Electronic states in GaAs-AlAs short-period superlattices: energy levels and symmetry

JOURNALOF

~-

LUMINESCENCE

--

ELSEVIER

Journal of Luminescence

59 (1994) 163—184

Electronic states in GaAs—AlAs short-period superlattices: energy levels and symmetry Weikun Gea,

W.D. Schmidti~1, M.D.

~,

Sturgea, L.N. Pfeiffer”, K.W. Westb

Department of Physics, Dartmouth College, Hanover, NH 03755-3528, USA AT&T Bell Laboratories, Murray Hill, NJ 07974, USA

(Received 2 June 1993; revised 20 August 1993; accepted 15 September 1993)

Abstract We have made a comprehensive photoluminescence (FL) study of the low lying conduction band states of (GaAs)m/(AlAs)~type-Il short-period superlattices (SL) with m, n ~ 4, in order to determine their energy and symmetry. The symmetry is found from the shift and splitting of the levels under uniaxial stress, and from the no-phonon oscillator strengths determined by time-resolved FL. Our samples are found to be true superlattices obeying the optical selection rules predicted by the space group symmetry, which determines whether a transition is indirect or pseudo-direct. However, selection rules depending on parity with respect to reflection in the growth plane are not obeyed. The results are compared with theoretical calculations from the literature. When strain due to lattice mismatch is taken into account, the ordering of the levels is found to agree with the most recent calculations, except for the case m = n = 1. In this SL the lowest conduction band state is found to derive from the bulk X valley, rather than from the L valley as predicted. We confirm that this discrepancy can be resolved if there is an ordered interchange of a certain fraction of the Ga and Al atoms, and our data support the theoretical prediction that such an ordered intermixed SL may in fact be more stable than either the perfect SL or the random alloy.

1. Introduction

the symmetry of the band edge states. These characteristics determine the optical and transport

Knowledge of the electronic structure of a semiconductor superlattice (SL) is fundamental to the understanding of the physics of the system and to the device applications of the material. Electronic structure here means the positions of the energy levels, particularly those near the band edges, the subbands, level crossing and mixing, and

properties of the materials. Advances in epitaxial growth techniques, such as molecular beam epitaxy (MBE) and metal-organic chemical vapor deposition (MOCVD), have permitted the growth of atomic-scale ultrathin-layered crystals, and very short-period superlattices (SPSL) have attracted much attention in the past few years. The most well-studied SPSL system is (GaAs)m/(AIAs)n, where m and n represent the num-

*

1

Corresponding author. Present address: Department of

Physics, Hong Kong University ofScience and Technology, Clear Water Bay, Kowloon, Hong Kong. Present address: American Optical Co., Precision Products Division, 14 Mechanic St, Southbridge, MA 01550, USA.

0022-2313/94/$07.00 © 1994

—

ber of monolayers in a GaAs layer and an AlAs .

layer respectively. Since the pioneer work of Dawson et al., Finkman et al., and Danan et al. in 1986 [1], it has been established that sufficiently

Elsevier Science B.V. All rights reserved

SSDI 0022-2313(93)E0223-K

W. Ge ci ul. Journal of Luminescence 59 (/994) 163 184

164

short period (GaAs)m/(AIAs)n SLs have an indirect energy gap, and that they are type-Il. This means that, whereas the valence band maximum (VBM) is at F and is predominantly located in the GaAs layer, the conduction band minimum (CBM) wave function is mainly derived from the bulk X states of AlAs. Furthermore, since the SL is uniaxial rather than cubic, the X valleys are no longer degenerate as they are in the bulk. We designate the valleys with their k vectors along [100], [010] and [001] (growth direction) as X~,X5. and X~respectively, According to the space group of the SL (which depends on whether m + n is odd or even) X. may or may not be folded in to the F point of the SL Brillouin zone (BZ), and X~and X~,may be mixed, and hence split, by the SL potential. We are particularly concerned here with the order of X~and X~5with the splitting, if any, between the X~and X~states; and with the space group symmetry of the states. The band edge alignment of a GaAs/AlAs superlattice is shown in Fig. 1.

AlAs

GaAs

AlAs

~

-

X~~

x~~/

—

—

-

— —

~z

X —

- - -

\ \

\ \

X~•

VO

Fig. 1. Energy band structure of a type-Il GaAs—AlAs shortperiod superlattice. The valence band offset for GaAs—AlAs hetero-structure is about 0.36—0.46 eV [66]. The energy position is not to scale in the figure.

This paper reports a detailed study of the energy levels and space group symmetry of the lowest conduction band states of the (GaAs)m/(AlAs)~ SPSL with m = n ~ 4 and (m + I) = n ~ 3. Hereinafter, when we use the terminology (GaAs)m/(AlAs)n SPSL, we mean that the numbers of rn and n are within the above mentioned range, unless specifically pointed out. The SL (GaAs)m/(AlAs)n will normally be abbreviated to rn/n. The reason that we confine our attention to such SPSL is that if(rn + n) is too large, a symmetry analysis which depends on the precise value of rn and n is no longer meaningful. This is not only because fluctuations in growth rate lead to uncertainties in layer width which are roughly proportional to the width. More seriously, surface electric fields and other parity breaking perturbations tend to mix states of different parity and to localize the exciton on the GaAs/AlAs interface. The SL 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 rn and n is necessarily broken. Such localization has been demonstrated very clearly for excitons in SLs with (m + n) 40: these show a linear Stark effect [2], proving that the exciton has a dipole moment even in the absence of an external electric field, and hence must be localized at an interface. In the SPSL which are the subject of the present study neither electrons nor holes are confined to single layers, and minibands are formed. The symmetry discussed here is that of the lowest exciton, which is the one that is observed in PL at low temperature. This exciton is associated with the conduction band minimum (CBM) and valence band maximum (VBM). Most of our results have been briefly reported before [3]: they are brought together in this paper in order to fill in detail and to give the complete picture. Information on the symmetry of the CBM is obtained from the shift and splitting of the exciton under uniaxial stress, from the intensity ratio between the no-phonon (NP) line and the phonon side bands (PS), and from the time dependence of the photoluminescence (PL) decay after pulse excitation. Fig. 2 shows the BZ of the [0011-oriented SL. (“mini BZ”) [4]. As the mini BZ is a result of folding from the bulk BZ, its shape is determined by the

W. Ge et a!.

/ Journal of Luminescence

w

x~

-

~_‘~

F (b) A

NZ

165

BZ. In the following, electronic states are referred to the symmetry points in the bulk BZ from which the SL states are derived, since the perturbation due to the SL potential is usually rather small and each minizone state retains most of the characteristics of its parent. The electronic structure of GaAs/AlAs SPSLs has been studied in the past by PL, photoluminescence excitation (PLE), photoreflectance, and ellipsometry, and various positions of the states denying from F and X have been reported [5—12].In the case of m = n, it has been established that rn = n = 11 is the critical number for the type-I to type-Il transition [13], and that the CBM derives from X~,in most cases [8] when rn = n ~ 11. To our knowledge, neither uniaxial stress experiments nor time-resolved PL have been made for SPSL, and the state symmetry has not been determined unambiguously. Some erroneous assignments of extrinsic lines in the PL of SPSL samples have been made, which are rectified in the present work. There has been much theoretical interest in

_____

zL.

59 (1994) 163—184

~

-

(c)

Fig. 2. (a) First Brillouin zone of the zincblende crystal (GaAs or AlAs). (b) Mini Brillouin zone for m + n = even SPSL. (c) Mini Brillouin zone for m + n = odd SPSL.

