Download Kasapoglu E, Yesilgul U, Sari H, et al. The effect of hydrostatic

ARTICLE IN PRESS
Physica B 368 (2005) 76–81
www.elsevier.com/locate/physb
The effect of hydrostatic pressure on the photoionization
cross-section and binding energy of impurities in
quantum-well wire under the electric field
E. Kasapoglua,, U. Yesilgüla, H. Saria, I. Sökmena,b
a
Physics Department, Cumhuriyet University, 58140 Sivas-Turkey
b
Physics Department, Dokuz Eylül University,İzmir-Turkey
Received 26 May 2005; received in revised form 20 June 2005; accepted 29 June 2005
Abstract
Using a variational approach, we have calculated the hydrostatic pressure and electric field effects on the donorimpurity related photoionization cross-section and impurity binding energy in GaAs/GaAlAs quantum well-wires. Both
the results of impurity binding energy as a function of the impurity position and photoionization cross-section for a
hydrogenic donor impurity placed at the center of the quantum well-wire as a function of the normalized photon energy
in the quantum well-wire under the hydrostatic pressure and electric field which are applied to the z-direction for two
different wire dimensions are presented.
r 2005 Elsevier B.V. All rights reserved.
PACS: 71.55.Eq; 71.55.i
Keywords: Photoionization; Quantum-well wires; Impurity; Hydrostatic pressure
1. Introduction
The problem of an electron bound to an impurity
atom in the presence of the external electric and
magnetic field plays a fundamental role in understanding the optical properties of impurities in
semiconductors, and the impurity-related photoCorresponding author. Tel.: +903462191010;
fax: +903462191186.
E-mail address: [email protected] (E. Kasapoglu).
ionization cross-section is mainly used in the
characterization of impurities in semiconductors.
The photoionization cross-section of hydrogenic
impurities in bulk semiconductors is firstly investigated by Lax [1]. In recent years, work has been
done on the photoionization cross-section of
hydrogenic impurities in structures of reduced
dimensionally such as quantum wells (QWs), wires
(QWWs) and quantum dots (QDs) [2–10].
The hydrostatic pressure effects on the
electronic and impurity states in low-dimensional
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.physb.2005.06.039
ARTICLE IN PRESS
E. Kasapoglu et al. / Physica B 368 (2005) 76–81
heterostructures such as QWs, QWWs and QDs
have been studied in several theoretical works
[11–17]. Elabsy [11] has calculated the effects of the
hydrostatic pressure on the binding energy of
donor impurities in QWs, finding that the binding
energy increases with increasing hydrostatic pressure for a certain well thickness and temperature.
Oyoko et al. [16] have studied the effects of an
unaxial stress on the binding energy of shallow
impurities in parallelpipe-shaped GaAs/GaAlAs
QDs. They have found that the binding energy
increases almost linearly with applied stress and
diminishes with the size of the structure. Correa et
al. [17] have calculated the effects of hydrostatic
pressure on the binding energy and photoionization cross-section in spherical QDs for different
dimensions of the structure and radial impurity
position. They have found that the binding energy
and photoionization cross-section are affected by
the pressure.
In this paper, we report a calculation of the
photoionization cross-section and the binding
energy of a shallow donor impurity in GaAs/
GaAlAs QWW under the electric field and
hydrostatic pressure as a function of the wire
dimension and impurity position, for incident light
polarized along the axis of the wire.
2. Theory
In low-dimensional electronic systems, the
photoionization process is described as an optical
transition that takes place from the impurity
ground state as the initial state to the conduction
subbands, which requires sufficient energy in order
for the transition to occur. The excitation energy
dependence of the photoionization cross-section
associated with an impurity, starting from Fermi’s
golden rule in the well-known dipole approximation, as in the bulk case is [1,8]
" #
xeff 2 nr 4p2
sð_oÞ ¼
aFS _o
x0
ðPÞ 3
X 2
c j~
dðE f E i _oÞ,
ð1Þ
i rjcf
f
77
where nr is the refractive index of the semiconductor, aFS ¼ e2 =_c the fine structure constant, and
_o the photon energy. xeff/x0 is the ratio of the
effective electric field xeff of the incoming photon
and the average field x0 in the medium [18].
ci j~
rjcf is the matrix element between the initial
and final states of the dipole moment of the
impurity.
In the effective mass approximation, the Hamiltonian describing the interaction of an electron
with a hydrogenic impurity placed at the position
~
r ¼ ð0; 0; zi Þ in a QWW in the presence of an
electric field F applied in the z-direction, may be
written as
H¼
p2y
P2x
P2z
þ V ðz; PÞ
þ
V
ðxÞ
þ
þ
2me
2me 2me ðPÞ
þ eFz e2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðPÞ x2 þ y2 þ ðz zi Þ2
ð2Þ
where, the first terms in Eq. (2) are independent
from pressure since the pressure is applied only to
the z direction. The quantities e(P) is the static
dielectric constant as a function of pressure, me is
the effective mass, e is the elementary charge, F is
the electric field and V(x) and V(z, P) are the
confinement potentials for the electron in the x
and z-directions, respectively.
