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Space group symmetry of the conduction band minimum in very short period GaAsAlAs superlattices
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Solid State Communications, eel. 82, No. 12, PP. 951-954, 1992. Printed in Great Britain.
0038-1098/9255.00+.00 Pergamon Press Ltd
SPACE GROUP SYMMETRY OF THE CONDUCTION BAND MINIMUM IN VERY SHORT PERIOD GaAs/AIAs SUPERLATTICES Weikun Ge, W.D.Schmidt, M.D.Sturge, Department of Physics. Dartmouth College, Hanover, NH 03755-3528 L. N. Pfeiffer an.d K. W. West AT & T Bell Laboratories. Murray Hill, NJ 07974 (Received February 28, 1992 by A. Pinczuk)
In a perfect (GaAs)m(AlAs)n type-II superiattice the symmetry of the conduction band states,and therefore the band mixing, depends on whether rn, n or (m+n) are odd or even. W e have investigated the symmetry of the conduction band minima in very shortperiod suporlattices (m < n _<4) by time resolved photolumincscence under applied uniaxial stress,and find that selection rules that depend on space group symmetry, through kconservation are well obeyed, but parity with respect to space inversion is apparently not a good quantum number in our samples.
minband rain/mum and valence mmiband max/mum. The symmetry of the conduction band states in (00 I) SLs has been exhaustively studied theoretically5,g. For even (re+n) the SL Bravals latticeis primitive tetragonal, while for odd (re+n) itis body-centered tewagonal. W e distinguish the symmetry points in the SL mini BZ from those of the bulk BZ by a bar beneath each letter9, such as I'. ~L etc, and write X x , X v , Xz for the bulk X minima with k directed along the x.y'and z directions respectively. • )r the purposes of this paper we take the CBM of bulk AIAs to be at X, and ignore any possible "camel's back" in the band structure at the X-point I0. Then if (re+n) is even. Xz folds to __F,Xx,y to IV[;while if m + n is odd. X z folds to 7., X x to ~, Xy to Y. I" always folds to F. We ignore states deriving from L since they have been shown not to give the CBM in any of our SLs f I. For any value of m or n, the VBM is the heavy hole Kramers doublet. Therefore our discussion mainly concerns the CBM. The physical meaning of. for example, the folding of X z into l" for even (re+n) is that the SL potential VsL has an [001] component which mixes the bulk X z state with the bulk r state. The transition from X z in the conduction band to F in the valence band is then pseudo-direct, and a nophonon (N'P) transition is possible in absorption and emission. For (m+n) odd. on the other hand. VSL has no [001] component, and any no-phonon line must be due to deviations from ideal symmetry. Besides theCBM's position in the mini BZ, its parity. with respect to z-inversion is also relevant, since in a perfect SL only states of the same parity can mix. The interface between AIAs and GaAs is a common As layer, so for even n the central atomic layer of the AlAs is made up of As atoms, while for odd n it is a layer of AI atoms. Since in the bulk the X wave funcdon has a node at the AI site, the parity of the CBM is determined8 by the parity (odd or even) of n. The parity, of the lowest (kz=0) state deriving from F is always even. Thus, if parity is a good quantum number, X and F cannot mix if n is odd, and the N'P transition should be forbidden, whatever m. Furthermore, for a perfect SL with m = n. only odd harmonics exist in its Fourier expansion of VSL. Thus in the case of m = n = even, VSL has no [001 ] component and
It is now established experimentallyI'4 and theoretically5 that short-period (GaAs)m(AlAS)n superlattices (SL) with m < n < 11 are type-II I-4, i.e. electrons and holes are separately confined at the X conduction band minimum (CBM) of AlAs and the F valence band maximum (VBM) of GaAs respectively. However, the precise symmetry. (i.e. the space group telxesentation and parity) of the lowest electron states has not been determined experimentally. "Parity" in this paper refers to parity under z-inversion (z is the growth direction, perpendicular to the layers) about the center of the AlAs layer (since we arc interested in stateswhich derive from the CBM of bulk AlAs). "Space group representation" refers to the space group of the SL: in particular, it determines whether the CBM (which derives from the X minimum of bulk AlAs) is folded back to the center of the Brillouin zone (BZ) of the SL or to some other point. In a perfect SL the sv ~ e t r y of the X electron is determined (a) by whether the AIAs layer thickness n, measured in units of one monolayer (2.83A), is even or odd. and (b) by whether the SL period (m+n) is even or odd. However, if (m+n) is too large, a symmetry analysis which depends on the precise value of m and n ceases to be meaningful. This is not only because fluctuations in growth ram lead to uncertainties in layer width which are roughly proportional to the width. More seriously, surface