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NUMERICAL ANALYSIS OF ELECTRO-OPTIC EFFECTS IN SUPERLATTICES COMPARISON
WITH PHOTOREFLECTANCE RESULTS
J. Cavailles, M. Erman, P. Frijlink
To cite this version:
J. Cavailles, M. Erman, P. Frijlink. NUMERICAL ANALYSIS OF ELECTRO-OPTIC EFFECTS
IN SUPERLATTICES COMPARISON WITH PHOTOREFLECTANCE RESULTS. Journal de
Physique Colloques, 1988, 49 (C2), pp.C2-97-C2-100. �10.1051/jphyscol:1988221�. �jpa-00227638�
JOURNAL DE PHYSIQUE
Colloque C2, Suppl6ment au n06, Tome 49, juin 1988
NUMERICAL ANALYSIS OF ELECTRO-OPTIC EFFECTS IN SUPERLATTICES COMPARISON WITH PHOTOREFLECTANCE RESULTS
J.A. CAVAILLES, M.
ERMANand P. FRIJLINK
Laboratoire d'E1ectronique et de Physique ~ppliquBe('), 3 Avenue Descartes, P-94451 Limeil-Brevannes Cedex, France
RBsumt5 : Nous prbsentons une analyse numerique des effets electrooptiques dans un superrbseau basbe sur une approche de type liaisons fortes. La prise en compte du couplage entre puits conduit B des effets trbs diffbrents du cas d'un puits isolb, caracterises notamment par un dbplacement effectif du seuil d'absorption vers le bleu
.
La thborie permet par ailleurs de prbdire de manibre satisfaisante la forme des spectres de PhotoreflectionAbstract : We present a numerical analysis of the electrooptic effects in superlattices based on a tight-binding analysis
.
These effects are very different from the case of isolated quantum wells (e.g. effective blue shift of the absorption edge ).
Theoretical predictions compare well with Photoreflectance results.
lntroductlon
Due to their possible applications in optoelectronic devices, electric field effects on opticai properties of isolated quantum wells have been extensively studied ,both theoretically and experimentally 11-21
.
By contrast, very few theoretical studies have been reported about these effects in the case of coupled Multiple Quantum Wells systems ,none of which ,to our knowledge .showing comparison whith experimental results 13-41.In the following, we describe a numerical method to determine the absorption coefficient and the refractive index of a superlattice submitted to a transverse electric field from the energy levels and the wavefunctions of the system
.
Typical results are analysed and compared with photoreflectance spectra of a GaAslGaAIAs superlattice
.
Description of the method
We first consider the case of an isolated Quantum Well submitted to an electric field in the direction perpendicular to the layers ( x Axis)
.
The well is supposed to be centered around the x=O position, corresponding to the origin of the electrostatic energy One must then solve, for an electron :
V,(x) corresponds to the Quantum Well potential ; it is equal to 0 in the well and to V,, in the barrier.
The solutions of (1) can be expressed as linear combinations of Airyfunctions . The coefficients of these combinations 1
cw
are determined by the continulty of Y(x) and
7 -
d
.
It must be noted that (1) does not admit strlctly%oun$solutions , since the electron has a non-vanishing probability to tunnel out of the well
.
However, for sufficiently low fields, long-lived bound states can be found ( the "quasi-bound")LEP a member of ths PHILPS Research Organization
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988221
C2-98 JOURNAL
DE
PHYSIQUEstates")
.
The energies of these states are determined as the resonance energies in the probability of transmission of an electron through the structure .For the sake of simplicity, we suppose that (1) only admits one quasi-bound state
.
Its wavefuction is denoted by Y , ( x . F ) and its energy by EJF)Let us now consider a N periods Superlattice (SL) consisting in identical Quantum Wells (QW) centered around the positions x,, with n = 1,
....
NTo find the wavefunctions of the system, we use a quasi-tight binding method, in which we expand the solution in terms of the localized wavefunctions in each well :
The Schrodinger equation leads to the pseudo-eigenvalue equation
where S,, is the overlap integral between wavefunctions located in the nm and mth wells ,and
B.,
is given byin (4) AV'") is the perturbation on the
n*
well due to the other wells in the SL.
Equation (4) is then solved numerically, leading to N eigenvectors.noted
0'
corresponding to N values of the energy E' with I = 1, ..., NThis procedure is successively applied to the electrons and to the holes, which enables to compute the absorption coefficient of the SL as
where C is a constant, hw is the photon energy, E, is the gap of the well, and H(x) is the Heavyside function
.
The subscripts 'e' and 'h' refer respectively to the electron and hole wavefunctions and energies.
