we calculate the configurational average of corrections to the photon self-energy in second-order perturbation theory,
Autocorrelation function 3oo
1oo --
rii/~!i I I
',
-1oo
-
Sl2)
d 2 p / / ( 3 h 2 T 2 " n ' 2 ) --- 0.2(P/kbar)(d/lO0
i -300
A ) 2,
where I± and Ill denote the oscillator strength for field vectors normal and parallel to the pressure direction. Du( = 2.55 eV), D~,( = 3.94 eV) arc deformation potentials and S ~ ( = 1.15 × 10 -" bar-l), St2(= -0.35 × 10 -6 b a r - l ) , and $44( = 1.66 x 10 -6 bar -~) are the elastic constants of the semiconductor (GaAs). For the light holes the above results should be multiplied by minus one. We see that relatively small pressures of the order of 1 kbar can cause large anisotropies. In ideal grid-inserted quantum wells on a GaAs substrate all strain from the A I A s / G a A s lattice mismatch of 0.2% is accommodated by the AlAs and optical effects should be very small. The o~t~rnnl pre~,,re w h l c ' h w n , ~ i d h a r p t n h o ~ n n l l e c l to cause the same strain in bulk AlAs is about 5 kbar, however, it should therefore be kept in mind that in imperfect structures observable optical anisotropies could be caused by only a small transfer of the strain from the AlAs to the GaAs strips.
100
iilii '
!,Iji! , -100 v
-300
A 100 -
-100
I!!//
J
\f ,'
-300 -300
-100
100
300
x (A) Fig. I. C o n t o u r plots o f the a u t o c o r r e l a t i o n f u n c t i o n s of Al/ks islands as used foi i h c xcsulis in fig. 2, which a~c pi.tlialiicici-
ized as C(x, y ) = e x p { - ( x / 1 , ) 2-(y/I,)e}C(L). where I , / , are Gaussian decay lengths and C(L) is the autocorrclation of a perfect grid withoPeriod L. Fromo above, the figures are obtained for L x = 50 A, L,.= 200 A, L = zc (upper figure. dotted line in fig. 2), L x = 5 0 A,, L , . = 200 .A,, L = 100 ,4, (middle figure, dashed line in fig. 2). and L , = 50 A, L , = 200 ,~,, L = 50 ,~ (lower figure, dashed-dotted line in fig. 2). _
o
444
G.E.W. Bauer, H. Sakaki / Polarization anisotropies it+ QW's
F
= 2.5 m e V
3
°~
~
~ ',
O
"'" e-
1
:, " :l
t-I,....
0
..~
.
.
I
',
".
t
' " ~. ~ .
.
." , .
,.
".,
• .
o~
-.-.:.,_ o-
L.
.1
7t :
0
0
sarily involve quantum-wire formation has recently been obtained by Dawson and Wentink [10]. A maximum in the anisotropy of MEE-grown structures as a function of the tilt angle of the substrate, and thus the period of the grid, as well as a finite anisotropy for quantum wells grown on untilted substrates is reported [10], whicn confirms the predicted trends.
, l
-2
L, 50
100
150
Note added in proof
Energy (meV) Fig. 2. Interband transition spectra of 100 .~ GaAs/AIAs quantum wells with an insertion of half a monolayer of AlAs for a broadening parameter F = 2.5 meV. The full curve is the result for unpolarized light and the other curves are the difference spectra ( x 4 ) for polarizations parallel to the xand y-directi'.ms. Fhe correlation functions of the scattering potential causing the anisotrapies are plotted in fig. 1.
taking the valence band structure into account by the Luttinger Hamiltonian. We treat higher-order scatterings by a phenomenological broadening constant and neglect exciton effects. Input to the theory are the autocorrelation functions of the island structures. The results for the optical anisotropies using the model-autocorrelation functions plotted in fig. 1 are displayed in fig. 2 for a 100 A wide quantum well. We see that optica w anisotropies which agree quite well with the experimental evidence [2] can be obtained without having to assume any superlattice or quantum-wire formation at all. We can draw the following conclusions: (1) Optical anisotropies as reported in refs. [1,2] are a necessary but not a sufficient condition for quantum-wire or lateral superlattice formation, even when strain effects can be excluded. (2) Short-range correlations between anisotropic islands can enhance the anisotropies, but only when the periodicity of thc "local" grid is chosen properly. The anisotropies in a given quantum well are large when the wave vector 7r/L hits the anticrossing between heavyhole and light-hole bands. Clear experimental evidence that optical anisotropies do not neces-
The optical anisotropies in (110) quantum wells predicted in section 2 have been observed by D. Gershoni, I. Brener, G.A. Baroff, S.N.G. Chu, L.N. Pfeiffer and K. West, Phys. Rev. B 44 (1991) 1939, and by Y. Kajikawa, M. Hata, T. Isu and Y. Katayama, Surf. Sci. 267 (1992) 501, these proceedings.