electric fields and other parity breaking perturbations tend not only to mix st~t~ of different parity but to localize the exciton on one A1As/GaAs interface. The superla~ce potential VSL seen by the exciton, which determines its symmetry and band mixing, is then effectively a step function, and any symmetry dependent on the precise value of m and n is necessarily broken. This locali,~tion has been demonstrated very. clearly for (n+m)~30 by electric field experimems6,7. which show that in these SI.,s the exciton has a dipole moment even in the absence of an externally applied field. We therefore confine our attention to short-period superlattices (SPSL's), which we define for our present purposes as SL's with periods (re+n) < g. In short-period SL's neither electrons nor holes ate confined, and minibands are formed. The symmetry discussed hem is that of the lowest exciton, which is the one that is observed in photolummescence (PL) at low temperature. This ¢xciton is associated with the conduction 951
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SPACE GROUP SYMMETRY
there can be no mixing between F and X to lowest order in perturbation theory. Symmetry and parity,considerations refer to the possible effects of VSL in a perfect SL. However, in an imperfect SL, the interface disorder potential Vd~, can also mix states of different symmeu7 and make the N P transition allowed. As we shall see, Vdi~ is an order of magnitude weaker than VSL, but its effects arc nev~cless observable. Impurity potentials and the electron (exciton)-phonon interaction also mix different bands, but they produce transitions shifted from the NP line. Throughout the following, a (GaAs)m(A1As)n SL is designated nVn. Seven samples were used for this study: SI..swith m/n = III, 2/2, 3•3, 4•4, i12, and 2/3, and a AI.sGa.~As alloy. The above symmetry, analysis tell us that: (1) For 2/2 and 4/4. Vst" mixes X z with F but not in lowest order. (2) For 1/1 and 3/3, VSL mixes X z with F in lowest order, but F - X mixing is forbidden by parity. (3) For 1/2 and 2/3, and for the alloy, mixing between X and F by VSL is forbidden by k-conservation, ff the random potential due to disorder is neglected. To testthese predictions, we have studied the P L decay ~ pulse excitation of the N P line of these SLs, using unlaxial stress to re-order the C B M states4, and thus fwA the relative oscillatorsmmgths of the transitions to the ground state. To see ff the predicted effects of symmetry in fact occur we have made the following measurements at 1.8K, for each of the samples listedabove: (I) The position of the N P line as a function of unlaxial stress.Xz moves to lower energy with increasing stress along [001] (the growth direction), while Xx,y moves to higher energy a. The reverse is true/'or[i I0] stressI 1. (2) The intensity ratio in PL of the NP line to the phonon sideband (PS) for the Xz and Xx,y transition, applying stress when necessary to bring the required state lowest. In longer period Type H AIAs/GaAs SL's (m+n ~ 20) ODMR2,12 has been used to distinguish the Xz CBM from the Xx,y by the symmetry of the g-tensor, and the O D M R resulis show that the NP/PS ratio is a reliable guide to the nature of the C B M in these SL's. [Note that for the reasons given above, at such large periods the effect of symmetry on oscillatorstrengths, with which this paper deals, do not occur. The Xz exciton is always pseudo-direct if(re+n) is large and n is not much larger than m]. A strong line for (re+n) even shows that the C B M is Xz , which is mixed with F by VSL in thiscase, while a relatively weak N P line indicates Xx,y, which is not mixed by VSL. However, we have found that the radiative rate of the PS depends somewhat on the state involved, being less for X z than for Xx,y, and also tends to decrease with increasing stress (we donot at present have an explanation for this). The NP/PS intensity ratio is thus only a qualitative measure of the N P ~aiative rate and hence of the strength of the F-X mixing. (3) The fluorescent decay curves after pulse excitation of the Xz and Xx,y NP lines. When allowance has been made for non-radiative decay, these give the r'ad!~tive rams and thus a quantitative measure of the F-X mixing. Figure 1 shows the effect of stress on the position of the NP line in four samples: the stress is applied along [110] for the 4/4 sample and along [001] for the others. The lineshffts show the changeover from the Xx,y state at zero stress ( Xz for the 4/4 sample) to Xz state at high stress (Xx,y for the 4/4 sample). Figure 2 shows the PL spectra in zero stress of the 4/4, I/l,and 2/3 samples (fulllines).The PL specwa of 3/3
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2.02 2.0C J, '
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' io','2
Stress(kbar)
Fig. 1 Position of the NP line as a function of uniaxial stress ( [110] stress for the 4/4 SL, [001] stress for the others): (a) 2/3 Co) alloy (c)3/3 (d) 4/4.
