HAL Id: jpa-00246994
https://hal.archives-ouvertes.fr/jpa-00246994
Submitted on 1 Jan 1994
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Rôle of dielectric effects in the red-green switching of porous silicon luminescence
J.-N. Chazalviel, F. Ozanam, V. Dubin
To cite this version:
J.-N. Chazalviel, F. Ozanam, V. Dubin. Rôle of dielectric effects in the red-green switching of porous silicon luminescence. Journal de Physique I, EDP Sciences, 1994, 4 (9), pp.1325-1339.
�10.1051/jp1:1994191�. �jpa-00246994�
Classification Physics Absii.acis
78.55D 71.50 77.30
Rôle of dielectric effects in the red-green switching of porous
silicon luminescence
J.-N. Chazalviel, F. Ozanam and V. M. Dubin (*)
Laboratoire de Physique de la Matière Condensée (**), CNRS-Ecole Polytechnique, 91128 Palaiseau, France
(Rec.eived 10 Maic.h 1994, ac.c.epied in final foi-ni 30 May 1994)
Résumé. Le piégeage d'un porteur par une impureté ionisée dans le silicium poreux peut devenir inefficace lorsque le matériau est plongé dans un milieu de grande constante diélectrique tel qu'un électrolyte aqueux. Cet effet est estimé pour une géométrie de filaments cylindriques de silicium, les deux milieux étant modélisés par des constantes diélectfiques indépendantes du vecteur d'onde.
Le potentiel image de l'électron est pris en considération, et la dépendance en fréquence de la
constante diélectrique du milieu extérieur est traitée de manière simple. Les résultats démontrent
que les états liés sur l'impureté ne sont pas accessibles en présence de l'électrolyte, simplement en
raison de la relaxation diélectrique de ce milieu. Ce résultat peut s'appliquer à différentes sortes
d'états électroniques localisés, en particulier ceux responsables de la luminescence rouge du silicium poreux sec. Ceci foumit une explication plausible du changement de la luminescence du rouge au vert lorsque le silicium poreux est mouillé, et suggère que l'utilisation de milieux
extérieurs de constantes diélectriques intermédiaires devrait permettre d'observer une transition
progressive entre les luminescences rouge et verte. L'observation de la luminescence du silicium poreux dans des solvants de diverses constantes diélectriques fournit un test préliminaire de cette
prédiction.
Abstract. Trappmg of a carrier ai an iomzed impufity in porous silicon may be significantly hindered when the material is embedded in a high-dielectfic-constant medium such as an aqueous
electrolyte. This effect is estimated for a geometry of cylindrical silicon wires, and by modeling the two media with wavevector-independent dielectric constants. The self-image potential of the electron is taken into account, and the frequency dependence of the outer dielectfic constant is treated in a simple manner. The results demonstrate that the impufity states are not accessible in the presence of the electrolyte, just due to the dielectnc relaxation of the embedding medium. This result may apply to different kinds of locahzed electronic states, including those responsible for the red luminescence in dry porous silicon. This provides a plausible explanation for the red to green
switching of the luminescence when the porous silicon is wet and suggests that using embedding
media of interrnediate dielectric constants should allow one to observe a progressive transition between red and green luminescence. Observation of porous silicon luminescence in solvents of
various dielectric constants provides a prelimmary test of this prediction.
j*) On leave from Minsk Radioengineefing Institute, Minsk, Republic of Belarus.
(**) URA 1254 du CNRS.
1. Introduction.
The photoluminescence (PL) of porous silicon (PS) has been reported to be reversibly
switchable between red and green upon successive rinses in ethanoic HF and propanol il ], or
upon successive cycles in acetic HF solution and air [2]. Furthermore, we have observed that PS, obtained from dark anodization of p-Si and kept in its HF anodization electrolyte, never
becomes red-luminescent, unless it is removed from HF and dried [3]. fn-situ photomodulated
infrared spectroscopy has shown that the weak green luminescence observed in HF-wetted PS
is associated with a large density of photoinduced free carriers, whereas the red luminescence of dried PS appears associated with some kind of localized states, characterized by an infrared
absorption in the 0.15 to 0.25 eV energy range [3]. These infrared observations may be taken
as a support in favor of the hypothesis of Koch et ai. [4] that the red luminescence is associated with a recombination through radiative surface states. However, at least for the materials that
we have studied, there is no infrared evidence for a change in surface chemistry when PS is removed from HF. Rather, the surface unavoidably appears covered with SiH bonds, 1-e-, no
evidence for oxidation is found, at least on an hour time scale. Moreover, when dried PS is reimmersed into aqueous HF, the red luminescence survives because of non-wetting of the surface [3]. Yet the chemistry of a silicon surface in the presence of saturated HF vapor is
expected to be rather similar to that of an HF-wetted surface. Therefore, the surface-state
hypothesis requires that very minor surface modifications would lead to generating an amount of surface states sufficient for governing the recombination. Although this is still possible, we
will propose here an alternate explanation, based upon simple electrostatic effects.