References [1] M. Tsuchiya, J.M. Gaines, R.J. Simes, P.O. Hoitz, L.A. Coldren and P.M. Petroff, Phys. Rev. Lett. 62 (1989) 466. [2] M. Tanaka and H. Sakaki, Appl. Phys. Lctt. 54 (1989) 1326. [3] K. Kash, J.M. Worlock, A.S. Gozdz, B.P. van d,: Gaag, J.P. Harbison, P.S.D. Lin and L.T. Florez. Surf. Sci. 229 (1990) 245. [4] J.C.M. Henning, Y.A.R.R. Kessener, P.M. Koenraad, M.R. Leys, W. van der Vleuten, J.H. Wolter, A.M. Frens and J. Schmidt, Semicond. Sci. Technol., to be published. [5] G.E.W. Bauer, in: Spectroscopy of Semiconductor Microstructures, eds. G. Fasol, A. Fasolino and P. Lugli (Plenum, New York, 1989). [6] L. Vifia, G.E.W. Bauer, M. Potemski, J.C. Maan, E.E. Mendez and W.I. Wang, Phys. Rev. B 41 (1990) 10767; Y. limura, Y. Segawa, G.E.W. Bauer, M.M. Lin, Y. Aoyagi and S. Namba, Phys. Rev. B 42 (1990) 1478. [7] P.M. Petroff, A.C. Gossard and W. Wiegmann, Appl. Phys. Lett. 45 (1984) 1071. [8] T. Noda, J. Motohisa and H. Sakaki, Surf. Sci. 267 (i992) p. i87; J. Motohisa and H. Sakaki, unpubli,:hed. [9] G.E.W. Bauer and H. Sakaki, Phys. Rcv. B 44 (1991) 5562. [10] P. Dawson and D.J. Wentink, private communication.
surface science
Polarization anisotropies in quantum wells Gerrit E.W. B a u e r 1 Philips Research Laboratories, 5600 JA Eindhocen, Netherlands
and Hiroyuki Sakaki Research Center for AdL,anced Science and Technology, Unicersity of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153, Japan Received 1 June 1991; accepted for publication 29 June 1991
Three mechanisms are discussed which can cause a dependence of t,~c optical absorption on the direction of the polarization vector in the plane of a quantum well (QW), i.e. the warping distortio~a, strain, and anisotropic scattering potentials. Measured optical anisotropies do not constitute proof of quantum-wire or lateral superlattice formation.
1. Introduction
Perfect quantum wells of tetrahedral semiconductors grown on (001)-oriented substrates belong to the symmetry group D2d (D4h in effective mass theory). If the growth direction is chosen parallel to the z-axis, x and y form a basis of its irrcducible representation E(E,). Therefore, the optical matrix elements for linearily polarized li, ht with wave vector normal to the interfaces do not depend on the field vector. However, when ~;-,e four-fold symmetry is broken intrinsically or by external perturbations, the optical properties become anisotropic with respect to a rotation of the polarization vector in the plane of the interfaces. These optical anisotropies have recently been observed in quantum-wire arrays [1] and grid-inserted quantum wells [2] and were interpreted as evidence of lateral quantum confinement of the carriers. Anisotropies have also beer: measured in strain-patterned quantum wells [3] Part of this work was carried out at the NTT Telecommunications Chair at RCAST, University of Tokyo, ~1-6-i Komaba, Meguro-ku, Tokyo 153, Japan.
and 6-doped GaAs [4]. Here we discuss theoretically the effects of some symmetry breaking perturbations on the optical anisotropies.