The refractive index is computed as the Kramers-Kronig transform of a(hw).Results and discussion
In equation (3). the main effect of the field lies i n the term eFx. : the individual energies of the single well localized states have to be corrected by the electrostatic potential at the center of the well
.
In presence of the field coupling between the wells thus ceases to be resonant leading to a strong localization of the SL wavefunctions in the different wells . At sufficently high fields (such that the potential drop across one period of the SL (eFd), is greater than the SL miniband width (W)) the properties of the SL have to become identical to those of isolated OW. Our results confirm this analysis in agreement with a recent paper 141Energy (eV)
Figure 1 : Computed absorption spectra of a 20 periods 40A x 40A Ga,al,As Superlattice at zero and 100 kVlcm fields .
Fig. 1 shows the absorption spectra for a 20 periods 40A x 40A Ga,AI,As superlattice at zero field and for a 100 kV/cm field
.
It can be seen that the characteristic 'sigmoid' absorption corresponding to the zero-field superlattice evolves towards a single step absorption curve . This illustrates very clearly the above analysis.
The net effect of the field is thus a steepening of the apsorption curve , leading to an effective blue shift of the absorption edge.
We have studied the spectral absorption and refractive index changes induced by growing electric fields : we have computed the difference a(hw, F)
-
a(hw, 0) and its Kramers Kronig transform.
Fig 2 shows typical results.Fig 2 : Computed differential absorption and refractive index (dotted lines) spectra of a 20 periods 20A x 20A Ga.,A!,As superlattice for a) F=lOkV/cm, b) F=50 kV/cm, c)
F =
100 kV1cm.
The Heavyside function in (5) is replaced by a smooth function with a broadening parameter of 10 meV .We can identify three domains of interest :
JOURNAL DE PHYSIQUE
Weak field regime (eFd < < W): (fig 2a) In this regime the major absorption and refractive index changes occur in the vicinity of the rniniband extrema
.
The lineshape is very similar to the third derivative form obtained for>dimensional critical points as it has been described for bulk material by Aspnes 151. This implies that for these weak fields the band structure is hardly affected by the field and the semi-classical approximation is valid
.
Intermediate field regime (eFd < W) : (flg2b) The absorption changes show marked oscillations in the spectral region corresponding to the rniniband
.
The precise lineshape of these oscillations depends upon the value of the electric field and the broadening parameter.
In particular the numbre of peaks does not seem to be related to the field in a simple way.High field reglme (eFd>
w)
: ( f i l 2 c ) The main spectral feature is centered around the center of the miniband ,since the absorbtion at high fields consists mainly i n a unique step at this energy.Photoreflectance results
Photoreflectance spectroscopy consists in measuring the changes in the reflectivity of the sample induced by ab- sorption of a pump light beam
.
The photogenerated carriers screen the built-in fields i n the structure (e.g. surface field ) I61.
It i s thus an indirect way to study the influence of an electric field on the optical constants of the structure . Fig. 3 shows a spectrum obtained for a 25 periods 20A x 20A GaAs/Ga.&!,As superlattice.
The spectrum presents very clear oscillations in the spectral region corresponding to the SL rniniband.
as predicted by the theory in the case of intermediate fields ( here F-
5 x l(rV/cm ).
1.800 1.100 1 .a00 1.900 Energy (eV)
Fig 3 : Experimental Photoreflectance spectrum spectra of a 25 periods 20bi x 20A Ga,,A!,As superlattice
.
Analysis of PR spectra thus allows for the determination of the miniband width and gives an estimation of the built-in fields in the structure . The exact form of the PR signal, however is difficult to reproduce by theory since it consits of an energy dependent linear combination of the changes i n absorption and refractive index due to optical interference effects
.
Inhomogeneous broadening should also be taken into account i n a more sophisticated way : as a matter of fact the lowest states of the miniband are much less sensitive to wells and barriers thickness variatlons than the states near the miniband maximum energy.
These points are currently under investigation in our laboratory.References
111 D A 8. Miller, J.S. Weiner and D.S. Chernla , IEEE J. of Quantum Electronics, QE-22, 9 1816,(1986) 121 G.D. Sanders and K.K. Bajaj , Phys. rev. B , 35 (5),2308,(1987)
131 P.A. Mc llroy , J. Appl. Phys. , 59 (10),3532,(1986)
I41 J Bleuse, G. Bastard and P. Voisin , Phys Rev. Lett. , 60 (3),220,(1988) 15/ D. Aspnes , Surf. Science 37 418.(1973)
I61 B.V. Shanabrook, O.J. Glembocki and W.T. Beard , Phys. Rev. B 35 2540, 1987