and 2/2 are similar to that of 1/1 and the spectrum of 1/2 is similar to that of 2/3. The strong ~ line of 4/4 at zero stress shows that the CBM in this case is Xz, while in the spectra of the other SL the relatively weak NP line indicates Xx,y. The dashed lines in Figure 2 show the PL specu'a under sufficiem uniaxial stress to reverse the order of the Xz and Xx,y states. Also shown is the spectrum of the AI.sGa.sAs alloy sample under [001] stress. In the 4/4 case, [ 110] stress brings the Xx,y state lowest, so that the NP line is greatly weakened. Similarly for 1/1, [001] su'ess brings Xz lowest, and the NP line is dominant. In the 1/2 and 2/3 SLs and in the alloy, [001] stress brings Xz lowest (see Fig. i). While the NP/PS ratio is greater than for Xx,y, the change is much less than for the even period SLs, and the strong resemblance between the spectra of the odd period SLs and that of the alloy suggests that the NP line is made allowed by the disorder potential Vaa, rather than by VSL. In all our samples, for a given state (Xz or Xx,y ), we fred an overall reduction in radiative efficiency with increasing stress, which we do not at present understand: this reduction is much more pronounced in the I/2 and 2/3 samples than in the others, and the PS intensity fallsoff faster than the NP. While this reduction in efficiency rules out quantitative inmrpretation of the intensities, the ratio of the Xz NP line to the Xx,y NP line gives a qualitative measure of the Xz - F mixing. For (re+n) odd, this ratio is about unity., as it is in the alloy, while for (re+n) even, the ratio is in the range 20 to 30, implying strong Xz - F mixing. On the other hand, we find no significant difference between the 1/1 and 3/3 (odd panty) and the 2/2 and 4/4 (even parity.) samples. The time decay was measured under the same conditions as the PL spectra. At zero stress, where the low energy tail of the Xz state 15-20 meV above the CBM sometimes overlaps the Xx,y emission, the data are subject to a small error due to this. It has been shownl, 13,14, and we find here, that when Xz is the CBM the decay of the NP line is given by I(t) = I(O~xp(-wot)(l~-2wt)':~
(i)
where w0 is the decay rate for phonon-assisted emission and w is the mean decay rate, This equation, originally developed to account for disorder-induced decay in random
SPACE GROUP SYMMETRY
Vol. 82, No. 12 •
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Fig. 2. CW PL spectra at 1.SK, excited at 514.5nm: (a) 4/4 at zero stress; (b) 4/4 under [ 110] stress of 5.96 kbar. (c) 1/I at zero stress; (d) 1/1 under [001] stress of 1.43 kbar. (e) 2/3 at zero stress; (f) 2/3 under [001] stress of 4.03 kbar. (g) Alloy under [001] stress of 3 kbar.
Fig. 3. NP PL decay after pulsed excitation. (a) 4/4 at zero su'ess; (b) 4/4 under [ 1I(3] stress of 5.96 kbar. (c) 1/I at zero stress; (d) 1/I under [001] stress of 1.43 kbar, (e) 2/3 at zero stress; (f) 2/3 under [001] stress of 4.03 kbar.
