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Anisotropic exchange splitting of excitons in „001…GaAs/Al0.3Ga0.7As

superlattice studied by reflectance difference spectroscopy

Z. Y. Zhou, C. G. Tang, Y. H. Chen,a?and Z. G. Wang

Key Laboratory of Semiconductor Materials Science, Institute of Semiconductors,

Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, People’s Republic of China

?Received 6 December 2007; accepted 22 April 2008; published online 3 July 2008?

Anisotropic exchange splitting ?AES? is induced by the joint effects of the electron-hole exchange

interaction and the symmetry reduction in quantum wells and quantum dots. A model has been

developed to quantitatively obtain the electron-hole exchange energy and the hole-mixing energy of

quantum wells and superlattices. In this model, the AES and the degree of polarization can both be

obtained from the reflectance difference spectroscopy. Thus the electron-hole exchange energy and

the hole-mixing energy can be completely separated and quantitatively deduced. By using this

model, a ?001?5 nm GaAs/7 nm Al0.3Ga0.7As superlattice sample subjected to ?110? uniaxial

strains has been investigated in detail. The n=1 heavy-hole ?1H1E? exciton can be analyzed by this

model. We find that the AES of quantum wells can be linearly tuned by the ?110? uniaxial strains.

The small uniaxial strains can only influence the hole-mixing interaction of quantum wells, but have

almost no contribution to the electron-hole exchange interaction. © 2008 American Institute of

Physics. ?DOI: 10.1063/1.2947602?

I. INTRODUCTION

The fine structures of excitons in low-dimensional semi-

conductor structures attract much attention due to their im-

portance in both fundamental physics and device applica-

tions in optoelectronics and quantum information systems.

The joint effect of electron-hole exchange interaction and the

symmetry reduction can split the optically allowed exciton

states into two polarized states: ?110? and ?11 ?0?. This split-

ting is called anisotropic exchange splitting ?AES?, which

has attracted a great deal of interest in recent years.1–9

The electron-hole exchange can separate the optically

allowed exciton states from the optically forbidden states.

Both the allowed states and the forbidden states are doubly

degenerate in ideal quantum wells ?QWs? and quantum dots

?QDs? with D2dsymmetry. Once the D2dsymmetry is re-

duced to C2vby some external perturbation or the roughness

of the interfaces, the optically allowed states will also be

split into two states with two different polarizations: ?110?

and ?11 ?0?.7Generally, AES can be given by ?E=EexZ/?lh,

where Eexis the electron-hole exchange energy, Z is the hole-

mixing energy arising from the symmetry reduction, and ?lh

is the energy spacing between heavy- and light-hole levels in

the quantum-confined structures.8Usually, Eexand Z are

lesser than ?lh, leading to an AES less than 1 meV.6,8–11

No method has been reported before to completely sepa-

rate the exchange energy Eexand the hole-mixing energy Z

quantitatively. Many recent reports only focus on the abso-

lute value of AES without further discussion of the behavior

of the electron-hole exchange and interface anisotropy.6,10–13

The degree of polarization ?DOP?, which reflects the aniso-

tropy of samples directly, is usually neglected. It is also very

difficult to quantitatively calculate the overlap integral of

electron and hole envelope functions in real QWs and QDs.

As a result, there are still no reports on quantitative study of

the detailed behavior of electron-hole exchange interaction in

real samples.

Furthermore, it is much more difficult to analyze the

AES of QWs than that of QDs. For QDs, the electron-hole

exchange energy is strongly enhanced by the complete three-

dimensional confinement, and the spectral line width of a

single QD at low temperature is usually extremely narrow.

