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BINDING ENERGIES AND CAPTURE OF ELECTRONS, HOLES AND EXCITONS ON
QUANTUM WELL INTERFACE DEFECTS
F. Gerbier, Gérald Bastard
To cite this version:
F. Gerbier, Gérald Bastard. BINDING ENERGIES AND CAPTURE OF ELECTRONS, HOLES
AND EXCITONS ON QUANTUM WELL INTERFACE DEFECTS. Journal de Physique Colloques,
1987, 48 (C5), pp.C5-219-C5-222. �10.1051/jphyscol:1987545�. �jpa-00226749�
BINDING ENERGIES AND CAPTURE O F ELECTRONS, HOLES AND EXCITONS ON QUANTUM WELL INTERFACE DEFECTS
F. GERBIER and G. BASTARD
Groupe de Physique des Solides de 1'Ecole Normale Superieure, 24, Rue Lhomond, F-75005 Paris, France
Nous avons calcule les energies de liaison des electrons, des trous et des excitons sur un defaut d'interface de puits quantique, modelise par une protuberance cylindrique. L'effet sur ces e n ~ r g i e s de I'application d'un champ electrique a ete etudie. Par ailleurs, nous avons examine les temps de capture d'un porteur ou d'un exciton par une distribution de defauts,
lorsque la capture est assistee par I'emission d'un phonon acoustique.
We have calculated the binding energies of electrons, holes and excitons on a quantum well interface defect, modelled by a cylindrical protusion. The effect of an electric field on these interface bound states has been studied. Besides, we have examined the capture time of a carrier or an exciton by uncorrelated defects when the capture is mediated by acoustical phonon emission.
I. INTRODUCTION
If quantum wells are nowadays high-quality heterostructures, interfaces are not yet plane. For a GaAs/Ga(AI)As QW, the interface defects are clearly seen as inhomogenous line broadening [ I ] or Stokes shift between photoluminescence and excitation spectroscopy [2] at low temperature. In this paper, we report on the effects of an attractive interface defect in GaAs/Gal _,AIxAs ( x-30% ) QW's. The defect is modelled by a cylindrical protusion of the well-acting material into the barrier-acting one and is caracterized by its radius and its depth. The binding energies on this defect will be studied, then the capture time by a defect distribution will be examined.
II. BINDING ENERGIES
To carry out the calculation of electron, hole and exciton binding energies, we have assumed that the movement in the layer plane ( noted I ) and the movement along the growth axis ( noted z ) are separable ( strong
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987545
C5-220 JOURNAL DE PHYSIQUE
quantification along the growth axis ) and that the movement along the growth axis is not modified by the defect. Finally, the binding energies are calculated by a variational method.
For the electron and the hole, the variational functions are chosen as:
9 = X, ( 2 ) F(
r,
) with F( r, ) =Nh
exp(-
r, 1 h )where ~ , ( z ) is the ground eigenstate of the one dimensional potential well, r, has its origin at the defect center,
Nh
is a normalisation constant and h the variational parameter which mesure the mean distance between the bound carrier and the defect center.For the exciton, the movement in the layer plan is decomposed into an internal motion and a mass center motion. First, the internal motion is determined for an exciton in a perfect QW [3]. Afterwards, the mass center is bound to the interface defect. The variational functions are then chosen as :
'Pexc=
X I
(2 h) X,l(ze) F( R,, rl ) with F( Rls r l )=Ng N h
ex~(-r,/5 ) e x ~ ( - R l J a ) where r, is the relative position vector between the electron and the hole,R,
the mass center position vector,5
and h the variational parameters.For all our calculations, the defect will be taken with a radius of 400
A
and with a depth of 5.66A
which is equivalent to two GaAs monolayers.The parameters used in the calculations are : Ve=243 meV, me=0.067m0, Vh=131 mev, mhl=(gl +g2)-lm0 , mhz= (gl -2g2)-1m0 where g1= 6.85 and g2=2.1
.
In Fig.1 are reported the binding energies of electron, hole and exciton versus the quantum well thickness. We observe i) that the binding energies increase when the QW thickness decreases (until the QW is so thin that the carrier starts to heavily leak in the barrier), ii) that due to its heavier masses the hole binding energy is smaller than the electron one, and iii) that the binding energies are especially large for narrow QW's ( L lower than about 70
A
). This last point confirms the importance of interface quality for narrow QW.
When a constant electric field is applied parallel to the growth axis, the binding energies are obtained by replacing X, in the variational functions by a ground-state wave function proposed by Brum et
a/.
I41
and which describes the displacement of the carrier along or opposite to the electric field.
In Fig.2 , the electric field dependence of binding energies is shown, for a QW thickness of 100
A.
For an electric field oriented from left toexciton instead, the energies increase whichever the location of the defect and they at least double when the electric field increases from 0 to 100 kV/m.
Ill CAPTURE
We have calculated the capture time of a delocalized particle in a QW by uncorrelated interface defects, assisted by the emission of one acoustical phonon. From the electron-phonon interaction Hamiltonien, written in the deformation potential approximation, and for acoustical phonons, we have used the Fermi golden rule between a delocalized initial state caracterized by its in plane wave vector K i and a final bound state, corresponding to the ground level of the defect. Then, we have multiplied the result by the number of defects on the interface.
In Fig.3 are reported the capture times of excitons, electrons and holes versus the quantum well thickness and for an initial wave vector equal to zero.
In Fig.4, the capture time of exciton is given for three Q W thicknesses, versus the initial wave vector.
All the curves have been plotted at T=5K, for a defect areal concentration Ndef=l
o1
O C ~ - ~ , with the deformation potentials ce=8 eV and ch=3 ev, and a sound velocity v=3.7 103 m/s.Acknowledgements
This work has been supported in part by the GRECO "Experimentations Numbriques
".
REFERENCES
[I] C. Weisbuch, R. C. Miller, R. Dingle, A. C. Gossard and W. Weigmann, Solid State Commun.
z,
219 ( 1981 )[2] G. Bastard, C. Delalande, M. H. Meynardier, P. M. Frijlink and M. Voos, Phys. Rev. I3 29,7042 ( 1984 )
[3] J. A. Brum and G. Bastard, Phys. Rev. B
a,
3893 ( 1985 )[4] J. A. Brum and G. Bastard, J. Phys. C : Solid State Phys.
18,
L789-L794 ( 1985 )C5-222 JOURNAL DE PHYSIQUE
Fig.1 : Binding energies of exciton (solid Fig.2 : Electric field dependence of exciton line), electron (dashed line) and hole (solid line). electron(dashed line) and hole (dotted line) on interface defect versus (dotted line) binding energies for a well the quantum well thickness. thickness of1 00 A. The defect can be on left
interface (L) or on right interface (R) for a field pointing from left to right.
-
-
Fig9 : The decimal logarithm of the capture times (in ns) of an exciton (solid line), an l , , , , I , n u , .---q---l- a a n 4 .J electron (dashed line) and a hole (dotted line) 50 100-
150 200 is plotted versus the well thickness L, for~ ( h )
initial wave vector equal to zero.Fig.4: The decimal logarithm of the exciton capture times (in ns) is plotted versus the exciton wave 0.005 0.01 vector Ki for three well thicknesses :
0