alloys 15, is valid for Xz when Xz and F are mixed by VsL, which is itself random because of random fluctuations in the interface position 13.14. When Xx,y is the CBM, or when VsL-induced Xz-r" mixing is forbidden, the phonon-assisted term dominates and we might expect exponential decay. However, because of the low radiative rate, non-radiative processes are also imlxaxant. Tunneling to non-radiative centers introduces a rapid initial decay in I(t), so that the decay is multi-exponential. [This process is found to be faster for Xx,y than for Xz ]. In this case, the taft of the decay curve, which is exponential should represent the true radiative decay of those excitons which are far from nonradiative centers. While this intm-pretation of the decay curves is plausible, itis not necessary in order to interpret the diffctmw.¢ between the decay curves of different samples in mrms of theirsyrnmctty, so long as radiative processes make a significantcontribution to the decay. Figure 3 shows the smoothed decay curves for the 4/4, I/1 and 2/3 SLs at zero stress (fulllines)and under sufficient uniaxial stzess (dashed lines} to reverse the order of Xz and Xx,y • (Note the change of timescale for the 2/3 sample). For (re+n) even, the crossover from Xx,y to Xz is accompanied by a large change in the decay curve, but the change for (m+n) odd is much less, and in the opposite sense. The results for all our S i s arc summarized in Table I. Here, Wz is the decay rate ofXz, Wx,y that of Xx,y • For (re+n) even, wz = w in the fit to eq.l, while Wx,y, and Wz for (re+n) odd, arc obtained by fitting the slow tail of the decay to the usual exponential decay law I o~ exp(-wt). AE is; the energy separation between F and Xx,y 16, and the fourth row gives w* - 10"4AE2Wx,y, whose approximate
constancy shows that the symmetry-breaking mechanisms which permit radiative decay of the Xx,y exciton are almost independentof sample for m,n < 3.
In curves e and f, the timescale should be multiplied by a factor of 16.
Table I Decay rates of the Xz and Xx,y states for different superlattices m/n Wz (ms"I) Wx,y(rns"!)
4/4 313 1,000 780 180 200
~ 340 180
AE (nmV) w*
159 455
118 251
120 288
I/I 2/3 1,200 49+10 160 57__.10 121 234
203 235
I/2 15+_.5 20+_5 356 253
Table I shows that for even (m+n), the ratio Wz/Wx,y is in the range 1.9 to 7.5, showing that the superlatticc potential VSL is contributing substantially to Wz. For odd (re+n), on the other hand, the ratio is less than one (the deviation from unity is within experimental error), showing that there is no appreciable contribution from VSL, in agreement wiiii the prediction fi'om the space group
symmetry.
The fact that wz is less for 2/2 than it is for 1/1 and : 3/3 is in agreement with the expectation that when m -- n = even, tbete can be no r-Xz mixing to lowest order. On the other hand. Wz is large for 4/4, as it is for longer period , Sis 1. It is possible that for (re+n) = 8 the exciton tends to localize on one interface, as mentioned above; or that growth
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SPACE GROUP SYMMETRY
rate fluctuations are sufficient to wash Out the precise value of m and n. Parity conservation predicts that X-F mixing should he forbidden for mfnffiodd. As can be seen from Table I, our data show no sign of this, and we conclude that parity is not a good quantum number in these Sis. This could be a consequence of f'mldsind-,c_~by surface charge, and we are planning to make measurements in an appliedelccu'icfieldto check thispoint.
Vol. 82, No. 12
We conclude that these Sis are sufficiently perfect that the derailed predictions of theory. Concerningthe dependence of the symmetry on the precise layer thickness are well obeyed, with the exception of those which are a consequence of parityconsideranons. Acknowledgements - One of us (MDS) has benefited greatly from discussions with Dr L.J.Sham. This work is supported by DOE grant #DEFG 0287 ER45330.
REFERENCES
1. E. Finkman, M. D. Sturge, and M. C. Tamargo, Appl. Phys. Lett. 49, 1299 (1986); E. Finkman, M. D. Sturge, M-H. Meynadier, R. E. Nahory, M. C. Tamargo, D. M. Hwang, and C. C. Chang, J. Lumm. 39, 57 (1987). 2. H.W. van Kesteren, E. C. Cosman, P. Dawson, K. J. Moore, and C. T. Foxon, Phys. Rev. B39, 13426 (1989). 3. G. Li, D. Jiang, H. Han, Z. Wang, and K. Ploog, Phys. Rev. B40, 10430 (1989). 4. W. Ge, M. D. Sturge, W. D. Schmidt, L. N. Pfefffer, and K. W. West, AppL Phys. Lett 57, 55 (1990); M.D.Sturge, Weikun Ge, W.D.Schmidt, L. N. Pfeiffer and K.W. West, Proc. 20th ICPS, ed. E.M.Anastassakis and J.D.Joannopoulos (World Scientific, Singapore, 1990). p. 1029. 5. L.J. Sham and Y-T Lu, J. Lumin. 44, 207 (1989). 6. M-H Meynadier, R E Nahory, J.M.Worlock, M C Tamargo, J.L. de Miguel and M D Sturge, Phys. Rev. Lett. 60, 1338 (1988). 7. B.A.Wilson, R.C.Spitzer, C.E.Bonner and L.N.PfeiHer, Bull. Am. Phys. Soc. 34, 958 (1989).