2. (110) quantum wells
In quantum wells grown on a (ll0)-oriented substrate the four-fold symmetry axis is broken. In the effective mass approximation the tetragonal distortion is described by the warping term of the Luttinger Hamiltonian, which has been worked out for quantum wells in ref. [5]. The mixing of the heavy-hole and light-hole states by the nondiagonal elements modifies the oscillator strengths anisotropically [1,2]. The order of magnitude of the polarization can be estimated "back-of-the-envelope" by assuming infinite barrier n n t e n t ; n l ~ nncl or'~nciAor;nn t~nl~, ~ho ~+~+,~,- .~ the band edge (k X= k~ = 0). We obtain for the relative modulation of the oscillator strengths for linearily polarized light: [ A l [ / i = [ Ayl/'(2~/) to first order in [Ay[/~, where Ay and ~/ are the difference and the average of the Luttinger parameters )'2 and )'3, respectively. The maximum
()()39-6()28/92/$05.00 !i~ 1992 - Elsevier Science Publishers B.V. and Yamada Science Foundation. All rights reserved
G.E. IV. Bauer, 14. Sakaki / Polar&ation anisotropies in QW's
of intensity of the heavy-hole absorption is achieved when the polarization vector is parallel to the zig-zag bonding chains, which are seen to cause effects similar to quantum wires in the same direction. Polarization spectra are very sensitive to perturbations of the four-fold symmetry. Independent of the well width, the anisotropy amounts to --17 percent for GaAs parameters (T2 = 1.9, T3 = 2.7). Experimental anisotropies in ( l l 0 ) - q u a n t u m wells would be direct evidence of the warping distortion, which continues to elude optical detection in (001)-quantum wells [6].
3. Uniaxial pressure Kash et al. [3] experimentally and theoretically demonstrated optical anisotropies in quantum wells in which the symmetry was broken by lateral strain patterning. Following ref. [3], it is easy to derive the lowest-order effect of a uniaxial pressure P applied to a [001] "hard-wall" quantum well of width d. For the heavy holes we obtain that w h e n P II[ll0]: (I I . ) / [ = 2m o S a a D u , d 2 p / ( 3 h 2 y 2 7 r 2 ) = 0.3 ((J/kbar)(d/lO0 i ) 2 and when PII[100]: (Iit-I ± ) / i = 2moD~(S ~ -
443
4. Anisotropk scattering
The epitaxial growth-controlled fabrication of lateral superlattices and quantum wires on tilted surfaces [7] is an important alternative to lithographic techniques. Recently, anisotropies in the mobilities of a heterostructure with an AlAs grid inserted close to the 2d-electron gas has been interpreted in terms of scattering from AlAs island structures in the inserted sub-monolayer, which are characterized by some degree of shortrange correlations [8]. We have investigated the consequences of such a model for the optical anisotropies [9]. Briefly~ we calculate the configurational average of corrections to the photon self-energy in second-order perturbation theory,
Autocorrelation function 3oo
1oo --
rii/~!i I I
',
-1oo
-
Sl2)
d 2 p / / ( 3 h 2 T 2 " n ' 2 ) --- 0.2(P/kbar)(d/lO0
i -300
A ) 2,
where I± and Ill denote the oscillator strength for field vectors normal and parallel to the pressure direction. Du( = 2.55 eV), D~,( = 3.94 eV) arc deformation potentials and S ~ ( = 1.15 × 10 -" bar-l), St2(= -0.35 × 10 -6 b a r - l ) , and $44( = 1.66 x 10 -6 bar -~) are the elastic constants of the semiconductor (GaAs). For the light holes the above results should be multiplied by minus one. We see that relatively small pressures of the order of 1 kbar can cause large anisotropies. In ideal grid-inserted quantum wells on a GaAs substrate all strain from the A I A s / G a A s lattice mismatch of 0.2% is accommodated by the AlAs and optical effects should be very small. The o~t~rnnl pre~,,re w h l c ' h w n , ~ i d h a r p t n h o ~ n n l l e c l to cause the same strain in bulk AlAs is about 5 kbar, however, it should therefore be kept in mind that in imperfect structures observable optical anisotropies could be caused by only a small transfer of the strain from the AlAs to the GaAs strips.
100
iilii '
!,Iji! , -100 v
-300
A 100 -
-100
I!!//
J
\f ,'
-300 -300
-100
100
300
x (A) Fig. I. C o n t o u r plots o f the a u t o c o r r e l a t i o n f u n c t i o n s of Al/ks islands as used foi i h c xcsulis in fig. 2, which a~c pi.tlialiicici-
ized as C(x, y ) = e x p { - ( x / 1 , ) 2-(y/I,)e}C(L). where I , / , are Gaussian decay lengths and C(L) is the autocorrclation of a perfect grid withoPeriod L. Fromo above, the figures are obtained for L x = 50 A, L,.= 200 A, L = zc (upper figure. dotted line in fig. 2), L x = 5 0 A,, L , . = 200 .A,, L = 100 ,4, (middle figure, dashed line in fig. 2). and L , = 50 A, L , = 200 ,~,, L = 50 ,~ (lower figure, dashed-dotted line in fig. 2). _
o
444
G.E.W. Bauer, H. Sakaki / Polarization anisotropies it+ QW's
F
= 2.5 m e V
3
°~
~
~ ',
O
"'" e-
1
:, " :l
t-I,....
