Polarizability of D+X complex in bulk semiconductors
The electric polarizability α of ionized-donor-bound exciton D+X in bulk semiconductor is calculated for all values of the effective electron-to-hole mass ratio σ included in the range of stability (σ<σχ). The calculation is performed within the variational method by using 56-term wave function....
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2006
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irk-123456789-1216392017-06-16T03:04:03Z Polarizability of D+X complex in bulk semiconductors Katih, M. Diouri, J. El Haddad, A. The electric polarizability α of ionized-donor-bound exciton D+X in bulk semiconductor is calculated for all values of the effective electron-to-hole mass ratio σ included in the range of stability (σ<σχ). The calculation is performed within the variational method by using 56-term wave function. An asymptotic behavior of α in the vicinity of the critical value σc is deduced. We have also calculated the limiting value σ for which the polarizability equals that of D− system. 2006 Article Polarizability of D+X complex in bulk semiconductors / M. Katih, J. Diouri, A. El Haddad // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 4. — С. 7-11. — Бібліогр.: 16 назв. — англ. 1560-8034 PACS 71.35.-y http://dspace.nbuv.gov.ua/handle/123456789/121639 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The electric polarizability α of ionized-donor-bound exciton D+X in bulk semiconductor is calculated for all values of the effective electron-to-hole mass ratio σ included in the range of stability (σ<σχ). The calculation is performed within the variational method by using 56-term wave function. An asymptotic behavior of α in the vicinity of the critical value σc is deduced. We have also calculated the limiting value σ for which the polarizability equals that of D− system. |
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Katih, M. Diouri, J. El Haddad, A. |
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Katih, M. Diouri, J. El Haddad, A. Polarizability of D+X complex in bulk semiconductors Semiconductor Physics Quantum Electronics & Optoelectronics |
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Katih, M. Diouri, J. El Haddad, A. |
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Katih, M. |
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Polarizability of D+X complex in bulk semiconductors |
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Polarizability of D+X complex in bulk semiconductors |
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Polarizability of D+X complex in bulk semiconductors |
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Polarizability of D+X complex in bulk semiconductors |
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Polarizability of D+X complex in bulk semiconductors |
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polarizability of d+x complex in bulk semiconductors |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Polarizability of D+X complex in bulk semiconductors / M. Katih, J. Diouri, A. El Haddad // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 4. — С. 7-11. — Бібліогр.: 16 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT katihm polarizabilityofdxcomplexinbulksemiconductors AT diourij polarizabilityofdxcomplexinbulksemiconductors AT elhaddada polarizabilityofdxcomplexinbulksemiconductors |
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2025-07-08T20:15:49Z |
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2025-07-08T20:15:49Z |
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1837111180818120704 |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 7-11.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
7
PACS 71.35.-y
Polarizability of D+X complex in bulk semiconductors
M. Katih, J. Diouri and A. El Haddad*
Faculté des sciences, Département de Physique, B.P. 2121, Tétouan, Morocco
*Correspondent address: a_haddad01@yahoo.fr
Abstract. The electric polarizability α of ionized-donor-bound exciton D+X in bulk
semiconductor is calculated for all values of the effective electron-to-hole mass ratio σ
included in the range of stability (σ<σχ). The calculation is performed within the
variational method by using 56-term wave function. An asymptotic behavior of α in the
vicinity of the critical value σc is deduced. We have also calculated the limiting value σ
for which the polarizability equals that of D− system.
Keywords: exciton, polarizability, wave function, variational method.
Manuscript received 02.10.06; accepted for publication 23.10.06.
1. Introduction
The existence of ionized-donor-bound exciton in
semiconductors was first predicted by Lampert [1] and
confirmed later by experimental works [2-4]. For direct
gap semiconductors with isotropic bands, the calculation
of the ground state energy of such a complex is reduced
in the effective mass approximation, to solve the
Hamiltonian of three bodies system formed by one
electron-hole pair (e,h) trapped by one donor centre D+.
This system is labelled D+X. It is clear that when the
energy
XD
E + is less than the neutral donor energy
0D
E , the excitonic complex forms and may affect, to
some extent, the optical spectra of the host material. The
stability of such a complex depends on the electron-hole
mass ratio he mm /=σ . Several works have been
devoted to this question [5-9]. Particularly, Skettrup et
al. [8] have shown that the D+X complex stabilizes for
all σ values lying lower than a critical point
426.0=cσ . Recently, dos Santos et al. [10] have
reconsidered again the question and calculated σc by an
original adiabatic approach using hyperspherical
coordinates and obtained 431.0=cσ . In the particular
case of 2D system, Stauffer and Stébé [9] have shown
that the range of stability extends to 88.02 =D
cσ .