As a result, AES in QDs is sufficiently large for experimental

studies of the fine structure and the relaxation dynamics be-

tween these states.3,6,11,14,15However, for QWs and superlat-

tices ?SLs?, the exchange energy is usually too small to mea-

sure because the AES ?usually around 10 ?eV? is much

smaller than the inhomogeneous line broadening ?usually

around 1–2 meV?.13Therefore the fine structure cannot be

clarified experimentally by conventional linear spectroscopy

method. There are studies on AES of QWs by other methods

such as quantum beats8,16and optically detected magnetic

resonance ?ODMR?.9,17However, these two methods are ap-

propriate only when the radiative lifetime of the studied ex-

citons is long enough. Therefore, only AES of type-II GaAs/

AlAs QW, in which excitons have a sufficiently long

lifetime, has been studied by these two methods.8,9,16There

are still many difficulties in studying the AES of type-I QWs.

In this paper, we develop a model to completely separate

the electron-hole exchange energy and the hole-mixing en-

ergy of QWs. This model is applicable for most ordinary QW

and SL systems without the conventional restrictions from

the linewidth and the lifetime. The electron-hole exchange

energy and the hole-mixing energy of QWs can be calculated

quantitatively by analyzing the reflectance difference spec-

a?Electronic mail: yhchen@red.semi.ac.cn.

JOURNAL OF APPLIED PHYSICS 104, 013106 ?2008?

0021-8979/2008/104?1?/013106/6/$23.00© 2008 American Institute of Physics

104, 013106-1

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troscopy ?RDS? of the samples. Using this model, we inves-

tigate a type-I 5 nm GaAs/7 nm Al0.3Ga0.7As SL. The

AES and DOP of the n=1 heavy- and light-hole related ex-

citons ?named as 1H1E and 1L1E excitons? have been deter-

mined. The electron-hole exchange energy ?Eex? and the

hole-mixing energy ?Z? for the 1H1E exciton have been re-

spectively obtained. The linear tuning of AES by uniaxial

strains has been demonstrated. We find that the electron-hole

exchange energy ?Eex? remains constant while the hole-

mixing energy Z shows a linear dependence on uniaxial

strains.

II. ANALYSIS MODEL

Taking into account the spin states of the electron and

hole, the exciton states in a QW are fourfold degenerate.

Electron-hole exchange interaction can lift the degeneracy

and split an exciton state into dipole-allowed and dipole-

forbidden states, in which the dipole-allowed states are dou-

bly degenerate. Considering both the electron-hole exchange

interaction and the mixing between heavy and light holes

arising from C2vsymmetry, the perturbation Hamiltonian can

be written as18

H? = aJ?? ? + g?z??JxJy?,

?1?

where Jxand Jyare the components of the hole total angular

momentum J, ? ? is the spin angular momentum of the elec-

tron and g?z? is the hole-mixing coefficient, ?JxJy?=?JxJy

+JyJx?/2, and the parameter a equals to 2?j0−j1? in the no-

tation of Cho.19The hole-mixing coefficient g?z? can be in-

fluenced by many factors caused by the symmetry reduction

from D2d to C2v, and it takes the form of g?z?=D?xy

+?n?p1??z−d1/2+nL?−p2??z+d1/2+nL??,20where d1is the

well width, L is the period of the SL, D is the deformation

potential, and ?xyis the applied shear strain. p1and p2are the

interface potential parameters related to the strength of the

C2vsymmetry of two interfaces. As discussed in Ref. 20, the

inequality of p1and p2can reduce the symmetry of QWs and

SLs from D2dto C2vas a whole.

Adopting a similar model presented by Gourdon and

Lavallard,8the wave functions of the split states take the

form of

??= ?1h??1HE1? ? ?1HE2?? + ?1l??1LE1? ? ?1LE2??, ?2?

where the notations + and − mean the ?11 ?0? and ?110? direc-

tions according to the interband transition matrix elements.

The bases ?1HE1?,?1HE2?,?1LE1?,?1LE2? are the same as in

Ref. 8 One has ??1h????1l? for heavy-hole related excitons

and ??1h????1l? for light-hole related excitons. The second

term in Eq. ?2? is neglected in Ref. 8 because it is only

concerned with the behavior of the heavy hole.