Table 1 Folding relationships between some symmetry points of the bulk BZ and mini-BZ (after Ref. [4]) m-t-n—~ in bulk BZ (parent)

even in mini BZ

odd in mini BZ

F X~

F F

F Z

L

R,

~

number of monolayers in each SL period. Fig. 2(b) is the simple tetragonal mini BZ for even (m + n), and Fig. 2(c) is the body-centered tetragonal mini BZ for odd (m + n). The symmetry points and lines in the mini BZ are marked by a bar to distinguish them from points in the face centered cubic BZ of the bulk crystal (Fig. 2(a)). Table 1 relates the symmetry points in the bulk BZ to those in the mini

SPSLs because of the breakdown of the effective mass approximation (EMA), and of the possibility of making “first principles” calculations of band structure. Ref. [4] provides a comprehensive review of the theory. Here we will only give some representative examples of the many different types of calculation which can be compared with our experimental results. Essentially there are two approaches to calculate the SPSL energy levels. One is a semi-empirical approach, based on the knowledge of the bulk band structure of the constituent compounds which make up the SL. The other is based, more or less, on first-principles. Within the former category, the most widely used technique is the EMA (e.g. the Kronig—Penney (KP) model), and its natural extension, the envelope function approximation (EFA) which is based on Kane’s kp model [14]. The success of the model, in which one necessarily confines oneself to a small region of the Brillouin zone, is perhaps surprising, since the short range potential at the interface might be expected to mix states of widely different k vectors. For this reason it cannot, for example, predict transition probabilities for pseudo-direct transitions, but as far as energy levels are concerned its shortcomings only become

166

W. Ge ci al.

Journal ot Luminescence 59 (1994) /63 184

apparent for SLs with periods less than 8 monolayers, i.e. the SPSL discussed here [15]. A more fundamental but still semi-empirical approach is the tight-binding method which was used by several authors including Ihm [16]. Ibm considered only the nearest neighbor interaction in his calculation, leading to an infinite transverse effec-

a self-consistent potential, usually within the local density approximation [21—24] for the actual SL configuration, thus taking care of the interface in a consistent manner. They are therefore more appropriate for SPSLs than bulk-based empirical methods. All these calculations, while differing in their precise results, agree that for in = n that X5~

tive mass rn1 in the X valley. This makes the lowest ~ conduction state identical to the bulk AlAs X state and lower than the X~state, which deviates significantly from the AlAs X level due to the strong confinement and band mixing effect. In a later calculation by Lu and Sham [17] the second nearest neighbor interaction is included. They find that in the absence of mismatch strain the X~level lies below the XX), levels, as predicted by the EFA: the case m = n = 1 is an apparent exception but the authors do not claim that their method is valid for such a short period. They correctly predict the transition of the energy gap from type-I to type-Il for the in = n SLs occurs at n = 12: this is also predicted by EFA calculations. However, these authors point out that their interface parameters are not fully self-consistent, so that their results for SPSLs may not be reliable. If atoms not actually at the interface are affected by its presence, the effective thickness of the interface increases and reduces the X~~—X~ separation. Disorder in the interface also increases the effective thickness of the interface, with the same effect. Calculations of rn/n SL energy levels tightwith 3S* empirical 2binding ~ (rn,n) ~ 22, based on an SP model and on the surface Green-function

is lower than X~for in = I to 3. but that R (derived from L: see Table I) is lower still in the in = I case. The discrepancy between this prediction and cxperiment is discussed in Section 5, which deals with the vexed question of the nature of the CBM in the case in = ii = 1. The breakdown of the EMA is mainly due to the fact that in an SL the potential varies rapidly over a distance much less than the scale of the envelope functions of electrons, holes, or excitons. When we compare our experimental results with the theoretical calculations, it is important to remember that the real samples are not perfect, as the theory usually assumes. For example, recent work has shown that the interface plays a role beyond just providing a potential step. In particular, disorder at the interface provides a mixing potential between states at different points in the BZ: for example, it makes the forbidden X, ~—F transition weakly allowed. On the other hand, as we shall see, this mixing due to disorder is small relative to that due to the SL potential. Interface disorder can also affect the ordering of different electron states [25,26], as will be shown later. In this paper we prove that for 1/1, 2/2, 3/3, 1/2 and 2/3 SPSLs, X 11 is the CBM, but that X~ is the CBM for in = n ~ 4. Section 2 describes the experimental methods. In Section 3. we discuss the symmetries of the conduction band states, how one can determine them experimentally, and the relevant experimental results. Energy levels are discussed in Section 4. Section 5 deals with the electron states in the specially interesting case 1/1, showing the discrepancy between the experimental observations and the theoretical calculations for a perfect SL, and how this discrepancy can be explained in a way consistent with the observed selection rules if one assumes, as Laks and Zunger suggested [26], some ordered intermixing at the interfaces. Section 6 summarizes the paper.

matching (SGFM) method, were made by Munoz et al. [18]. Their results agree well with our experimental result for rn/rn SLs, that the cross-over from X.. to X~. CBM occurs between m = 4 and 3. On the other hand, Fujimoto et al. [19] using the same tight-binding technique, concluded that the CBM for in = 4 is ~ The third semi-empirical approach is the pseudopotential method, as used, for example, by Andreoni and Car [20]. Like the tight-binding method, this has the advantage over the EMA that it does not have to match wave functions at the interface, Unlike the above mentioned semi-empirical approaches, “first-principles” calculations generate ~.

W. Ge eta!.

/ Journal of Luminescence 59

2. Experimental The SLs were grown by MBE without notation, and the SLs are 0.5—1 ~.tmthick. In some samples, growth was interrupted for a few seconds between each layer: this had no observable effect on the spectra. Electron diffraction from a piece of the layer adjacent to the sample used for PL, showed well resolved superlattice spots [27], although in the 1/1 samples the spots were weaker than in the other samples. The sample was cooled in a gas immersion Dewar capable of giving temperatures between 1.5 and 60 K. All measurements were made at 4.2 K unless otherwise noted. PL was excited with a 514.5 nm argon laser. Normally the excitation intensity was low, 10mW cm 2 except when otherwise noted. The PLE spectra were measured at high resolution between 2.03 and 2.17 eV with a Rhodamine 590 dye laser whose output power is kept constant by a feedback system, and at lower resolution (10—20 meV) to 2.5 eV with an f/3 monochromator, using a tungsten—iodine lamp as the light source. PL was measured with a Spex double (in some cases, single) f/8 monochromator with typical resolution 0.3 meV, fitted with a cooled Hamamatsu 928 photomultiplier. CW data were digitized and stored in a MacIl computer. Uniaxial stress up to about 10 kbar could be applied during the PL measurement, either parallel (“longitudinal”) or perpendicular (“transverse”) to [001] (the growth direction). In the longitudinal —

‘~

case sapphire anvils were used to allow light to enter and exit the sample along the direction of stress. Samples for the transverse measurements were either cleaved (for stress along [110]) or cut and polished (for stress along [100]). The equipment and methods are described in more detail by Schmidt [28]. Time-resolved PL data were obtained by modulating the laser beam with a pulsed 400 MHz quartz acousto-optic modulator, which could give pulses from 10 ns to 10 ,.ts in length, at a duty factor of 1%. The PL was measured as a function of time with a standard single photon counting system using a Ortec 934 constant fraction discriminator, followed by an Ortec 457 time to amplitude converter and a Hewlett-Packard 5416/22 pulse height

(1994) 163—184

167

analyzer which was used in the 256 channel mode and interfaced to a Macli computer. The count rate was always kept well below 10% of the laser repetition rate, so that connection for pulse pile-up was unnecessary.