Localized states may consist of surface states, but also of excitons confined by the irregular geometry of PS, or alternately of carriers or excitons bound to an impurity. As soon as some
Coulombic interaction gets involved, one might expect dielectnc effects to play an important rôle. For example, the electnc field produced by an iomzed impunty in a small silicon particle
is dramatically affected by the dielectric constant of the embedding medium, as it has already
been noticed in the literature [5-7]. If the embedding medium is taken with a dielectnc constant lower than that of silicon, it results in significant increase of the binding energy of a carrier
trapped on the impurity potential [6, 7]. Here, in constrast to calculations in the literature, we will consider the geometry of cylindrical silicon wires rather than silicon spheres
-, which may be more suitable for the case of unoxidized porous silicon. In section 2, we will show that
an embedding medium with a high dielectnc constant leads to a screening of the field of the ionized impunty, and to a reduction of the binding energy of a trapped carrier, which can be calculated in a simple fashion. In section 3, we will include the self-image potential of the carrier, which brings a significant correction to the preceding result. This will lead us to a
prediction for recombination energies of, e-g-, a free electron with a hole trapped on an
acceptor site. Next, we will consider the time dependence of the dielectric constant of the
electrolyte. This is a mandatory step, as far little explored, and which comphcates the picture significantly. Finally, in section 4, we will discuss the results obtained and report on
preliminary measurements of PS luminescence in solvents of vanous dielectric constants, showing the progressive transition between red and green luminescence. We will keep in mind
that the case of a carrier trapped on an ionized impunty is taken here as a model calculation,
but that rather similar effects are to be expected from the more complicated though possibly
more relevant case of a confined exciton.
2. Impurity potential m the presence of an embedding medium.
We mortel porous silicon as a set of infinite cylindrical wires of radius R, of dielectnc constant
~ embedded in a medium of dielectnc constant F~. (Within the framework of the constant-
F, approximation, the small size of the silicon wire implies an F, value lower than that of bulk silicon [6]). Using cylindrical coordinates (1, 9, z), and introducing the relative dielectric
constant of the embedding medium, F
= F/F,, the electnc potential 4~ generated by a point charge e at the origin (1,e., on the axis of the cylinder) must fulfill Laplace's equation
A4~
=
0 (everywhere except at the origin and cylinder surface), and the following boundary
conditions
4i~ ~
~ for I,z-0, (1)
4wF, Fo r +z
~~
=
F~~ (2)
ôr
,
ôi
e
where the and e indices refer to the inner and outer limit for r
- R.
The solution has been computed numerically by separating out the singular part, 1e., q~ ~ F (I', Z) + 1(3)
4 WFi F0 ^~
F (1, = was determined by using the relaxation method with an explicit recursion scheme [8].
The domain was taken as a square box (0
~ z ~ L, 0
~ i ~ L with a 160 x 160 square mesh, where L was taken large enough (L/R
=
10 so that the boundary conditions at the far edges are
not too cntical. The boundary conditions at the edges 1= 0 and z
= 0 were taken as
ôF/ôz [, o = ôF/ôr
= o =
0, expressing the symmetry of the problem. The boundary
condition at the far
r edge was taken as ~P(L, z)
=
0 (metallic wall) and two alternate conditions at the far z edge were used 1) W (r, L) = 0 (metallic wall) and ii) ôW/ôz
~ ~ = 0.
Condition ii) is equivalent to the assumption that there is a periodic array of charges
e along the cylinder axis, of penod 2 L. Condition i) amounts to taking an array of charges of altemate sign + e at 0, e at 2 L and 2 L, + e at 4 L and 4 L, ...). The results were found to depend slightly upon which boundary condition was taken, especially for the lowest values of F. The arithmetic average of these two solutions (1.e., assummg a penodic array of charges
e, of period 4L) was taken as the final solution.