However, if one reviews the literature in the area, one is
surprised by the insufficiency of works carrying on the
effect of the electric field on D+X complex, in particular,
the calculation of polarizability. To our knowledge, the
unique work dealing with this question is that of
Essaoudi et al. [11] in which the specific case of
GaAs/Ga1−xAlxAs quantum well with the electric field
applied parallel to the growth direction is studied. It has
been shown in this work that the D+X complex is
sensitive to the action of the field only for well widths
higher than 10 nm. The numerical method used in this
calculation cannot be generalized to the bulk limiting
case because of the axial character of the used trial
function inherent in the specific case of the 2D
symmetry.
Let’s recall that in a previous paper, we have
calculated the polarizability of −X and +
2X complexes
[12, 13]. But for these systems, the range of stability
covers all σ-values whereas for D+X, the range of
stability is limited. This is why we were interested in the
present study. In what follows we calculate the electric
polarizability of D+X complex in the framework of the
variational method by using a trial function including 56
terms which gives an accurate numerical result.
This paper is organized as follows: in section II we
outline our method to determine the polarizability of
D+X, in section III we explain our numerical method
using a 56 terms trial wave function. Finally, in the last
section we discuss our results.
2. The model
In the effective mass approximation, the Hamiltonian of
an ionized donor bound to an exciton in the presence of
a constant electric field F directed along to the z-axis can
be written as:
H = H0 + W , (1)
where H0 is given by
H0 = T + V . (2)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 7-11.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
8
Here, T is the kinetic energy and V is the Coulomb
interaction between the particles of the system
heT Δ−Δ−=
22
1 σ
, (3)
ehhe rrr
V
111
−+−= , (4)
and W is the electric energy operator
FzzW he )( −= . (5)
Note that in the previous expressions, we have used
the atomic units (a.u.); 22 / ema eD hε= as the unit of
length, DD aeE ε/2= as the unit of energy and
DD eaEF /0 = as the unit of electric field strength. ε is
an appropriate dielectric constant taking into account
possible polarization effects. The parameter
he mm /=σ defines the electron-to-hole effective mass
ratio. re and rh are the distances from the ionized donor
to the electron and the hole, respectively, while reh is the
distance between the electron and the hole. ze and zh
denote the coordinates of the electron and the hole along
the electric field direction, respectively. eΔ and hΔ are
the Laplacian operators with respect to the hole and
electron coordinates.
In order to calculate the polarizability α of the
system, we develop the wave function Ψ and the energy
E of the system in power series with respect to the
electric field intensity. So we have
...2
210 +Ψ+Ψ+Ψ=Ψ FF , (6)
...2
210 +++=
ΨΨ
ΨΨ
= FEFEE
H
E , (7)
where Ψ0, Ψ1 and Ψ2 are F-independent functions, Ψ0
and E0 being the wave function and the energy of D+X in
the absence of the field. Substituting H and Ψ in
equation (7) and taking into account the spherical
symmetry of the ground state in absence of electric field
(F=0), we obtain:
⎪⎩
⎪
⎨
⎧
−=
=
α
2
1
0
2
1
E
E
(8)
where the polarizability α is given by
.8
)(422
2
00
2
10
0
00
10110101
ΨΨ
ΨΨ
+
+
ΨΨ
Ψ−Ψ−ΨΨ+ΨΨ−
=
E
zzEH he
α
(9)
One may ensure, as established in the appendix,
that this entity is essentially positive for all σ values,
what proves the stability of the complex for any weak
electric field. Furthermore, equation (9) shows that the
polarizability α depends only on E0, Ψ0 and Ψ1, the
terms including Ψ2 simplify. E0 and Ψ0 are the well-
known energy and wave function of the ground state of
D+X in absence of electric field which are determined
variationally by several authors [5-9]. As we can remark,
the determination of the polarizability requires the
knowledge of the wave function part Ψ1. On the other
hand, since we are interested with the calculation of the
polarizability, we consider that the electric field is
sufficiently low, so, we can restrict the development of
the energy to its quadratic form:
E = E0 + E2F2. (10)
Then it is convenient to use the variational method
for calculating the energy E of the ground state of the
system. With account of the symmetry of the problem,
the trial function Ψ is chosen in the following way [12]:
),,(),,,,(1 ehheheehhe rrrfzzzrrr =Ψ , (11)
where
z = ze – zh , (12)
f is a function that contains the variational parameters.
Hence,
ΨΨ
ΨΨ
=
H
E min . (13)
Such a choice allows considerable simplifications.
First, we can ensure that the integral 10 ΨΨ
vanishes. In addition, we may establish the following
relations whatever the choice of the function f(re,rh,reh):
,)1()(
)()(
0
00
f
z
fzHf
zz
fHzzfzzH
eh
hehe
∂
∂
+−=
∂
∂
−
∂
∂
+
+−=−
σσ
(14)
fHrffHzf eh 0
2
0
2
3
1
= , (15a)
,
3
1
.