3. Symmetry of electron states 3.1. General discussion of electronic state symmetry and parity We distinguish in this paper between the space group representation (SGR) of a state and its “parity”. SGR refers to the space group of the SL: in particular, it determines whether the CBM (which, in the present case, derives from the X minimum of bulk AlAs) is folded back to the center of the mini BZ, or to some other point. By “parity”, following Ref. [4], we refer only to parity under z inversion (z is the SL growth direction, perpendicular to the layers) about the center of the AlAs layer, since we are interested in states which derive from the CBM of bulk AlAs. The symmetry of the conduction band states in [001] SPSLs has been exhaustively studied theoretically [4]. For any value of m or n, the VBM is the heavy hole Kramers doublet. Therefore our discussion mainly concerns the CBM. The question of the SGR and the F—X mixing can be approached from a perturbation theory point of view. We define a space dependent potential V5L, with amplitude equal to the band edge difference between AlAs and GaAs (see Fig. 1) and averaging to zero over the whole SL. If VSL has a [001] component which mixes the bulk X~state with the bulk F state, X~is folded into 1’. This is the case when (rn + n) is even. The transition from X~in the conduction band to F in the valence band is then pseudo-direct, and a no-phonon (NP) transition is possible in absorption and emission. On the other hand, as discussed in Ref. [4], for (rn + n) odd, VSL has no [001] component, and any NP line must be due to deviation from ideal symmetry. This is also true for the ~ states for any (m + n). We have remarked before [3g] that the X—F mixing may still be small even if (m + n) is even. This is because, for a perfect SL with m = n, VSL(z) is a symmetric square wave,

168

14’. Ge ci al. / Journal at Luminescence 59

and has only odd harmonics in its Fourier expansion. Thus in the case of in = n and ii even, the period is 2ia0, where i is an integer and a0 is the lattice parameter, so VSL has no [001] component and there can be no mixing between F and X to lowest order in perturbation theory, if the SL is perfect. Besides the CBM’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 AlAs and GaAs is a common As layer: hence, for even n the central atomic layer of the AlAs is made up of As atoms, while for odd nit is a layer of Al atoms. Since in the bulk the X wave function has a node at the Al site, the parity of the CBM is determined [17] by the parity (odd or even) of n. The parity of the lowest (k~= 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 NP transition should be forbidden, whatever in. However, this argument depends on the assumption that k, is well defined within the exciton minizone. If the minizone bandwidth is small, quite a small deviation from perfect translational symmetry in the z direction can mix the different k0 states, and parity in our sense ceases to be a good quantum number, According to the above discussion, in a perfect SL the highest X—T NP transition probability should be found in samples for which m and n are both even but in ~ n. Unfortunately we do not have such samples. Table 2 lists the SGR and parity restrictions for the samples which we have studied. It must be emphasized that the above symmetry and parity considerations refer to the possible effects of VSL in a perfect SL. In a real SL, the interface disorder potential V~5can also mix states of different symmetry and make the NP transition partially allowed. As we shall see, VdI. is one or two orders of magnitude weaker than VSL, but its effects are nevertheless observable. 1/dj. We in have distinguish the toplane, which between the component of mixes X~ and F, and the component of 17dj. perpendicular to the plane. In the case of selection rules that depend on whether (in + n) is odd or even, such interface fluctuations along the z direction can be very significant. For example, in the case of a 2/3 SPSL, a one layer fluctuation could

1/994)

163 /84

make a certain region 2/2 or 3/3, i.e. changing from odd period to even period. Thus the X~--F mixing (forbidden for m + n odd) would be greatly enhanced. The situation is different for X~,~---Tmixing due to disorder in the x,~’plane, since the “islands” of constant layer width are known to be quite large [29], giving relatively weak [100] and [010] cornponents to VdIS. We will see this effect in the PL spectra of X. and X5, emissions from 1/2 and 2/3 SPSLs (Section 3.3.1). 3.2. Si’rnrnetrt’, paritt’ and optical

tran.sitioi1,~

Within second-order perturbation theory, the matrix element for an indirect NP optical transition is [30] M~

=

< ~ E1-

~

+

cc..

(I)

—

where V is a short range mixing potential, p is the dipole operator. The transition probability is then essentially determined by the nature of V. which may be VSL or V~l5.The Fourier components of V51, have wave vectors only along the SL growth direction, i.e. the z axis, so that only the X states can be mixed with F states by V5L. As discussed in Section 3.3, in a perfect SL the existence of nonzero components of V5L connecting F and X depends on the SGR and parity of the X state. For our samples, these considerations predict the following grouping of the transitions of the X~states (Table 2). In Section 3.3 we will compare these predictions with our experimental results. Impurity or defect potentials and the detron(exciton)—phonon interaction also make the forbidden transition weakly allowed by mixing X and F, but they produce transitions shifted from

Table 2 x~r transition probability mn SGR

Parity

1/I and 3/3 2/2 and 4,4 12 2,. 3

forbidden allowed allowed forbidden

-~

allowed allowed in second order forbidden forbidden

____________________________

_____

W. Ge ci a!.

/ Journal of Luminescence

the intrinsic NP line. Specifically, if F—X mixing is due to phonons, with interaction potential Hep, then within the momentum conservation (MC) model (i.e. second order perturbation theory) the corresponding phonon side band (PS) transition matrix element is [30] M ph 05

59 (1994) 163—184

169

optically detected magnetic resonance (ODMR). ODMR results give unambiguous assignments of the CBM which are in good agreement with those based on the methods used here. To our knowledge no ODMR measurements have been made on SPSLs.

T /(JJX \ c r~t ep (11r\/111r c/\ cP W v =

E +h E r W~i~ X .

(2)

—

Since the phonon scattering potential H and hence the transition rate M~,is similar for all of the SPSL samples, the intensity ratio NP/PS between the NP line and the PS line is a good indication of the NP transition mechanism. This will be further discussed in Section 333 ,

3.3. Symmetry determination The selection rules reveal themselves in the NP oscillator strength, i.e. the radiative rate. At low temperature, where the quantum efficiency is high, we can obtain the oscillator strength of the transition between the lowest state and the ground state from the corresponding PL decay rate after pulsed excitation (see Section 3.3.2). Using uniaxial stress, we can re-order the CBM states, and thus find the decay rate of states which are not the actual CBM at zero stress. The shift and splitting of a state under stress depends directly on the symmetry, and has been particularly helpful in distinguishing between states originating from X and from L of the bulk crystal. In Section 3.3.1 we discuss the effect of uniaxial stress on the conduction and valence band states. The integrated intensity ratio of the NP to the PS lines provides a useful qualitative cross-check on the decay data, since it does not depend on the nonradiative decay rate. However, for this to be useful, the PS rate must depend only weakly on the symmetry. This is only true for PS associated with momentum conserving (MC) phonons. The question of the assignment of the PS to particular lattice phonons is discussed in Section 3.3.3. A very powerful technique for state symmetry determination, which has been used to great effect on longer period SLs by the Philips group [31], is

3.3.1. Uniaxial stress Under uniaxial stress, the shift AE of the lowest transition, which is the only one observable in PL at low temperature, has three components: the change in splitting of the valence band due to the shear strain, the change in band gap due to the hydrostatic strain, and the splitting of the conduction band by the shear strain. The VBM is the heavy-hole (HH) band, and the light-hole (LH) band is well separated in these SLs. The first term AE5, the HH valence band shift, in .

the limit of small stress, and neglecting HH—LH mixing away from k = 0, is, for [001] stress: A

— —

~ —

~Jfl

—

~

and for [110] or [100] (in plane) stress: AE5

=

0.5b(S~1 S12)a, —

(3b)

where b is the (negative) deformation potential for strain of tetragonal symmetry, S are elastic compliance coefficients. The applied stress a, in kbar, is defined here as positive for compression. For in-plane stress, the symmetry is no longer uniaxial. HH—LH mixing at k = 0 can be taken into account [32] by adding a semiempirical term nonlinear in a to AE~.This was calculated in Ref. [3f]; the correction is small and does not significantly affect the comparison of theory and experiment. In-plane stress also induces coupling between HH and the spin—orbit split-off valence band [3 3,34], but its effect is negligible in our case, since the spin—orbit splitting is large (0.33 eV). The second term in the shift comes from the hydrostatic component of the strain, It depends on the parent valley of the CBM, but not on the stress direction: A i L.xEh=a(Sll+Si2)a=~—a,

(4)

170

W. Ge ci a!.

/

Journal 01 Luminescence 59 (1994)163-- /84

where a’ is the hydrostatic deformation potential for the conduction band, i indicating the parent valley (L, X or F), ~E’/ep is the corresponding hydrostatic pressure coefficient. Finally we come to the most significant term for our purposes, the splitting of the CBM by the shear

states derived from X: AE ~ ) AE (X )

AE5(X5)

=

—

15/3,

A L’

..‘~ -

— )—

±

44a1

(Sb)

~.

—

ó/6.

(7)

a piezoelectric field [36], but Fig. 23 of Ref. [36] shows that this is negligible in very short-period superlattices. In summary, the overall shift of the transition is given by AEioiai = iSE,~+ AEh + AE5. Comparison between the predicted shift under various stresses and the experimental results permits assignment of the state symmetry associated with the transition. We use the following deformation potential parameters [37] (note that those parameters are those of AlAs for state and1.43 of GaAs F and L 1~X-related = 8.49 eV, ax= eV, aLfor = 4.37eV, related): a

( )

,

=

X3 and X~splitting in AlAs is about 0.4 eV [35], this term is extremely small. In addition, the [110] stress produces

—

L’L 2

AE (X.)