The resulting maps of the potential are shown in figure1, for two typical values of
F. For
F =
0.1 (porous silicon embedded in a medium of smaller dielectric constant), the electnc field tends to be channelled along the cylinder axis, and the obtained potential is much larger than it would be in an mfinite medium of dielectric constant F,. This point has already been mentioned by others in a different geometry [6, 7]. On the other hand, for the opposite
case of an external medium of higher dielectric constant (here F
=
10), the potential appears to be significantly lowered. Figure lc shows a plot of the potential along the cylinder axis, for
vanous values of F. The screenmg of the potential for the largest values of F clearly appears in
this plot.
Next we have considered the fundamental and first excited state of an electron bound by
such a potential (1,e., donor impunty ; the problem of a hole trapped on a e charge (acceptor)
would be exactly similar). In a first approach, we have just solved the eigenstate problem
HV/
=
EV/ in the effective mass approximation, usmg infinite-wall boundary conditions
(P (R, z)
=
0) and as the potential either 0 (no charge) or V,~~ =
eW (electron bound on a fixed positive « impunty » charge). We have used a relaxation method similar to that exposed
in [9] with further incorporation of the potential. The fundamental (resp. first excited) state was
obtained by looking for a solution V/(r, z even (resp. odd) with respect to the z variable. The calculations were performed using the natural units of the problem (effective Bohr radius and
(a) (b)
r/lt
4
(c)
g
5 /
°
o 5 io wR
Fig. l. Maps of the electrostatic potential ~ jr, z) for a dielectric cylinder of axis z and radius R with a charge +e ai the origin; the relative dielectric constant of the embedding medium IF
= F~/F,) is F 0.1 la) and F 10 16) : the equipotential hnes are labelled with the value of the
potential, expressed in units of [em wF, FOR]. (c) plots of the potential along the cylinder axis for
F =
0.1 F
=
and F 10.
effective Hartree of the medium, here of the order of 15 À and 0.1eV respectively). The results for different values of R and F are represented in figure 2. In the absence of potential,
the increase in energy is just associated with quantum-confinement effects, and coincides with
the analytical prediction E~ = (~h~/2 m* R~, with (
=
2.4048.. (first zero of the modified Bessel function Jo) il 0]. When the potential is taken into account, the electron ground state is
seen first to decrease as the cylinder radius is decreased, then to increase for smaller values of R il1] However the binding energy taken with respect to the shifted level of the extended
states always mcreases upon decreasing R. The quantitative values are seen to depend quite
sensitively upon F. For example, quantum confinement, which is the only effect for
e = 1, increases the binding energy of the fundamental by a factor of 3 for R = 1. This factor
becomes as high as 11 for F
= 0,1, and is only 1.7 for e
=
10. The effect of s on the first
excited state is still more dramatic: the increase in its binding energy, for the same value R
= 1, is about 30 for F
=
0,1 and the state merges with the extended state continuum for
F = 10.
Notice that in principle, the value of F, should not be taken as a constant. Even within the framework of this simple approximation, it would be better to assume that F; is a function of R [6]. For simplicity, we have considered this effect neither here nor m the following. The
information in figure 2 is yet sufficient for extracting the energy values for any F, even if it
depends upon R.
Interestingly, all the curves represented in figure 2 can be fitted very accurately by
expressions of the form A B/R + C/R~. The physical meamng of C is obviously related to
quantum confinement effects, whereas B anses from Coulombic effects, namely, the potential generated by the polarization charges on the cylinder boundary. At this point, it becomes clear that the polarization charges generated by the electron itself (self-image effects) should also be
taken into account. This is what we have clone in a second step.
12 12
(a) 16)
8 8
~
3.
~ ~
~ E
0
~ ~
l 10 10
CYLWDERRADIUSR CYLfIiDERRADIIJSR
Fig. 2. Electromc energy levels (unit
= effective Hartree) as a function of the cylinder radius (unit
=
effective Bohr radius), for vafious values of the relative dielectric constant F p~le,. The heavy top curve
represents the bottom of the conduction band E~, which is rised by quantum confinement effects (a) ground state of a hydrogenic impurity Eo (curves from bottom to top F 0.1, 0.3, 1.0, 3.0, 10.0), 16) first excited state Ej jsame values of F). The self-image potential of the electron has not been taken mto
account.