3
1
f
r
rf
frff
z
zf
eh
eh
eheh
∂
∂
=
=∇=
∂
∂ rr
(15b)
frffzf eh
22
3
1
= , (15c)
frfz eh
2
0
2
0 3
1
Ψ=Ψ . (15d)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 7-11.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
9
In these conditions, the expected value of E2 is
written as:
00
2
0
3
2
min
ΨΨ
Ψ+
=
frfGf
E
eh
, (16)
where the operator G is given by
eh
eheheh
r
rrEHrG
∂
∂
+−−= )1(2
00
2 σ . (17)
The expression (16) shows that the calculation of
E2 involves only the distances re, rh, reh.
Next we use for Ψ0 the expression given by
Stauffer and Stébé [9] :
∑ −=Ψ ++
nml
nmlnml
lmn
ks
tuskCuts
,,
0 )
2
exp(),,( . (18)
The exponents l, m, n are positive or zero integers
and the elliptical coordinates s, t, u are given by
s = re + reh, t = re – reh, u = rh . (19)
The scaling factor k and the linear coefficients Clmn
are determined variationaly for each σ value in the range
of stability. We choose the function f in the following
form:
∑ −= ++
nml
nmlnml
lmnehhe
ks
tuskArrrf
,,
)
2
exp(),,( , (20)
where Almn are the linear variational parameters.
Following the Hasse variational method [15], we use the
values of k and Clmn that we obtain by minimizing the
mean values of the Hamiltonian H0 in the absence of
electric field. The variational parameters Almn are
obtained by minimising the following expression (21)
obtained after substituting equations (18) and (20) into
(16).
CSCk
ARCAGA
E ~~~
3
~~~
2
~~~
min
22 +
++ +
= , (21)
where A
~
( C
~
) denotes the column matrix of the
coefficients Almn(Clmn) and +A
~
( +C
~
) its transposed
matrix. G
~
, R
~
and S
~
are the squared matrices of the
coefficients defined by:
,'''))1(
)((
2
0
22'''
nml
r
rk
EkVTkrlmnG
eh
eh
eh
nml
lmn
∂
∂
+−
−−+=
σ
(22a)
'''2''' nmlrlmnR eh
nml
lmn = , (22b)
'''''' nmllmnS nml
lmn = . (22c)
The basic functions lmn are given by
)
2
exp(
s
tuslmn nml −= . (23)
Equalling to zero the derivative of the expression
involved in equation (21) with respect to A
~
, we find the
following secular equation for Almn,
CRAGG
~~~
)
~~
(
2
1
−=++ , (24)
which yields
CSC
ARC
k
~~~
~~~
3
2 0
2 +
+
−
=α , (25)
where 0
~
A is the solution of the equation (24).
3. Numerical steps
The calculation of the scaling factor k, the coefficients
Clmn and the energy E0 is derived from the solution of the
generalized eigenvalue problem [9]
CQkCP
~~~~
= (26)
with
STQnmlVlmnP nml
lmn
~~~
and'''''' β+=−= , (27)
where
CSC
CTC
nmlTlmnT nml
lmn ~~~
~~~
and''''''
+
+
== β . (28)
Starting from 4/)2/1(0 σβ += that corresponds
to the asymptotic behavior, we calculate the upper
eigenvalue k, and then we deduce the corresponding
vector C
~
which gives the next value of β and so on until
the desired convergence on β, k and C
~
. Consequently,
the energy E0 is deduced from the relation
2
0 kE β−= [9]. Let’s note in passing that the solution of
equation (26) requires to solve the eigenvalue problem
of the real symmetric matrix 2/12/1 ~~~~ −−= QPQM . This
calculation is performed by using the Jacobi numerical
method [14]. The system of linear equations (24) is
solved numerically by using the LU-decomposition
method [14]. Practically, we have limited the
development of the functions Ψ0 and Ψ1 to 56 terms
corresponding to the condition 5≤++ nml . Within this
approximation, we obtain a rather good value of the
critical mass ratio σc= 0.367.
4. Results
The variations of the polarizability of D+X versus the
mass ratio σ is presented in Fig. 1 (solid line). It is seen
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 7-11.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
10
that the polarizability increases with increasing σ up to
the critical value σc in the vicinity of which an
asymptotic behaviour is observed. Such a result may be
interpreted physically by the weak bonding of the hole to
the neutral donor in this point. Indeed, it is well known
that the binding energy of D+X decreases with increasing
σ in the absence of the electric field [8]. In the presence
of a weak field, the hole is removed far from the electron
along the direction of the field. This behaviour is
confirmed by the variations of the electron polarizability
(αe) and the hole polarizability (αh) defined by 〈ze,h〉 =
αe,hF for weak electric field. This result is illustrated in
Fig. 1 (dashed lines). The polarizability of D+X is then
given by α = αh − αe. This asymptotic trend attests the
consistency of our method because it is compatible with
physics of such systems.