—

where ~5= E~(S11 Si2)a, E~being the tetragonal shear deformation potential for the X valley (of AlAs). For [100] stress. X2 and X.,~ are interchanged in Eqs. (5). Under [110] stress, the lower state R1, derived from L, is split, giving shifts —

=

Mixing between X3 and X~states induced by in-plane [110] stress will also produce a nonlinear term [33]: 2/[E(X ) E(X )] (8) AE = (~S a) I where is a shear deformation potential. Since the

(Sa)

AE5(X~)= ~//3,

~

-

component of the strain. This depends both on the symmetry of the CBM and on the stress direction, Obviously shear shift there for the and under there [001] isorno[100] stress, is F no valley, shear shift or splitting of the L valleys. The shifts of X. and X 5,5 for [001] stress are =

—

S

where E~is the rhombohedral shear deformation potential for the L valley (of GaAs), while for the

—

Table 3 Comparison of theoretical and experimental line shifts under stress Stress Dir. [001] X~: Xi,:

L:

Theoretical.

—3.31 4.74 4.52 6.26

Experimental I/I 2,”2

3-3

—6.8

—6.8

—4.4

—4.2

—5.1

4.4

I 2

2,3

alloy

—4.0

—-3.8

—5.4

—4.2

[110] X: X~,:

L:

0.44 —3.54 — 3.96 1.96

1.1

[100] X: X~:

0.61 —7.34 0.39 2.13

—7.1

—4.1

W. Ge ci al.

/

Journal of Luminescence 59 (1994) 163—184

liz lower at o stress Energy

Stress

Xxy lower at

Xxy

0 stress

__________________________ Stress

___________________________________ 2.10

a

‘

(b)

‘

2.08

neglected in the above discussion. This mixing has the effect indicated by the dashed lines. Because of the slight lattice mismatch between the GaAs substrate and AlAs, even in an A1GaAs alloy the X valleys are split in zero applied stress, with X~,~lowest. Fig. 3(b) shows the observed PL peak shifts under [001] stress for some SPSL and the alloy samples. It clearly shows that for 4/4 the CBM is X. at all stress, but for the other SPSL, and the alloy, the CBM is at zero stress and changes over to X. atspectrum a criticalofstress of about 2 kbar.inThis PL the SPSL, as shown Fig. 4changewhere over is accompanied by a dramatic change in the 1/1, 2/2, 1/2, and alloy PL spectra at low and high [001] stress are depicted. A similar change-over

______________________ —

Fig. 3(a) shows schematically the expected effect of [001] compressive stress on X, and X~,~ states, ignoring AE5 mixing expect some and AEh. by When disorder, levels which cross,haswebeen can

~

Energy

(a)

171

a

under stress has been observed in longer period SLs by Lefebvre et al. [37]. The large NP/PS ratio at high stress is indicative of the X~state. Note that in the alloy there is, and can be, no significant change in this ratio, and that even the 1/1 SPSL has a much

Ce

~2.06

a

~co,~oo

a

30

.~2.04 ~2.02

~•

.

L

.

e2/3 2.00

alloy o 3/3 .4/4 0

• 2

4

~

I

•

_1

/

I’ l~

6

[OOilStress (kbar)

2

Fig. 3. (a) Schematic energy level shift for X3,0 and X~states under [001] stress; the dashed lines show the mixing of the two levels at cross over. (b) PL peak shift of 3/3, 4/4, 2/3 and alloy samples under [001] stress.

—,

f

b = —1.76 eV, d= eV, E~= 15 eV, and E~= 5.0 eV. The elastic compliance for 1, coefficients S GaAs are: S11 = 1.16 Mbar 12 = —0.37 Mbar 1, and 544 = 1.67 Mbar 0; and for 5i2 those = —0.39 AlAs are Sii = 1.20 Mbar’, Mbar and S 44 = 1.7 Mbar With these parameters, we obtain the predicted shifts for various PL lines under different uniaxial stresses given in Table 3. —4.55

—

—.—,

‘

1.95

200

\ ‘-—

2.10

Photon eneo/ry tcV)

~,

~.

Fig. 4. FL spectra at liquid helium temperature, excited at 514.5 nm: (a) 4/4 at zero stress; (b) 4/4 under [110] stress of 5.96 kbar; (c) 1/1 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 (x = 0.5) under [001] stress of 3 kbar.

W. Ge ci al.

172

,/

Journal of Luminescence 59 (1994) 163 184

The difference between the effect of disorder parallel and perpendicular to the growth axis, discussed in Section 3.1, can be seen by comparing the spectra for stress below and above the X~,~—X. cross-over in the 1/2 sample, for which the X. NP line is forbidden. While the decay rate is dominated by phonon side band emission and of the same order for X, and X~.5(see Section 3.3.2), the NP/PS ratio is considerably larger for X, (see Fig. 4), showing that the short range component of VdI. is larger along the z direction than in the x—y plane. Fig. 5(a) shows the PL spectra of 4/4 under various [110] stresses, clearly showing the predicted changeover of the CBM from X. to X~, Fig. 5(b) shows the PL peak shift of 4/4 as a function of [110] and [001] stress. It clearly shows the CBM varies continuously from X, (under [001] stress and zero stress) to X~ (under high [110] stress). The NP to PS intensity ratio as a function of stress is also consistent with the change of symmetry at

_~/\,~~4l)k~r

I’

~

410k/or

~ I 94

92

k/or

1%

/1kb,,

19)1

2(11)

2(12

2)4

(Or

~.

Plrrrirr,r orrerir~ e~1

------————----~

—

—----

•

Ib)

Fig. 5. (a) PL spectra of4/4 at T = 2 K and excited at 514.5 nm. under various [110] stresses;(b) PL peak shift of4.4 at T = 2K and excited at 514.5 nm, under [001] and [110] stresses,

cross-over. The PL spectrum of the 1/1 SL is similar to that of 2/2 and of 3/3 [3a—3c].The same NP—PS separations, similar PS/NP intensity ratios, and similar lifetimes, suggest the CBM is the same for I/I, 2/2, and 3/3, and is derived from X. The PS energy positions are also consistent with this assignment, as discussed below (Section 3.3.3). To confirm the CBM symmetry of the 1,/I SL. stress was applied along three directions, [001], [110], and [100]. Figs. 6(a) and (b) show the FL spectra of 1/I under uniaxial stress parallel to [110] ~nd [100] respectively. The data for [001] stress are similar to those obtained [3b,3c] on 2/2 and 3/3 SLs: the X~state is pushed down and becomes the CBM. Unlike X, or L, X~is mixed with F by the SL potential, so that the transition from X. to F k pseudo-direct, with a strong NP line, high quantum

smaller NP/PS ratio than the alloy at zero stress. These results indicate that these samples are good SLs, and that disorder plays a relatively minor role in the optical transitions. This is confirmed by the time decay measurements described in Section 3.3.2, which show that the SPSLs have disorderinduced radiative rates which are several orders of magnitude less than the random alloy.

efficiency, and short lifetime. On the other hand, [110] or [100] stress does not change the spectral shape but only shifts its peak position. The spectrum retains the well-established shape associated with the X,~CBM [3b,3e,28,37,38]. Under moderate [100] stress, at 30 K, a very weak shoulder is seen on the high energy side of the line, as shown in the inset of Fig. 6(b). This is not seen for [110] stress. We assign this shoulder to the Xr state, split from X5 by the stress. Note that

•

.

• ,

--

•

,

-

,--

I))]

~

I,~or/

0

,,.