8. Y-T Lu and L. J. Sham, Phys. Rev. B40, 5567 (1989); L. J. Sham and Y-T Lu, unpublished. 9. G.P. Srivastava, R. J. Gordon, and Z. Ikonic, Superlatticesand Microstructures,9, 43 (1991). i0. G.F.Glinski and A.A.Kopylov, Soy. Phys. SoL State 23, 1883 (1981). 1I. Weikun Ge, M. D. Sturge, W. D. Schmidt, L N. Pfefffer, and K. W. West, Phys. Rev. B44, 3432 (1991); J. Lumin. 48/49, 759 (1991). 12. H. W. yon Kcstcren, E. C. Cosman, W.J.A van der Poel, and C. T. Foxon, Phys. Rev. B39, 13426 (1989). 13. F. Minami, K. I-Iirata,R. Era, T. Yao, and Y. Masumoto, Phys. Rev. B36, 2875 (1987). 14. M.D.Sturge, Janet t Mackay, Colette Maloney and J.K.Pribram, J. Appl. Phys. 66, 5639 (1989) 15. M. V. Klein, M. D. Sturge, and E. Cohen, Phys. Rev. B2S, 4331(1982). 16. Weikun Ge, Janet L. Mackay, L. N. Pfeiffer, and K. W. West, J. Lumin. 50, 133 (1991); Weikun Ge, unpublished.
Solid State Communications, eel. 82, No. 12, PP. 951-954, 1992. Printed in Great Britain.
0038-1098/9255.00+.00 Pergamon Press Ltd
SPACE GROUP SYMMETRY OF THE CONDUCTION BAND MINIMUM IN VERY SHORT PERIOD GaAs/AIAs SUPERLATTICES Weikun Ge, W.D.Schmidt, M.D.Sturge, Department of Physics. Dartmouth College, Hanover, NH 03755-3528 L. N. Pfeiffer an.d K. W. West AT & T Bell Laboratories. Murray Hill, NJ 07974 (Received February 28, 1992 by A. Pinczuk)
In a perfect (GaAs)m(AlAs)n type-II superiattice the symmetry of the conduction band states,and therefore the band mixing, depends on whether rn, n or (m+n) are odd or even. W e have investigated the symmetry of the conduction band minima in very shortperiod suporlattices (m < n _<4) by time resolved photolumincscence under applied uniaxial stress,and find that selection rules that depend on space group symmetry, through kconservation are well obeyed, but parity with respect to space inversion is apparently not a good quantum number in our samples.
minband rain/mum and valence mmiband max/mum. The symmetry of the conduction band states in (00 I) SLs has been exhaustively studied theoretically5,g. For even (re+n) the SL Bravals latticeis primitive tetragonal, while for odd (re+n) itis body-centered tewagonal. W e distinguish the symmetry points in the SL mini BZ from those of the bulk BZ by a bar beneath each letter9, such as I'. ~L etc, and write X x , X v , Xz for the bulk X minima with k directed along the x.y'and z directions respectively. • )r the purposes of this paper we take the CBM of bulk AIAs to be at X, and ignore any possible "camel's back" in the band structure at the X-point I0. Then if (re+n) is even. Xz folds to __F,Xx,y to IV[;while if m + n is odd. X z folds to 7., X x to ~, Xy to Y. I" always folds to F. We ignore states deriving from L since they have been shown not to give the CBM in any of our SLs f I. For any value of m or n, the VBM is the heavy hole Kramers doublet. Therefore our discussion mainly concerns the CBM. The physical meaning of. for example, the folding of X z into l" for even (re+n) is that the SL potential VsL has an [001] component which mixes the bulk X z state with the bulk r state. The transition from X z in the conduction band to F in the valence band is then pseudo-direct, and a nophonon (N'P) transition is possible in absorption and emission. For (m+n) odd. on the other hand. VSL has no [001] component, and any no-phonon line must be due to deviations from ideal symmetry. Besides theCBM's position in the mini BZ, its parity. with respect to z-inversion is also relevant, since in a perfect SL only states of the same parity can mix. The interface between AIAs and GaAs is a common As layer, so for even n the central atomic layer of the AlAs is made up of As atoms, while for odd n it is a layer of AI atoms. Since in the bulk the X wave funcdon has a node at the AI site, the parity of the CBM is determined8 by the parity (odd or even) of n. The parity, of the lowest (kz=0) state deriving from F is always even. Thus, if parity is a good quantum number, X and F cannot mix if n is odd, and the N'P transition should be forbidden, whatever m. Furthermore, for a perfect SL with m = n. only odd harmonics exist in its Fourier expansion of VSL. Thus in the case of m = n = even, VSL has no [001 ] component and