In this model, the splitting energies can be given by

?E1h= − 4EexZ/?lh,

?3a?

?E1l= 4EexZ/?lh,

?3b?

where Eexis the exchange-related term determined by the

overlap integral of the envelope functions of heavy- and

light-hole related excitons F1h1eand F1l1e, Z is the hole-

mixing term derived from the symmetry reduction from D2d

to C2vwhich is determined by ?F1h1eg?z?F1l1edr, and ?lhis

the energy separation between the heavy- and light-hole

bands, usually Eex,Z??lh.

The transition probabilities for 1H1E and 1L1E excitons

are

M1h?? ?f1h1e?2+2?Eex? Z?

?lh

?f1h1e

?

f1l1e?,

?4a?

M1l?? ?f1l1e?2−2?Eex? Z?

?lh

?f1h1e

?

f1l1e?,

?4b?

where ?f1h1e?2and ?f1l1e?2, which are determined by the enve-

lope functions F1h1eand F1l1e, are the oscillator strengths of

the 1H1E and 1L1E excitons. The notations + and − mean

the ?11 ?0? and ?110? polarized states, respectively. Generally,

?f1h1e?2:?f1l1e?2=3:1, so we let ?f1h1e?2=3?f1l1e?2=3 and

?f1h1e

Equation ?4a? and ?4b? shows that there is also a small

difference between the transition probabilities of the two di-

rections. But it is ignored in the discussions in Ref. 8. De-

fining the DOP as

?

f1l1e?=?3.

Ps= ?Ms+− Ms−?/?Ms++ Ms−?,

?5?

where s means 1H1E or 1L1E. From Eq. ?4a? and ?4b?, we

can obtain the DOP for 1H1E and 1L1E excitons

4Z/?3?lh

2?1 + 2Eex/?3?lh?,

P1h=

?6a?

P1l= −

4Z/?3?lh

2?1/3 − 2Eex/?3?lh?.

?6b?

Equation ?6a? and ?6b? indicates that the ratio of the DOPs of

the heavy and light excitons is not rigidly 1:3 due to the e-h

exchange interaction, but has a little deviation which is de-

termined by the value of 2Eex/?3?lh. According to g?z?

which is contributed by both interface asymmetry ?p1?p2?

and uniaxial strain, a linear tuning of AES and DOP by

uniaxial strain is expected when the strain is not too large.

Because both polarized absorption and RDS measure-

ments are related to the anisotropy of the dielectric

function,21it is convenient to approach the problem in terms

of dielectric functions instead of transition probabilities. The

dielectric function arising from an exciton with energy of Es

can be written as ?s=?Ms?S?E−Es?, where S?E−Es? accounts

for the spectral line shape of the exciton and ?Ms? accounts

for the total transition probability of ?110? and ?11 ?0?. Mean-

while, the anisotropic dielectric function between the two

directions can be defined by ??s=Ms

−Es

+S?E−Es

+?−Ms

−S?E

−? which can be rewritten as

??s= − ?Es

d?s

dE+ 2Ps?s,

?7?

where Psis the DOP defined by Eq. ?5?. According to Eq.

?7?, the line shape of ??sshould be a mixture of the line

shape of d?s/dE and ?s. We can determine the splitting en-

ergy ?Esand Psby fitting the line shape of ??swith ? and

013106-2 Zhou et al.J. Appl. Phys. 104, 013106 ?2008?

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d?/dE

measurements.22With the splitting energy ?Esand the DOP

Ps, the exchange energy Eexand Z can be easily calculated

according to Eqs. ?3a?, ?3b?, ?6a?, and ?6b?.

which canbe determinedbyreflectance

III. EXPERIMENTS AND ANALYSIS

Our sample consists of a 300 nm GaAs buffer layer and

30 periods of 5 nm GaAs/7 nm Al0.3Ga0.7As layers which

are grown on ?001? semi-insulating GaAs substrate by MBE.