After calculating the polarizability, we have
deduced the binding energy of the complex in the
presence of a weak electric field, which is defined as
W = E(D0) – E . (29)
-500
0
500
1000
1500
2000
2500
3000
3500
4000
4500
0,0 0,1 0,2 0,3
σ
Po
la
riz
ab
ili
ty
(a
t.u
ni
ts
)
(α)
(αe)
(αh)
αh = hole polarizability
αe = electron polrizability
α = αh - αe
Fig. 1. The polarizability of D+X complex (in a.u. = 3
Daε ) as a
function of the electron-to-hole effective mass ratio (solid
line). Electron and hole polarizabilities in D+X (dashed lines).
0,000
0,005
0,010
0,015
0,020
0,025
0,030
0,035
0,040
0,045
0,050
0,0 0,1 0,2 0,3
σ
Io
ni
za
tio
n
fie
ld
(a
t.u
ni
ts
)
Fig. 2. The variation of ionization electric field (in a.u.)
versus σ.
The calculation was made in the weak field
approximation: F<<FI, where FI is the ionization field
defined by
000 2
1
EWFr Ieh −−== . (30)
As an illustration, the variations of FI versus σ are
reported in Fig. 2.
In this condition, the neutral donor energy can be
written as follows:
20
4
9
2
1
)( FDE −−= (31)
and in the same way:
2
0 2
1
FEE α−= . (32)
Practically, we have restricted our calculation to the
strength field value F = 0.3FI which may be considered as
consistent with the quadratic approximation in calculating
the binding energy. Recall that it has been shown [16] that
the perturbative calculation of the binding energy of the
exciton for the strength field values up to 0.5FI gives
good results. In Table, we report the values (in a.u.) of α,
W0, and W for the strength field value F = 0.3FI in the
range of σ values between 0 and σc.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 7-11.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
11
Table. Listing of α, W0, and W for the strength field value F
= 0.3FI, in the range of σ values between 0 and σc (in a.u.)
σ α F = 0.3FI W0 W
0 22 0.01381 0.08378 0.08545
0.05 39 0.00832 0.05767 0.05886
0.1 77 0.00496 0.03946 0.04035
0.15 143 0.00285 0.02618 0.02674
0.2 289 0.00153 0.01645 0.01678
0.25 630 0.00074 0.00941 0.00958
0.3 1488 0.00029 0.00444 0.00450
0.35 4483 0.00005 0.00096 0.00096
σc ∞ 0 0 0
As an indication, we have calculated the value σD
of σ for which the polarizability equals that of D−
system. To do that, analogy of D− with the negative
hydrogen ion [17] is made and gives 1765.0=Dσ with
the corresponding polarizability α = 206 (a.u.). That
means that for this limiting value of σ, the shift of both
D− and D+X lines in the optical spectra when a weak
electric field is applied is the same. Hence, for
semiconductors with Dσσ < , the D+X shift is lower than
that of D− while the situation is inverted for Dσσ > .
In summary, we have presented a variational
calculation of the polarizability of D+X as well as the
binding energy in the presence of a weak electric field.
This study shows an asymptotic behaviour of the
polarizability in the vicinity of σc. This behaviour is
principally due to the contribution of the hole which is
weakly bound to the neutral donor D0. It has been
established also that the effect of a weak electric field is
more pronounced for σ values lower that σ = 0.3. As a
comparison, confrontation of the polarizability of D+X
with D− system is made and shows that it is possible to
range the semiconductors in two classes following the
relative shift of D+X and D− lines.
Appendix
We establish in what follows the effect of the
stabilization of D+X complex in the presence of weak
electric field. The expansion in power series of the
dipolar electric moment in terms of the electric field
strength yields:
.......
)(2
)(
00
10
+=+
ΨΨ
Ψ−Ψ
=
=
ΨΨ
Ψ−Ψ
FF
zz
zz
eh
eh
α
(A1)
By identifying to α as given by Eq. (9), we obtain:
.4
)(
00
2
10
0101
11010
ΨΨ
ΨΨ
+ΨΨ−
−ΨΨ=Ψ−Ψ
EH
Ezz he
(A2)
Substituting then (A2) in (9) gives
.8
2
2
00
2
10
0
00
110101
ΨΨ
ΨΨ
−
−
ΨΨ
ΨΨ−ΨΨ
=
E
EH
α
(A3)
It is evident that the first term in (A3) is positive
because of the variational principle, and regarding to the
negative value of E0, the sign of α is always positive
which asserts the property of the stability of the complex
as advanced above.
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