It is now established experimentallyI'4 and theoretically5 that short-period (GaAs)m(AlAS)n superlattices (SL) with m < n < 11 are type-II I-4, i.e. electrons and holes are separately confined at the X conduction band minimum (CBM) of AlAs and the F valence band maximum (VBM) of GaAs respectively. However, the precise symmetry. (i.e. the space group telxesentation and parity) of the lowest electron states has not been determined experimentally. "Parity" in this paper refers to parity under z-inversion (z is the growth direction, perpendicular to the layers) about the center of the AlAs layer (since we arc interested in stateswhich derive from the CBM of bulk AlAs). "Space group representation" refers to the space group of the SL: in particular, it determines whether the CBM (which derives from the X minimum of bulk AlAs) is folded back to the center of the Brillouin zone (BZ) of the SL or to some other point. In a perfect SL the sv ~ e t r y of the X electron is determined (a) by whether the AIAs layer thickness n, measured in units of one monolayer (2.83A), is even or odd. and (b) by whether the SL period (m+n) is even or odd. However, if (m+n) is too large, a symmetry analysis which depends on the precise value of m and n ceases to be meaningful. This is not only because fluctuations in growth ram lead to uncertainties in layer width which are roughly proportional to the width. More seriously, surface electric fields and other parity breaking perturbations tend not only to mix st~t~ of different parity but to localize the exciton on one A1As/GaAs interface. The superla~ce potential VSL seen by the exciton, which determines its symmetry and band mixing, is then effectively a step function, and any symmetry dependent on the precise value of m and n is necessarily broken. This locali,~tion has been demonstrated very. clearly for (n+m)~30 by electric field experimems6,7. which show that in these SI.,s the exciton has a dipole moment even in the absence of an externally applied field. We therefore confine our attention to short-period superlattices (SPSL's), which we define for our present purposes as SL's with periods (re+n) < g. In short-period SL's neither electrons nor holes ate confined, and minibands are formed. The symmetry discussed hem is that of the lowest exciton, which is the one that is observed in photolummescence (PL) at low temperature. This ¢xciton is associated with the conduction 951
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there can be no mixing between F and X to lowest order in perturbation theory. Symmetry and parity,considerations refer to the possible effects of VSL in a perfect SL. However, in an imperfect SL, the interface disorder potential Vd~, can also mix states of different symmeu7 and make the N P transition allowed. As we shall see, Vdi~ is an order of magnitude weaker than VSL, but its effects arc nev~cless observable. Impurity potentials and the electron (exciton)-phonon interaction also mix different bands, but they produce transitions shifted from the NP line. Throughout the following, a (GaAs)m(A1As)n SL is designated nVn. Seven samples were used for this study: SI..swith m/n = III, 2/2, 3•3, 4•4, i12, and 2/3, and a AI.sGa.~As alloy. The above symmetry, analysis tell us that: (1) For 2/2 and 4/4. Vst" mixes X z with F but not in lowest order. (2) For 1/1 and 3/3, VSL mixes X z with F in lowest order, but F - X mixing is forbidden by parity. (3) For 1/2 and 2/3, and for the alloy, mixing between X and F by VSL is forbidden by k-conservation, ff the random potential due to disorder is neglected. To testthese predictions, we have studied the P L decay ~ pulse excitation of the N P line of these SLs, using unlaxial stress to re-order the C B M states4, and thus fwA the relative oscillatorsmmgths of the transitions to the ground state. To see ff the predicted effects of symmetry in fact occur we have made the following measurements at 1.8K, for each of the samples listedabove: (I) The position of the N P line as a function of unlaxial stress.Xz moves to lower energy with increasing stress along [001] (the growth direction), while Xx,y moves to higher energy a. The reverse is true/'or[i I0] stressI 1. (2) The intensity ratio in PL of the NP line to the phonon sideband (PS) for the Xz and Xx,y transition, applying stress when necessary to bring the required state lowest. In longer period Type H AIAs/GaAs SL's (m+n ~ 20) ODMR2,12 has been used to distinguish the Xz CBM from the Xx,y by the symmetry of the g-tensor, and the O D M R resulis show that the NP/PS ratio is a reliable guide to the nature of