All epilayers are intentionally undoped. The sample is

cleaved along the ?110? and ?11 ?0? directions into a 4.8

?2.0?0.4 mm strip. We glue it on one side face of a

stacked PbZrTiO3?PZT? piezoelectric actuator with an active

length of 8 mm and a 5?5 mm2cross section, which is

attached to two Cu caps on the two heads. The setup is

illustrated in Fig. 1.

We fix the setup in a cryogenic Dewar bottle. One cap is

fixed on the probe of the Dewar bottle directly while the

other is connected to the probe with a braid of soft copper

wires to allow the PZT to elongate freely without weakening

the thermal contact. According to the band edge position in

the RDS spectra and the Varshni relation Eg?T?=E0

−?T2/??+T?,23the sample’s temperature is 83 K and stable.

Estimated from the stroke of the PZT, the strain in the

sample at 83 K is about 1.4?10−4under a bias of 90 V

?−50 V is regarded as zero strain according to the discussion

below?. The precise value of the strain can be measured by

resistance strain gauges.24However, in the present work we

are only concerned with the linear dependence of the

uniaxial strain on the bias voltage at 77 K as demonstrated in

Ref. 24 but ignore the rigorous calibration of the uniaxial

strain. Considering the different thermal expansion coeffi-

cients of PZT and the sample, we glue the sample on a 150

V-biased PZT at room temperature before cooling it. Other-

wise, the sample would probably crack at low temperatures.

In addition, the RD spectra under zero strain have also been

obtained for reference from the same sample glued on the

heat sink instead of PZT.

The RD and reflectance spectra under different bias volt-

ages are measured simultaneously at low temperature. De-

noting the anisotropic dielectric function of the SL layer be-

tween the x and y directions as ??=?x−?y, according to our

previously published work,22the anisotropic dielectric func-

tion ?? can be deduced from the RD spectra ?r/r by con-

sidering the substrate, SL layer, and vacuum as a three-phase

model.25The exciton-related dielectric function ? can be also

FIG. 2. ?Color online? ?a? ?R/R induced from 1H1E and 1L1E excitons

under the max strain ?90 V bias?. ?b? Imaginary part ?r/r related to 1H1E

and 1L1E excitons under different strains. ?c? The dielectric function ?

induced from 1H1E and 1L1E excitons under the max strain ?90 V bias?,

calculated by the three-phase model ?after subtracting the one without

strain?. ?d? Imaginary part ?? related to 1H1E and 1L1E excitons under zero

strain and the strain-induced Im???? under different strains.

FIG. 1. ?Color online? Illustration of the low temperature strain setup.

013106-3 Zhou et al.J. Appl. Phys. 104, 013106 ?2008?

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deduced from ?R/R which can be directly determined by the

dc part of the RD spectra ?Ref. 22? ?R is the reflectivity of

the sample while r is the reflection coefficient, thus R=?r?2?.

IV. RESULT AND DISCUSSION

The RD and reflectance spectra are obtained under PZT

bias varying from −50 V to 90 V with a step of 20 V. Figure

2?a? shows the 1H1E and 1L1E exciton-related ?R/R of the

sample, while Fig. 2?b? shows the imaginary part of ?r/r

related to 1H1E and 1L1E excitons under different strains.

The reflectance almost does not change with the PZT bias.

Compared with the RD reference spectra subjected to no

strain, the sample subjected to −50 V bias voltage is ap-

proximately free of strain. The dielectric function and aniso-

tropic dielectric function related to 1H1E and 1L1E excitons

are obtained by the three-phase model which are shown in

Fig. 2?c?. We then calculate the strain-induced ?? by sub-

tracting the nonstrain ?? spectra from each spectra. All the

results are shown in Fig. 2?d?.