W. Ge ci a!. I6~

/ Journal ofLuminescence ‘

Ia)

59 (1994) 163—184

173

1)

-—

‘3kb 4

12

(001] ‘I ‘

(1110)

-

:

~

~

0

10 a I

~20

-~

,8 .95 tSar

2

4

6

Stress (tSar)

Fig. 7. [001], under PL peak [110] shiftand of [100] 1/1 at stresses. T = 2 K and excited at 514.5 nm

Okbar 2.00

2.02

‘

2.04

2,06

2.00

This assignment is confirmed by a quantitative

Phoronenergyle9)

evaluation of the line shifts under stress. For a pre-

__________________________________-

cise evaluation of the peak shift, we have to include the nonlinear shift in the VBM (see above). Cardona et al. [11] estimate that for 1/1 the LH—HH splitting is 20 meV or more, with HH highest. In this situation, as discussed by Jagannath

_J’~”4”\~__/\,53()kbar~31)kbar

~bar

.8

2.00

2.02

2.04 2.06 Photon energy(eV)

2.08

2.10

Fig. 6. (a) PL spectra of 1/1 at T= 2K and excited at 514.5 nm, under [110] stress; (b) PL spectra of 1/1 at T= 2 K and excited at 514.5 nm, under [100] stress. The inset is PL at 30 K and 2.3 kbar, with contributions of X, subtracted, showing the line assigned to X 0 (gain of 50).

L cannot be split by [100] stress, but can be split by [110]. Splitting under [100] stress is strong evidence that the CBM derives from X, not from L.

et al. evaluated [32], the nonlinear fairly small. have this term term at ouris average stressWe of 3 kbar and corrected the predicted slope accordingly, as we did in Ref. [3f] for the 1/1 case. A correction has also been made for the AEh term using the generally accepted value for E~of 5.0 eV, instead of 7.7 eV which we used in Ref. [3f] to get the best fit to the data. Fig. 7 shows the shift of the 1/1 NP line under uniaxial stress along [001], [100] and [110]. To compare these experimental data with theory we must allow for the fact that the various CBMs are mixed by the [011] and [101] components of the random potential VdI. due to interface roughness. When different CBMs are nearly degenerate, this mixing gives to the E(a) plot a pronounced curvature which is much greater than the contribution of the nonlinear term mentioned above. To estimate the true SE/a we take the slope in the linear region at large stress, where the CBMs are well separated. In summary, [001] stress has been applied to all samples. [110] stress has been applied to the 1/1 and 4/4 samples to measure the X or L valley

174

W, Ge ci a!.

Journal of Luminescence 59 (1994) 163 --184

related FL lines and also to the 2/2 sample to measure the F line. [100] stress has applied to the 1/1 sample to distinguish the PL from X or L. In Table 3 we show the theoretically predicted PL line shifts (in the left column) comparing with the observed values (all in meV/kbar). Because the parameters in the theoretical calculation are uncertain, and also because it is difficult to be sure that the limiting slope has been reached. too much significance should not be attached to precise numerical agreement between the observed and calculated shifts. For this reason also, the approximations made in evaluating the nonlinear terms in the VBM splitting are not significant. However, it can be seen that for the transverse stress ([110] or [100]) the data are in good agree-

citon states at X with those at F and cause the NP transition to become weakly allowed [40]. It is shown in Ref. [39] that if the CBM is exactly at X, i.e. K = K~, the X--F scattering matrix element V~ V(K)I has a Gaussian distribution. The radiative decay rate w of an exciton localized at a particular site is proportional to V~,and has a probability density given by: (9) i.’2exp( w/2r(w>), where <‘iv> is the mean decay rate due to disorder, averaged over all possible sites for the exciton. The FL intensity as a function of time after pulsed excitation is then [39]: p(w) =(2itw)

—

-

.

.

.

=

1(0)exp(

.

1(t)

—

w 0t)(l + 2t)

ment with theory. There is a serious discrepancy between theory and experiment for [001] stress: while the shift has the predicted sign, the measured coefficient for some samples is up to a factor of two larger than expected, and varies in an unexpected way from sample to sample. The reason for this is not at present understood. It could be due to error in the stress calibration: it is more difficult to achieve uniform stress in the longitudinal stress apparatus than in the transverse and it is possible that the center of the sample, where the measurements were made, was subject to a higher than average stress, For the particular interesting 1/1 sample, the key observations are: (1) the shifts under [001] and [100] stresses are both negative; and (2) the shift for [100] stress is much larger than the shift for [110] stress. These relations are predicted if the CBM is derived from X,5~,,and in both cases are opposite to what is predicted if the CBM is derived from L. The predictions are essentially a consequence of symmetry, and are independent of any reasonable choice of parameters. 3.3.2. Tune-resolved PL Since the transition rate is directly related to the state symmetry, time-resolved PL is a powerful tool for determining the state symmetry. Klein et al. [39] considered the radiative decay of excitons in an indirect gap semiconductor alloy A~ ~B5C. The potential fluctuations created by the random placement of A and B atoms mix ex-

32,

(10)

where w0 is the decay rate due to nonstochastic processes (in the present case, phonon-assisted decay). It has been pointed out by Minami et al. [41], and discussed in more detail by Sturge et al. [42] and by Angell [43], that this decay equation applies also to the X.—F pseudo-direct transition in Type-lI SLs, in which the X,--F mixing is produced by the SL potential VSL. The decay is nonexponential because of short range fluctuations in the interface position. Because of these fluctuations, the matrix element connecting X.. and F is the sum of many contributions of random sign, and therefore has the Gaussian distribution which leads to Eq. (12). The average rate will be the same as for a perfectly abrupt interface, and it is this rate which we will use to estimate VSL. Eq. (12) only applies to low intensities and low temperatures, where the excitons are localized: these conditions apply for all the time-resolved data reported in this paper. The intensity and ternperature dependence of the PL decay has also been studied and is reported elsewhere [42,43]. Fig. 8(a) and (b) show some typical decay curves (smoothed) under different stresses, for SPSL sampies with m + n = even and in + n = odd respectively. If the X—F mixing by VSL is forbidden, so that the NP/PS ratio is small and the dominant contribution to the decay is from phonon assisted transitions, we would expect exponential decay. Experimentally, whenever the X—F mixing is not

W. Ge cial.

/ Journal of Luminescence

efficiency

\\ \\ \ ‘N~ 5’,,

I

—._~

,

~

j

59 (1994) 163—184

175

for these transitions is relatively low and we attribute the rapid initial decay to tunneling to nonradiative centers, which competes with the slow radiative decay for these forbidden transitions. If this interpretation is correct, the long time decay is characteristic of excitons far from a nonradiative center, and is probably largely radiative. This assumption is consistent with the observed quantum efficiency; but note that even if it is incorrect, and there is a significant nonradiative contribution at all times, the long time decay rate still gives an upper limit to the true radiative rate. Fig. 8(a) shows the smoothed decay curves for the 4/4 and 1/1 samples at zero stress (full lines) and under sufficient uniaxial stress (dashed lines) to reverse the order of X~and X~ Fig. 8(b) shows the corresponding data for the 2/3 sample (note the change of the time scale). For m/m SLs, i.e. (m + n) even, the cross-over from X~,~to X~is accompanied by a large increase in the average decay rate, but the change for (m + n) odd is much less, and in the opposite sense, showing that in these odd period samples the X~ F transition is forbidden. The significance of the fact that for odd (in + n) the decay rate decreases at cross-over, rather than staying constant, is discussed in the next section. The results for all our SPSLs are summarized in Table 4. Here, w~is the decay rate of X~,w,,~,~that of X~.For(m + n)even, w~= in the fit of the X~ decay to Eq. (12), while w~, and w~for (m + n) odd, are obtained by fitting the long time decay to the usual exponential decay law 1(t) cc exp( wt). AE is the energy separation between F and X,,~,~(see Table 6 below), and the fourth row gives w°” 104AE2w~,~,whose approximate constancy shows that the symmetry-breaking mechanisms (primarily phonon assisted transitions) which ~.

-

T4nrrr lOs)

______________________________ I

—*

-

j

\ \

~—-..-. ~‘.—..

a’.-..,

—

______________________________ 20

40 Tin,r~rs)

(rO

00

Fig. 8. NP PL decay after pulse excitation (514.5 nm) at T = 2K: (a) for rn + n = even SPSL (i) 4/4 at zero stress; (~) 4/4 under [110] stress of 5.96 kbar; (iii) 1/1 at zero stress; (iv) I/I

Table 4 Decay rates of the X, and X, 6 states for different superlattices

under [001] stress of 1.43 kbar. (b) form + n = odd SPSL (i) 2/3 at zero stress; (ii) 2/3 under [001] stress of 4.03 kbar.

due to VSL we find an approximately bi-exponential form with a well-defined exponential decay only at long times (see Fig. 8). The quantum

rn/n

4/4

3/3

2/2

1/!