the C B M in these SL's. [Note that for the reasons given above, at such large periods the effect of symmetry on oscillatorstrengths, with which this paper deals, do not occur. The Xz exciton is always pseudo-direct if(re+n) is large and n is not much larger than m]. A strong line for (re+n) even shows that the C B M is Xz , which is mixed with F by VSL in thiscase, while a relatively weak N P line indicates Xx,y, which is not mixed by VSL. However, we have found that the radiative rate of the PS depends somewhat on the state involved, being less for X z than for Xx,y, and also tends to decrease with increasing stress (we donot at present have an explanation for this). The NP/PS intensity ratio is thus only a qualitative measure of the N P ~aiative rate and hence of the strength of the F-X mixing. (3) The fluorescent decay curves after pulse excitation of the Xz and Xx,y NP lines. When allowance has been made for non-radiative decay, these give the r'ad!~tive rams and thus a quantitative measure of the F-X mixing. Figure 1 shows the effect of stress on the position of the NP line in four samples: the stress is applied along [110] for the 4/4 sample and along [001] for the others. The lineshffts show the changeover from the Xx,y state at zero stress ( Xz for the 4/4 sample) to Xz state at high stress (Xx,y for the 4/4 sample). Figure 2 shows the PL spectra in zero stress of the 4/4, I/l,and 2/3 samples (fulllines).The PL specwa of 3/3
Vol. 82, No. 12
2.1( 2.01 ~2.06 2.04 c
2.02 2.0C J, '
'
' io','2
Stress(kbar)
Fig. 1 Position of the NP line as a function of uniaxial stress ( [110] stress for the 4/4 SL, [001] stress for the others): (a) 2/3 Co) alloy (c)3/3 (d) 4/4.
and 2/2 are similar to that of 1/1 and the spectrum of 1/2 is similar to that of 2/3. The strong ~ line of 4/4 at zero stress shows that the CBM in this case is Xz, while in the spectra of the other SL the relatively weak NP line indicates Xx,y. The dashed lines in Figure 2 show the PL specu'a under sufficiem uniaxial stress to reverse the order of the Xz and Xx,y states. Also shown is the spectrum of the AI.sGa.sAs alloy sample under [001] stress. In the 4/4 case, [ 110] stress brings the Xx,y state lowest, so that the NP line is greatly weakened. Similarly for 1/1, [001] su'ess brings Xz lowest, and the NP line is dominant. In the 1/2 and 2/3 SLs and in the alloy, [001] stress brings Xz lowest (see Fig. i). While the NP/PS ratio is greater than for Xx,y, the change is much less than for the even period SLs, and the strong resemblance between the spectra of the odd period SLs and that of the alloy suggests that the NP line is made allowed by the disorder potential Vaa, rather than by VSL. In all our samples, for a given state (Xz or Xx,y ), we fred an overall reduction in radiative efficiency with increasing stress, which we do not at present understand: this reduction is much more pronounced in the I/2 and 2/3 samples than in the others, and the PS intensity fallsoff faster than the NP. While this reduction in efficiency rules out quantitative inmrpretation of the intensities, the ratio of the Xz NP line to the Xx,y NP line gives a qualitative measure of the Xz - F mixing. For (re+n) odd, this ratio is about unity., as it is in the alloy, while for (re+n) even, the ratio is in the range 20 to 30, implying strong Xz - F mixing. On the other hand, we find no significant difference between the 1/1 and 3/3 (odd panty) and the 2/2 and 4/4 (even parity.) samples. The time decay was measured under the same conditions as the PL spectra. At zero stress, where the low energy tail of the Xz state 15-20 meV above the CBM sometimes overlaps the Xx,y emission, the data are subject to a small error due to this. It has been shownl, 13,14, and we find here, that when Xz is the CBM the decay of the NP line is given by I(t) = I(O~xp(-wot)(l~-2wt)':~
(i)
where w0 is the decay rate for phonon-assisted emission and w is the mean decay rate, This equation, originally developed to account for disorder-induced decay in random
SPACE GROUP SYMMETRY
Vol. 82, No. 12 •
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Fig. 2. CW PL spectra at 1.SK, excited at 514.5nm: (a) 4/4 at zero stress; (b) 4/4 under [ 110] stress of 5.96 kbar. (c) 1/I at zero stress; (d) 1/1 under [001] stress of 1.43 kbar. (e) 2/3 at zero stress; (f) 2/3 under [001] stress of 4.03 kbar. (g) Alloy under [001] stress of 3 kbar.