3. Taking the self-image potential into account.
We first restrict our scope to the case of a surrounding medium with a dielectric constant that would keep the same value up to the high frequencies relevant for electron motion. This condition will obviously be fulfilled by vacuum or an oxide, but not by electrolytes whose case
will have to be considered in a later step.
3.1 CASE OF A FREQUENCY-INDEPENDENT F. If F keeps its value up to the highest frequencies of the problem, the electron expenences the potential of the polarization charges
that it generates at the cylinder surface. For example, for F
~ l, these charges will be negative,
and the potential will be repulsive. In pnnciple, the determination of the self-image potential
requires taking the electron at an arbitrary position inside the cylinder, then calculating the electric potential map W(r, 9, z), where 9 is the azimuthal angle in cylindrical coordinates, (this cannot in general be reduced to a 2-D problem), next deducing the interface polarization charges
~~~~ ~~ ~°Î Î Î
Î'Î ~
~~~
and finally calculating the interaction potential by summing over the cylinder surface eOE(9, Z')
R d9 dz'. 15)
~'~~~~' ~
4 wfo N/(r R
)~ + (z z')~
The self-image potential is actually V,~
= V,~~/2. The 1/2 factor anses because the dielectric
polanzation is due to the electron itself. It accounts for the progressive buildup of the
polarization [building the electron charge at location jr, z) or, equivalently, bringing it
-e
from mfinity costs an energy (V,~le~) q dq
= V,~/2]. We have not actually performed
o
this calculation for every position of the electron. For an electron on the cylinder axis,
ce (9, =) becomes independent of 9, and the result can be straightforwardly deduced from the
Coulomb calculation of section 2, yielding V,~(r =0)= TIR, where T is a numencal
constant, obtainable for any given value of F. On the other hand, the asymptotic expression of
V,~ is known for i
i R (flat interface), i e.,
p p ~2
~'~ Î
+
Î
16 wF, Fo(R rl' ~~~
Since the wavefunctions vanish at the cylinder surface, the image corrections are thought to be
essentially governed by TIR. We have then simply interpolated between the two limits, 1e.,
we have taken
F, Fe e~ R Î Y
~'~ (~)
Fj + Fe 8 wF, Fo R~ -1~ R ~ R
The eigenenergies were then calculated after incorporating V,~ into the eigenenergy
equation. The results for e
= 0.2 are shown in figure 3. It is seen that the extended states and impunty states are modified by about the same contribution, which is roughly equal to
TIR. This will be a point of importance when considenng the shift in luminescence energy.
Assuming a perfect symmetry between conduction band and valence band, and callmg E~ (resp. Eo) the above calculated band energy (resp. impunty-ground-state energy), the band-
to-band and impunty-to-band recombination energies should be increased by respective
amounts 2 E~(R) and E~(R) + Eo(R) Eo(co). These quantities are represented in figure 4,
Interestingly, the band-to-band recombination energy is very sensitive to the image-potential
contribution. This had to be expected, since both carriers undergo a self-image-potential repulsion. This brings in a blue-shift contribution, proportional to 1/R, which becomes
dominant at the highest values of R. On the other hand, the impunty-to-band recombination is
with
self-image potential
10 ~'~°'~~
Î
5 uà
' '
' '
'
Ù
' "-_ _--'-"
i io
CYLII4DER RADIUS R
Fig. 3.-Electronic energy levels, in effective Hartrees, (from bottom to top, ground state
Eo, first excited state E~, and conduction band E~), for p
= 0.2, as a function of cylinder radius lin effective Bohr umts). The solid (resp. dashed) curves are for a calculation induding (resp. excluding) the
self-image potential of the electron.
à~s bond-to-band
vÉ
il , ',
~/ ,
o ,
w ,
+ ',
il
,
~/ ,
u ,
w ,
',
~- ',
~ 0,1 iJnpudty-to-bond ",
P1 ce
0.01
10 CYLINI>ER RADIUS R
Fig. 4. Predicted energy shift~ of the band-to-band energy [2 E~(R)] for F
=
0.2 as a function of
cylinder radius (solid curve), and of the impurity-to-opposite-band energy [E~(R) + E~(R) Eo(oc), dashed curve], representing the energy shifts of two possible luminescence pathways. Here the bands
were taken with identical effective masses. Units
are effective Bohr radius and Hartree.