2/3

1/2

w4 [ms’] w00 [ms~] LiE [meV] ~

1000 180 159 455

780 200 120 288

340 180 118 251

1200 160 121 234

42 ±10 57 ±10 203 235

15 ±5 20 ±5 356 253

176

14’. Ge ci al. -

Journal of’ Luminescence 59 (1994) 163---- 184

permit radiative decay of the X,~,v exciton are almost independent of sample for in, ii ~23. Table 4 shows that for even (in + n), the ratio w,/’iv~5.is in the range 1.9 to 7.5, showing that the

-

I)

superlattice potential VSL is contributing substantially to w. For odd (in + n), on the other hand, the ratio is less than one, showing that there is no appreciable contribution to ‘iv, from VSL, in agreement with the prediction from the space group

-

symmetry.

2

The fact that ‘iv. is less for 2/2 than it is for 1,/I and 3/3 is in agreement with the expectation that when in = ii = even, there can be no F—X, mixing to lowest order. On the other hand, ‘it’. is large for 4/4, as it is for longer period SLs [lb]. It is possible that for (in + n) ~ 8 the exciton tends to localize on one interface, as mentioned in the introduction; or that growth rate fluctuations are sufficient to wash out the precise value of in and fl. Parity conservation predicts that X—F mixing should be forbidden for in = n = odd. As can be seen from Table 4, our data show no sign of this, and we conclude that parity with respect to z inversion is not a good quantum number in these SLs, There are several possible explanations for this. One is an electric field due to surface charge, and measurements in an applied electric field are needed to check this point. Another is that the true interface has a finite thickness, and its effective center may not coincide with a layer of As atoms, as assumed by the parity argument. A third possibil-

-

I)

/2 -

,

-

-

2/2

I)

(

—

.~q

--

21)11

I’)r,Orrl,,r,rrs,,VI

-

2I11

Fig. 9. (‘omparison of CW PL spectra o14,-4, 2 2. 1 2 and alloy

(x

=

0.5) at liquid helium temperature and excited at 514.5

respectively. They are assigned to bulk-like longitudinal-optic LO(X) modes, which are the predominant MC phonons coupling to X~--F~ transitions in GaAs and AlAs [44]. The 31 meV mode is close to the LO(X) phonon of GaAs (29.9 meV [45]) and is in good agreement with the ab initio calculations of Molinari et al. [46]. The 48 meV mode is close to LO(X) of AlAs [47]. For I/I,

ity, which will be discussed later, is that fluctuations are uncorrelated between adjacent interfaces. The dominant layer width fluctuation would then be from in/in to in + I/rn I and the period would be unchanged, but any selection rule based on in being

a linear chain model with nearest neighbor interactions [48] predicts that the GaAs-like LO phonon should be at 30.1 meV. Sturge et al. [49] found 31

odd or even would be washed out.

Phonon side bands are broader than Raman lines and hence are less precise measures of phonon

CBM derived from L, LO(L) and the longitudinalacoustic LA(L) phonons would be involved in assisting the L1—F1 transition [44]. The LA(L) phonon is at 26 meV for both GaAs [45], and AlAs [47], but no PS is found at this energy for any of

frequencies, but nevertheless they provide useful information. Fig. 9 shows the low temperature PL spectra of some SPSLs. The spectra of 3/3 and 1/1 are similar to that of 2/2, and 2/3 is similar to 1/2. A PL spectrum from the x = 0.5 alloy is also shown for comparison. Two main phonon modes are ob-

our samples. A weak line is observed 12 meV below the NP line, close to the AlAs TA(X) mode, but it is usually obscured by an impurity (or defect) related exciton at almost the same position, which can be distinguished from the intrinsic exciton by its short life-

served in all our samples, at 31 ±1 and 48 ±1 meV

time and different temperature dependence [Id].

—

3.3.3. Phonon sude bands

meV for LO(X) in a random Al0 5Ga0 5As alloy.

Both are close to our data. On the other hand, if the

W. Ge ci a!.

/ Journal of’ Luminescence

There are two commonly used models in the calculation of the electron—phonon coupling, the MC and the CC (configurational coordinate). These are in fact [50] two different approximations to the full quantum mechanical treatment of the problem. The CC model is good for the case of relatively large lattice relaxation and multiphonon emission, but in its usual form uses the Condon approximation in which the effect of the phonons on the electron wave functions is neglected, and so cannot deal with phonon-induced transitions. In the CC model, multiphonon (MP) transitions in PL are possible and the integrated peak intensities at low temperature should follow the Huang—Rhys distribution [51]: 1(p) cc S”/p!,

(11)

where S is the Huang—Rhys parameter which characterizes the lattice relaxation and p is the number of phonon emitted. The NP line corresponds to p = 0. Eq. (11) assumes a single phonon frequency, but can readily be extended to multiple frequencies and to nonzero temperature. In our case, as can be seen from Fig. 9, the relative intensities of the LO PS with p = 0, 1 and 2 cannot be fitted to Eq. (11)~ the p = 2 peak is much too weak. Thus the PS are predominantly MC in nature. However, the standard second order treatment of MC transitions [52] permits only the p = 1 PS. Hence the existence of p = 2 PS is evidence for a weak CC contribution from a phonon or phonons around 48 meV: probably LO (F) of AlAs. If we neglect the small CC contribution to the PS, the radiative rate for phonon assisted transitions bears approximately the same ratio to the observed decay rate as the integrated one phonon PS to the total emission line. The rates obtained, corrected to constant 1’—X separation AE of 120 meV (that is roughly the corresponding value for2 with the phonon x = 0.5 energy alloy hv[49]) by factor = 40 meV (aver(~E+ hv) age value of the AlAs-like and GaAs-like modes), are given in Table 5. Note that the rates for the X,,~, state are almost consistently larger by a factor 2 than that for the X~state, and that this difference becomes more marked for larger rn/n. The weighted (2/3 for that coupled to X,,~ and 1/3 for that coupled to X~)values are comparable to the value

59 (1994) 163—184

177

Table 5 Transition rates [ms ‘] of PS rn/n

I/I

2/2

3/3

4/4

1/2

2/3

Coupled to X, Coupled to X,,

144

38

90

149

55

41

119’ 127

128 98

148 129

242 211

104 86

108 88

0 Weighted average

of 90 ms~ observed in the alloy [53]. This is evidence for anisotropy of the electron—phonon coupling: LO (X) phonons with momentum in the plane couple more strongly than those perpendicular to the plane. This shows that although the coupled phonon wave vectors are at the Brillouin zone boundary, the quantum confinement effect still has to be taken into account; while in the case of 1/1, the confinement has less effect, and the anisotropy effect is not obvious. Another interesting quantity is the intensity ratio

between the AlAs-like LO(X) line and the GaAslike LO(X) line. This ratio is about 1.8 for all rn/rn samples, but is about 3 for the 2/3 sample and 3.6 for 1/2. Thus the ratio is roughly proportional to the ratio of layer widths, implying that the X valley electron, like the F electron, is completely delocalized in the growth direction over the range of superlattice periods studied.

4. Energy levels 4.1. Theoretical predictions Much theoretical attention has been paid to the mixing and splitting of the X valley [4]. Confinement along z (growth) direction makes the X~lower in energy than X,, since the longitudinal effective mass at X is than the transverse. On due the other hand, thelarger AlAs layers are slightly strained to the fact that the lattice constant of AlAs is larger than that of the GaAs substrate (by a fraction 1.410 3)~This strain splits X 0 and X~,0in the opposite sense by about 23 meV [38]. We found, by extrapolating the high stress part of the PL energy versus stress curve to zero stress, that the separation between X. and X~ for the alloy ~,

—

178

W. Ge ci al. / Journal of Luminescence 59 (1994) 163—184

Al0 5Ga0 5As is about 15 meY. This splitting comes entirely from the mismatch strain and by Vegard’s law should be half of the value for A1As. Theory predicts that X. and X~are mixed and hence split by the SL potential for in and n both even [17]. Interface disorder scattering may also couple different valleys [l,2a,l3]. It can mix X, and X5. and push them further down relative to X, [16,54].