Fig. 3. NP PL decay after pulsed excitation. (a) 4/4 at zero su'ess; (b) 4/4 under [ 1I(3] stress of 5.96 kbar. (c) 1/I at zero stress; (d) 1/I under [001] stress of 1.43 kbar, (e) 2/3 at zero stress; (f) 2/3 under [001] stress of 4.03 kbar.
alloys 15, is valid for Xz when Xz and F are mixed by VsL, which is itself random because of random fluctuations in the interface position 13.14. When Xx,y is the CBM, or when VsL-induced Xz-r" mixing is forbidden, the phonon-assisted term dominates and we might expect exponential decay. However, because of the low radiative rate, non-radiative processes are also imlxaxant. Tunneling to non-radiative centers introduces a rapid initial decay in I(t), so that the decay is multi-exponential. [This process is found to be faster for Xx,y than for Xz ]. In this case, the taft of the decay curve, which is exponential should represent the true radiative decay of those excitons which are far from nonradiative centers. While this intm-pretation of the decay curves is plausible, itis not necessary in order to interpret the diffctmw.¢ between the decay curves of different samples in mrms of theirsyrnmctty, so long as radiative processes make a significantcontribution to the decay. Figure 3 shows the smoothed decay curves for the 4/4, I/1 and 2/3 SLs at zero stress (fulllines)and under sufficient uniaxial stzess (dashed lines} to reverse the order of Xz and Xx,y • (Note the change of timescale for the 2/3 sample). For (re+n) even, the crossover from Xx,y to Xz is accompanied by a large change in the decay curve, but the change for (m+n) odd is much less, and in the opposite sense. The results for all our S i s arc summarized in Table I. Here, Wz is the decay rate ofXz, Wx,y that of Xx,y • For (re+n) even, wz = w in the fit to eq.l, while Wx,y, and Wz for (re+n) odd, arc obtained by fitting the slow tail of the decay to the usual exponential decay law I o~ exp(-wt). AE is; the energy separation between F and Xx,y 16, and the fourth row gives w* - 10"4AE2Wx,y, whose approximate
constancy shows that the symmetry-breaking mechanisms which permit radiative decay of the Xx,y exciton are almost independentof sample for m,n < 3.
In curves e and f, the timescale should be multiplied by a factor of 16.
Table I Decay rates of the Xz and Xx,y states for different superlattices m/n Wz (ms"I) Wx,y(rns"!)
4/4 313 1,000 780 180 200
~ 340 180
AE (nmV) w*
159 455
118 251
120 288
I/I 2/3 1,200 49+10 160 57__.10 121 234
203 235
I/2 15+_.5 20+_5 356 253
Table I shows that for even (m+n), the ratio Wz/Wx,y is in the range 1.9 to 7.5, showing that the superlatticc potential VSL is contributing substantially to Wz. For odd (re+n), on the other hand, the ratio is less than one (the deviation from unity is within experimental error), showing that there is no appreciable contribution from VSL, in agreement wiiii the prediction fi'om the space group
symmetry.
The fact that wz is less for 2/2 than it is for 1/1 and : 3/3 is in agreement with the expectation that when m -- n = even, tbete can be no r-Xz mixing to lowest order. On the other hand. Wz is large for 4/4, as it is for longer period , Sis 1. It is possible that for (re+n) = 8 the exciton tends to localize on one interface, as mentioned above; or that growth
954
SPACE GROUP SYMMETRY
rate fluctuations are sufficient to wash Out the precise value of m and n. Parity conservation predicts that X-F mixing should he forbidden for mfnffiodd. As can be seen from Table I, our data show no sign of this, and we conclude that parity is not a good quantum number in these Sis. This could be a consequence of f'mldsind-,c_~by surface charge, and we are planning to make measurements in an appliedelccu'icfieldto check thispoint.
Vol. 82, No. 12
We conclude that these Sis are sufficiently perfect that the derailed predictions of theory. Concerningthe dependence of the symmetry on the precise layer thickness are well obeyed, with the exception of those which are a consequence of parityconsideranons. Acknowledgements - One of us (MDS) has benefited greatly from discussions with Dr L.J.Sham. This work is supported by DOE grant #DEFG 0287 ER45330.
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