According to Eq. ?7?, we can determine the splitting en-

ergy ?Esand the DOP Psby fitting the ?? line shape from ?

and d?/dE. Generally 1H1E and 1L1E excitons have differ-

ent splitting energies and DOPs, so we must divide the ? and

??/dE spectra into two parts to fit the ?? with two groups of

?Esand Ps. The fitting results of zero strain and the maximal

strain ?after subtracting the spectra of zero strain? are shown

in Fig. 3. The line shape of the fitted ?? agrees well with the

experimental data.

Figures 4?a? and 4?b? summarize the AES and DOP of

1H1E and 1L1E excitons at different PZT biases. Both the

AES energy and DOP show linear dependence on strains as

expected. However, the AES of 1L1E shows deviation from

the prediction given by Eq. ?3a? and ?3b?. According to Eq.

?3a? and ?3b?, the AES of 1L1E should have the same mag-

nitude as 1H1E but with opposite signs. Instead, in our ex-

perimental results, the AES of 1L1E reaches more than four

times that of 1H1E. The possible reason is that only the hole

mixing between n=1 heavy and light holes ?named as 1H

and 1L? has been taken into account in the presented model.

In fact, the 1L subband can also be coupled with the n=2

heavy-hole subband ?2H? via the interface related term in

g?z?,20and coupled with electron subbands via the term

?2/3P0kzin the k·p method.26Taking the coupling between

2H and 1L subbands into account, Eq. ?3b? would have the

form ?E1l=4EexZ/?lh+4Eex2Z2/??lh−?hh?, where Eex2is

the exchange energy of 2H and 1L subbands, Z2is the hole-

mixing energy of 2H and 1L subbands, and ?hhis the energy

difference between the 1H and 2H subbands. Usually ?lh

???lh−?hh?, so 4Eex2Z2/??lh−?hh?, which is even larger

than 4EexZ/?lh, cannot be neglected. The contribution of the

electron subband to AES takes a similar form 4Eex3Z3/??lh

+Eg?, where Eex3, Z3are respectively the exchange energy

and the mixing energy between the electron and 1L subband,

and Egis the band gap of the GaAs. Because ?lh+Eg??lh,

the contribution of the electron subband can be neglected.

Thus, it is reasonable that the actual AES values of 1L1E

obtained from experiments show deviation from the values

FIG. 3. ?Color online? ?a? Spectra of the exciton-

induced dielectric function. ?b? Spectra of the exciton-

induced d?/dE function. ?c? Comparison between fit-

ting data and experimental data. The dotted line is the

fitting data while the solid line is the experimental data;

notations 1 and 2 mean the max ?after subtracting the

zero strain spectra? and zero strains, respectively.

013106-4Zhou et al.J. Appl. Phys. 104, 013106 ?2008?

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Page 5

predicted by Eq. ?3b?. However, since the 2H and the elec-

tron subbands have no direct coupling with the 1H subband,

the AES and the DOP of 1H1E in Fig. 4?a? can be respec-

tively regarded as ?E1hand P1hin Eqs. ?3a? and ?6a?. Below

we only discuss the results of 1H1E.

According to the model, the exchange energy Eexand the

strain-induced hole-mixing energy Z of 1H1E exciton can be

obtained by Eqs. ?3a? and ?6a?, which are shown in Fig. 4?c?.

The energy separation between the 1H and 1L subbands can

be obtained from the imaginary part of ? in Fig. 3. The

exchange energy Eexremains constant, while the hole-mixing

energy Z shows a linear dependence on uniaxial strain as

expected. The independence of Eexon the uniaxial strain

implies that the applied strain has little changes in the wave

functions of the 1H, 1L, and 1E states, which is consistent

with the fact that the resonant structures of 1H1E and 1L1E

in the reflectance spectra show almost no change with the

strain. Averaging the seven exchange energies under differ-

ent strains, we can obtain the exchange energy Eex=

−2.7?0.6 meV. Meanwhile, by fitting the nonstrain spectra,

we can obtain that without strain the exchange energy of

1H1E exciton is Eex=−2.5?0.5 meV. The results from the

two methods are nearly the same which verify the self-

consistency of the model and our calculations. Thus, theAES

can be linearly tuned by the ?110? uniaxial strain. The

uniaxial strains contribute only to the hole-mixing energy but

not to the electron-hole exchange energy. So the AES, which

is directly proportional to the product of the hole-mixing

energy and the electron-hole exchange energy, also shows

linear dependence on the ?110? uniaxial strain.