RRS selects out intrinsic excitons localized by potential fluctuations [55], since their dephasing time is much longer than delocalized ones [56], so the agreement confirms that the PL is from such excitons. 4.3. Present results’

Spectroscopic characterization of type-lI SPSLs is more readily achieved by PL than by absorption,

Temperature dependent FL, PLE and PL under uniaxial stress were used to determine the energy levels. Sharp PLE peaks, when observed, are a reliable measure of the energy level positions, but broad peaks and thresholds in PLE are difficult to interpret precisely, since their positions depend on

due to the forbidden nature of the lowest optical

poorly known wavelength dependent factors such

transition. Since 1976 [5], many reports of PL in SPSL have appeared in literature. Because of the difficulties in manufacturing good quality SPSLs, these studies, while numerous, are not altogether consistent. In most previous work on SPSL the PL is usually very broad. If a line is reported, comparison with our spectra shows that it is the impurity or defect related line mentioned in Section 3.3.3. Nagle et a!. [6] appear to be the first to have correctly identified a shoulder in the PL spectrum of a 2/2 SPSL as the intrinsic X—F transition. bestofpreviously data we areThe aware are those ofpublished ToriyamaPL et al. [7],that which appeared contemporaneously with our first reports [3]. They obtained clean FL spectra of SFSL from 1/1 to 4/4 in the temperature range of 4.6 to 300 K. They distinguished the X—F and F—F emission, and their spectra agree well with ours, but they incorrectly assigned all the X—F FL lines to X. Previous experimental data on the F—F transition are mainly from PLE, resonant Raman scattering (RRS) or ellipsometric [6,9] measurements. PL assigned to the F—F transition has also been reported in the literature, but some of these assignments are suspect, and no systematic data have yet been available to the authors’ knowledge. In Ref. [3d] we compare our results with those obtained by other techniques. Reasonable agreement is obtained with most of the RRS data, while ellipsometry seems to give consistently low results, probably because of incorrect assignment. This question is discussed in more detail in Ref. [3a].

as energy transfer and surface recombination rates, as well as on total SL thickness. Figs. 10(a) and (b) show typical low temperature FL and PLE spectra (for 2/2 and 4/4 respectively). FL and PLE spectra obtained from other SPSL with the ~ CBM are similar to those of 2/2. In both of the figures, a strong and very broad PLE peak occurs well above the NP line in FL: this is assigned to the F—F direct HH exciton. F—F FL is very weak in type-IL SPSLs since it competes with fast nonradiative transfer from F to X. The dashed curve in Fig. 11 shows the FL2)from SFSL at and the high2/2 gain. Simhighspectra excitation (~ 500 W/cm were also obtained ilar at high excitation from other SPSL samples. A PL line due to the F—F transition is seen, with the results given in Table 6. Localization, energy-dependent F—X transfer, and self-absorption of the luminescence all tend to lower the energy of the measured peak. Comparison with the PLE data suggests that the effective Stokes shift is similar to the 27 meV found in a 7/7 SL [Ib]. The integrated F—F FL intensity is 10 to 10 of the X intensity, and depends strongly on excitation intensity and wavelength. If we assume that radiative decay of the F exciton (rate l0~s’’) competes directly with nonradiative F—X transfer, this branching ratio implies a transfer time 0.1 ps, in order of magnitude agreement with results on longer period SLs [57]. However, our time-resolved measurements [3d] show that besides the expected rapid initial decay ( < 200 ps), the F—F FL has a relatively long-lived (~5 ns) component. In

4,2.

Previous experimental results

‘~

W. Ge ci a!. / Journal of Luminescence 59 (1994) 163—184

7

‘

/

A

r’

Rngron I

-~

RegIon 2

2.15 Photon energy

-

~-

-

-

-

/ __________ 2.0

2.20

teV)

179

2.1

2,2 -

Photon energy )oV)

Fig. 11. PL spectrum of 2/2 at liquid helium temperature under 2) at 514.4 nm. high excitation (500 W/cm

: ‘7 9

I

a’

j

~

a

I

•

s~

~

: 1

:‘

Reg~orI

~

Rn

effect will produce a decrease in energy of the direct

0ior 2

I/

/ . 0

(b) 2.0

2.1

2.2

2.3

Photon energy (eV)

Fig. 10. PLE and PL spectra at liquid helium temperature. PL was excited at 514.5 nm; PLE spectra were measured with a Rhodamine 590 dye laser for region 1 and with an f/3 monochromator, using a tungsten—iodine lamp as the light source, for region 2: (a) PLE and PL spectra of 2/2; (b) PLE and PL spectra of 4/4.

Ref. [3d] this was tentatively attributed to back transfer of hot electrons from the X minimum: but whatever the mechanism, the existence of the slow component and the strong excitation dependence of the intensity ratio invalidate the simple branching ratio argument used above. Sham and Lu [4] predict that the F—F gap for n/n SLs will decrease monotonically from n = 2 to n = 25. (They state that their calculation is unreliable for n = 1, 50 it is not listed in the table.) Wei and Zunger [21] have calculated the F—F gap for n = 1, 2 and predict that the energy level repulsion

Fgapfromn=2ton=1.Wefindthatthedirect F gap of SPSLs does not change substantially from n=lton=4. At lower energy there is a shoulder in the PLE spectrum, which we assign to X~.For 4/4, the PLE threshold coincides with the NP peak in FL, as shownin Fig. 10(b),andweattributethe ~20meV separation of the PL peak from the PLE “peak” to localization of the excitons by disorder. For n ~ 3, on the other hand the PLE threshold is 20 meV .

.

above the NP line in FL, giving a rough measure of the X2—X~,~ separation. Another measure of the X,—X~~ separation can be obtained from the stress data of Figs. 3(b) and 4, by extrapolation of the high stress portion to zero stress. The results are in within 1—2 meV of the results obtained from the FLE threshold. A third way to find the energy position of X. for n ~ 3 is from the PL at a higher temperature, where X~is thermally populated. Fig. 12 shows the FL for 2/2 in zero stress, at two temperatures: at 42 K emission from the X, state is clearly visible at the predicted position. The uncertainty here arises from a possible line shift due to exciton delocalization at the higher temperature: the agreement with other measurements indicates that this correction is only 1—2 meV.

W. Ge eta!.

180

/ Journal of

Luminescence 59 (1994) 163—184

Table 6 Energy levels in SPSL inn

State

This work

I~I 2/2 3/3 4/4 1/2 2/3 Alloy

F F F F F F F

2.195 2.185 2.178 2.188 2.47 2.30 2.189

1/I 2/2 3/3 4.4 1/2 2/3 Alloy

X X X X, X, X, X,

2.093 2.082 2.062 2.041 2.133 2.125 2.086

I/I

X,

2/2 3,3 4/4 1,2 2/3 Alloy

0 X,0 X,~ X,,0 X,~ X,,0 X,,,

Ref. [4]

2.12

2.08 2.08 2.03 1.95

Ref. [22]

2.168 2.151 2.069 2.022

2.18 2.23 2.16 2.09

1.93 2.03 1.97

2.073 2.036 2.009

2.156 2.084 2.078 2.082

2.17 202

1.99 1.85 1.79 2.50 1.88

2.10 2.08 2.08 2.05 2.03

2.00 2.00 2.00

2.061 2.060 2.037 1.963

2.10 2.10 2.03

77 1.88 1.86

2.10

at different ‘I’

—~‘

match), and compared with some theoretical predictions. Note that all the theoretical calculations, except those of Ref. [26], assume abrupt interfaces in the SLs. When different experimental methods were used the table gives the average value. It can be seen that the energy levels of these SPSLs vary more slowly with layer thickness than predicted by theory. Note that the lattice mismatch-induced strain effect, which shifts the X~

.-

.-

‘I PIrrrIrr~’~ergy (cv)

-

-

—,

I Os

Ref. [21]

2.22

-

I 9(1

Ref. [19]

2.03 2.13

--

‘~°

Ref. [18]

2.20 2.16 2.09

2.074 2.066 2.054 2.060 2.114 2.097 2.071

2/2 Pt.