0
..~
.
.
I
',
".
t
' " ~. ~ .
.
." , .
,.
".,
• .
o~
-.-.:.,_ o-
L.
.1
7t :
0
0
sarily involve quantum-wire formation has recently been obtained by Dawson and Wentink [10]. A maximum in the anisotropy of MEE-grown structures as a function of the tilt angle of the substrate, and thus the period of the grid, as well as a finite anisotropy for quantum wells grown on untilted substrates is reported [10], whicn confirms the predicted trends.
, l
-2
L, 50
100
150
Note added in proof
Energy (meV) Fig. 2. Interband transition spectra of 100 .~ GaAs/AIAs quantum wells with an insertion of half a monolayer of AlAs for a broadening parameter F = 2.5 meV. The full curve is the result for unpolarized light and the other curves are the difference spectra ( x 4 ) for polarizations parallel to the xand y-directi'.ms. Fhe correlation functions of the scattering potential causing the anisotrapies are plotted in fig. 1.
taking the valence band structure into account by the Luttinger Hamiltonian. We treat higher-order scatterings by a phenomenological broadening constant and neglect exciton effects. Input to the theory are the autocorrelation functions of the island structures. The results for the optical anisotropies using the model-autocorrelation functions plotted in fig. 1 are displayed in fig. 2 for a 100 A wide quantum well. We see that optica w anisotropies which agree quite well with the experimental evidence [2] can be obtained without having to assume any superlattice or quantum-wire formation at all. We can draw the following conclusions: (1) Optical anisotropies as reported in refs. [1,2] are a necessary but not a sufficient condition for quantum-wire or lateral superlattice formation, even when strain effects can be excluded. (2) Short-range correlations between anisotropic islands can enhance the anisotropies, but only when the periodicity of thc "local" grid is chosen properly. The anisotropies in a given quantum well are large when the wave vector 7r/L hits the anticrossing between heavyhole and light-hole bands. Clear experimental evidence that optical anisotropies do not neces-
The optical anisotropies in (110) quantum wells predicted in section 2 have been observed by D. Gershoni, I. Brener, G.A. Baroff, S.N.G. Chu, L.N. Pfeiffer and K. West, Phys. Rev. B 44 (1991) 1939, and by Y. Kajikawa, M. Hata, T. Isu and Y. Katayama, Surf. Sci. 267 (1992) 501, these proceedings.
References [1] M. Tsuchiya, J.M. Gaines, R.J. Simes, P.O. Hoitz, L.A. Coldren and P.M. Petroff, Phys. Rev. Lett. 62 (1989) 466. [2] M. Tanaka and H. Sakaki, Appl. Phys. Lctt. 54 (1989) 1326. [3] K. Kash, J.M. Worlock, A.S. Gozdz, B.P. van d,: Gaag, J.P. Harbison, P.S.D. Lin and L.T. Florez. Surf. Sci. 229 (1990) 245. [4] J.C.M. Henning, Y.A.R.R. Kessener, P.M. Koenraad, M.R. Leys, W. van der Vleuten, J.H. Wolter, A.M. Frens and J. Schmidt, Semicond. Sci. Technol., to be published. [5] G.E.W. Bauer, in: Spectroscopy of Semiconductor Microstructures, eds. G. Fasol, A. Fasolino and P. Lugli (Plenum, New York, 1989). [6] L. Vifia, G.E.W. Bauer, M. Potemski, J.C. Maan, E.E. Mendez and W.I. Wang, Phys. Rev. B 41 (1990) 10767; Y. limura, Y. Segawa, G.E.W. Bauer, M.M. Lin, Y. Aoyagi and S. Namba, Phys. Rev. B 42 (1990) 1478. [7] P.M. Petroff, A.C. Gossard and W. Wiegmann, Appl. Phys. Lett. 45 (1984) 1071. [8] T. Noda, J. Motohisa and H. Sakaki, Surf. Sci. 267 (i992) p. i87; J. Motohisa and H. Sakaki, unpubli,:hed. [9] G.E.W. Bauer and H. Sakaki, Phys. Rcv. B 44 (1991) 5562. [10] P. Dawson and D.J. Wentink, private communication.