According to the form of g?z? mentioned above, g?z?

=D?xy+?n?p1??z−d1/2+nL?−p2??z+d1/2+nL??, the hole-

mixing energy Z has two components: one is determined by

the interface roughness which is reflected by the parameters

p1and p2, and the other one is determined by the external

strain which is reflected by the deformation potential D and

the shear strain ?xy. It is hard to quantitatively estimate the

former one because the relation between the interface rough-

ness and the interface parameters p1and p2is still not clear.

However, it is possible to estimate the strain-related compo-

nents. According to our model, the strain-related hole-mixing

energy Zstraincan be written as Zstrain=

=2.7 eV ?Ref. 27? and the max ?110? uniaxial strain ?110

=1.4?10−4, we obtain Zstrain=0.16 meV, which agrees well

with our fitting data Zfitting=0.14?0.03 meV.

It is interesting to compare our results under zero strain

with the reported data. According to the ODMR results of

Gourdon and Lavallard,8the splitting of a type-II 2.3 nm

GaAs/4.1 nm AlAs QW is less than 1 ?eV, and the splitting

of a type-II 2.3 nm GaAs/1.1 nm AlAs QW is about 9 ?eV.

In addition, AES has been directly observed for the localized

1H1E excitons in a 2.8 nm GaAs/Al0.3Ga0.7As QW, which

is regarded as a GaAs QDs.13The obtained AES of 1H1E is

about 25 ?eV for a single GaAs QD ?Ref. 13? and 70 ?eV

for an ordered GaAs QD.28In our experiment ?type-I

5 nm GaAs/7 nm Al0.3Ga0.7As?, splitting under zero strain

is 35 ?eV which is much larger than that of type-II GaAs/

AlAs QWs in Ref. 8 even though the well width in our

sample is larger and very close to that of GaAs QDs natu-

rally formed in type-I GaAs/AlGaAs QWs. The AES of QWs

and SLs depends greatly on the type of the carrier confine-

ment. In a type-I QW, the electrons and holes are confined in

the same layer so the integral of their wave functions is much

larger than that in a type-II QW, where the electron and hole

are confined in different layers. Therefore, type-I structures

have an exchange energy much larger than type-II structures.

Under similar hole-mixing intensity, the AES of type-I QW

should be much larger than that of type-II.

As emphasized in Sec. I of this paper, it is impossible to

deduce the Eexand Z directly from only AES. Gourdon and

Lavallard point out that Z is independent of the well width

and they estimate Z=14.5?1.5 meV for their type-II GaAs/

AlAs SLs after numerical calculations of the exchange en-

ergy Eex.8This value is larger by three orders than Z

=0.064?0.013 meVfor

5 nm GaAs/7 nm Al0.3Ga0.7As?, which can be attributed

again to the different localizations of the excitons in two

types of carrier confinement. For type-I QWs, the electrons

and holes are localized in the same layer with the center of

mass at almost the center of the well layer. In this case, the

hole-mixing energy is determined by interface asymmetry,

?3

2D?xy. Using D

oursample

?type-I

FIG. 4. ?Color online? ?a? AES of 1H1E and 1L1E excitons. ?b? Degree of

polarization of 1H1E and 1L1E excitons. ?c? Electron-hole exchange energy

Eexand hole-mixing energy Z of 1H1E exciton under different strains—zero

strain, Eex,0=−2.5?0.5 meV, and Z0=0.064?0.013 meV.

013106-5Zhou et al.J. Appl. Phys. 104, 013106 ?2008?

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