Ref. [26]

-

2.05

IlK

“‘~“T 2.111

state down relative the X~state by about 23 meV, is not taken into account in some of the theoretical calculations. When it is, theory and experiment show much better agreement [26].

Fig. 12. CW PL spectra of 2/2 at two different temperatures, excited at 514.5 nm

We found it impossible to determine from our data whether the splitting of X~,~ predicted by theory for ~n = n = even does in fact occur. In prin-

Our results for the F~—F~ and X,—F~and X~,s—F. PL transition energies are listed in Table 6 together with those for the alloy (for which the difference between the X valleys is due solely to lattice mis-

ciple it should be possible to detect this splitting by raising temperature and thus populating the higher split level (of X,,, ±Xv). However, as shown in Fig. 12, the much stronger transition from X0. which is at about the same energy, may mask the

W. Ge ci a!.

/ Journal of Luminescence 59

luminescence from the higher X~~level. This splitting might also be detected by measurements under [100] stress: if X~and X~are mixed and already split, no further splitting (detected as broadening of the spectral line) can occur. In practice such a stress splitting is difficult to detect in PL (see Fig. 6(b) and the discussion thereof in Section 3.3.1), and the fact that it cannot be detected in any particular case is not evidence that it does not exist. ODMR measurements on 2/2 or 4/4 SLs are needed to settle this point,

5. Electron states of the 1/1 SPSL: ordered intermixing? 5.1. Theoretical predictions for the perfect SL In the 1/1 case the unit cell is sufficiently small that is accessible to standard band structure techniques. Self-consistent band structure calculations have been made [21,23] for SL with m = n = 1,2,3. These calculations, whose results agree quantitatively with each other and qualitatively with many others [22,26,58], appear to be the most reliable in that in principle they do not depend on parameters derived from the bulk band structure. As mentioned above, they predict that for m = n = 1 the CBM is at the R point, deriving from the bulk L point. These recent calculations disagree with earlier ones [11,20,59] that found a CBM derived from X. On the other hand, all recent calculations and experimental data agree that for 2 z~m = n ~ lithe CBM derives from the bulk X point, or precisely X~or X~ as discussed above, The prediction that L is lowest for m = n = 1 might seem rather surprising since in the A10,5Ga0,5As alloy the L point is well above the X point. Band structure calculations [60] put L about 0.3 eV above X in Al0 5Ga0 5As. That L is at least 0.1 eV above X for A104Ga0,6As has been confirmed by stress experiments [61]. However, in the SL the superlattice potential VSL tends to localize L in the GaAs layer. The four bulk L conduction valleys at (1/2, ±1/2, ±1/2) project onto R1 and R4 in a n/n SL (n odd), since VSL contains a (0,0,1) component which connects (1/2, ±1/2,1/2) to (1/2, ±1/2, 1/2). These have different energies, ~,

—

(1994) 163—184

181

and for 1/1 the splitting is large enough to bring R1 below the lowest state derived from X. Wei and Zunger [21] find an R~—R4splitting of 1.1 eV, while Zhang et al. [23] obtain 0.93 eV. 5.2. Influence of intermixing on the electron states As shown in Section 3.3.1, our data show unambiguously that the CBM derives from X. Recently Laks and Zunger [26], in an attempt to resolve this discrepancy, have extended their first-principles pseudopotential calculations [21] on perfect SLs to include the effect of atomic intermixing on the total energy and on the band structure. Their calculation for the [001] 1/1 SL shows that a partial and ordered intermixing of the Al and Ga atoms lowers the total energy, making this SL more stable than either the “perfect” 1/1 SL or the random Al0 5Ga0 5As alloy. The intermixing reduces the R1—R4 splitting and the CBM reverts from the GaAs L-derived state to the X,,, p-derived AlAs state. Laks and Zunger [26] calculated the total energy, diffraction pattern, and band structure where the abrupt (11) interface in the xy plane is replaced by larger (21), (31), and (41) interfacial unit cells and a fraction (1/2, 1/3, or 1/4) of the Al atoms on the AlAs side is exchanged with Ga atoms on the GaAs side. They first calculate the changes in total energy due to this local atomic intermixing. The excess enthalpy [62] of an abrupt 1/1 SPSL taken with respect to bulk GaAs + AlAs is 13.7 meV per A1GaAs2 unit, while the (positive) mixing enthalpy of the random Al0 5Ga0,5As alloy is calculated to be 10.6 meV. Hence, the ordered, abrupt 1/1 SPSL is less stable at T = 0 than the random alloy. However, total-energy calculations for the 1/2 fraction intermixed (21) 1/i SPSL show that the excess enthalpy is lowered to 5.5 meV. The 1/3 and 1/4 fraction exchanges give mmimum excess enthalpies of 7.3 and 8.8 meV respectively. Hence, local atomic mixing at the interface stabilizes the 1/1 [001] SPSL with respect to the random alloy. However, it is not clear that these very small energy differences can be significant at the growth temperature of about 500°C, unless there are kinetic factors also favoring ordered intermixing. Some possible kinetic factors which affect

182

14”. Ge et a!. / Journal of Luminescence 59 (/994)163- 184

ordering in alloys have recently been discussed by Zunger [63]. Laks and Zunger also calculated the energy levels for the (31) intermixed 1/1 SPSL, and show that this reconstruction removes the symmetry constraint that led to the large L splitting. The position of the CBM, taking lattice mismatch into account, is now ~ at 2.08 eV, in agreement with our experimental result. The F level moves up in energy, into better agreement with experiment, The question arises as to why such an intermixed 1/1 SPSL should still show the characteristic superlattice X-ray and electron diffraction lines observed [27,64] in such samples. Laks and Zunger made X-ray structure factor calculations and showed that this ordered intermixing in the [001] plane still exhibits the observed superlattice [OOn]diffraction features. Diffraction in other planes has not been investigated. Similarly, the spectroscopic selection

6. Summary and conclusion

rules and strain behavior remain the same as for the perfect SL. In conclusion, the apparent contradiction between theory for a perfect SL and experiment cannot be attributed to sample disorder, since the optical selection rules are well obeyed and the disorder potential VdIS is much less than the superlattice potential VSL, as shown by the fact that the ~ NP transition (which is made allowed by V~~5) is much weaker than the X, NP transition, which is made allowed by VSL. One would expect that interchange of Al and Ga atoms across the interface (leading to Vdt~)would mix F and Xjust as much as an ordered alternation along z (leading to VSL). However, as pointed out by Laks and Zunger [26], this would not be true if the interchange is ordered and preserves translational symmetry. They assume that, while there is indeed intermixing at the interface, it occurs in an ordered manner: every third A! atom is exchanged with a Ga. It is not clear whether our results are consistent with the observation of Ourmazd et al. [65], who used chemicallattice imaging of quantum well structures (from a different source than those used here) to show that interfaces that had been characterized optically as being perfectly abrupt were actually intermixed over two to four layers around the interface. This technique does not distinguish between random and ordered mixing.

tatively attributed to ordered intermixing of the Ga and Al atoms. It is still not clear whether this intermixing is an intrinsic property of these superlattices, or if it can be prevented by further improvement in the growth technique.

We have described a comprehensive study of the low lying conduction band states of type-lI (GaAs)m/(AlAs)n superlattices with in, n ~ 4. We find that if a meaningful comparison of experiment and theory is to be made it is essential to know the symmetry of the states involved, and we have found this from the effect on the energy levels of uniaxial stress and from the oscillator strengths determined by time-resolved photoluminescence. We find that not only is the corresponding NP transition stronger, but also the optical selection rules for a superlattice, as opposed to the corresponding alloy, are well obeyed even when in = 1, showing that our SLs are of better quality than any that have previously been reported. A discrepancy between theory and experiment regarding the ordering of the levels in the (GaAs)i/(AlAs)1 superlattice is ten-

7. Acknowledgement The work at Dartmouth is supported by the US Department of Energy (DOE) under grant no. DE FG 0287 ER45330.

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