The pressure dependence of the effective mass of
the electron in GaAs is determined from expression [19–21]
me ðPÞ ¼
1þ
E Gp
h
m0
ð2=E Gg ðPÞÞ
þ ð1=E Gg ðPÞ þ D0 Þ
i,
(3)
where m0 is the mass of the bare electron,
E Gp ¼ 7:51 eV, D0 ¼ 0:341 eVand E Gg is the variation of the energy gap (in eV) for a GaAs QW at
the G-point with the hydrostatic pressure in units
of kbar, which in turn is expressed [19–22] as
E Gg ðPÞ ¼ a þ bP þ cP2 ,
(4)
where a ¼ 1:425 eV, b ¼ 1:26 102 eV=kbar and
c ¼ 3:77 105 eV=kbar2 .
ARTICLE IN PRESS
E. Kasapoglu et al. / Physica B 368 (2005) 76–81
78
The pressure dependence of static dielectric
constant is expressed as [23]
ðPÞ ¼ ð0ÞedP
(5)
3
where d ¼ 1:73 10 k=bar and ð0Þ ¼ 13:18 is
the value of the dielectric constant at the atmospheric hydrostatic pressure(static dielectric constant is assumed to be same GaAs and GaAlAs,
charge image effects have not been considered).
The barrier potential which confine the electron
in the QW in the z-direction, given by [24]
8
V ðPÞ; zo LðPÞ=2;
>
< 0
0;
LðPÞ=2ozoLðPÞ=2;
V ðz; PÞ ¼
(6)
>
: V ðPÞ; z4LðPÞ=2;
0
where
V 0 ðPÞ ¼ Qc DE Gg ðx; PÞ
(7)
is the barrier height and Qc( ¼ 0.6) is the conduction band offset parameter [24], DE Gg ðx; PÞ is the
band gap difference between QW and the barrier
matrix at the G-point as a function of P, which for
an aluminum fraction x ( ¼ 0,3) is given by
DE Gg ðx; PÞ ¼ DE Gg ðxÞ þ PDðxÞ
(8)
where
DE Gg ðxÞ
2
¼ ð1:155x þ 0:37x Þ eV
(9)
is the variation of the energy gap difference and
D(x) is the pressure coefficient of the band gap
[25], given by
DðxÞ ¼ ½ð1:3 103 Þx
eV=kbar
(10)
In Eq. (6)
LðPÞ ¼ Lð0Þ½1 ðS 11 þ 2S12 ÞP
,
(11)
where S11 ¼ 1:16 103 k=bar and S12 ¼ 3:7 104 k=bar are the elastic constants of the GaAs
[19–21] and L(0) is the original width of the
confinement potentials in the z-direction.
The investigation of the photoionization
cross-section first needs to know the envelope
functions of the initial ground state and the final
state. In order to get the binding energy, we follow
a variational method and we take the following
trial wave function that satisfies the Hamiltonian
in Eq. (2)
ci ðrÞ ¼ wðxÞwðzÞfðy; lÞ,
(12)
where the wave function in the y-direction f(y,l) is
chosen to be Gaussian-type orbital function
[26–28]
1 2 1=4 y2 =l2
fðy; lÞ ¼ pffiffiffi
e
(13)
l p
in which l is variational parameter. With the
choice of f(y, l), the degrees of freedom are
limited to one dimension along the axis of the wire.
Our experience with variational calculations of the
hydrogenic binding energy in QWs and quantum
wires suggests that very simple Gaussian-type
function gives quite accurate results in the case
of the moderate and strong fields [28 and
references there in]. w(x) and w(z) are the first
subbands wave functions of the electron, which are
exactly obtained from the one-dimensional Schrödinger equation in the x and z-directions, respectively. The ground state impurity binding energy is
given by
E B ¼ E x þ E z min ci jH jci
(14)
l
where Ex and Ez are the lowest donor electron
subband energies related to the w(x) and w(z) wave
functions, respectively.
Because the electron motion along the y-axis is
free, without the impurity potential the eigenstates
associated with the Hamiltonian (2) of an electron
emitted to the subbands nx, nz relative to the x
and z-directions of the quantum-well wire, are
given by
1
cf ðrÞ ¼ pffiffiffiffiffiffi wnx ðxÞwnz ðzÞeiky y ,
Ly
(15)
where Ly is the length of the wire and ky is the onedimensional wave vector of the electron along the
axis of the wire. wnx ðxÞ and wnz ðzÞ are chosen to be
solutions of one-dimensional Hamiltonian for
electron in the x and z-directions, respectively.
The final state energy corresponding to the wave
function Eq. (15) is
Ef ¼
_2 2
k þ E nx þ E nz .
2me y
(16)
ARTICLE IN PRESS
E. Kasapoglu et al. / Physica B 368 (2005) 76–81
where x ¼ _o=E S and E S ¼ E nx þ E nz E i ¼ E B ,
ðxeff =x0 Þ ¼ 1. In producing the sð_oÞ expression,
we accounted for all ky values in the final state by
converting the summation to a one-dimensional
integral along the axis of the QWW of length Ly.
40
P = 30 kbar
P = 10 kba r
P=0
20
10
-1.0
-0.5
0.0
zi / L
(a)
0.5
1.0
24
P = 10 kbar
3. Results and discussion
20
Eb ( meV)
The variation of the binding energy as a
function of the normalized impurity position for
different hydrostatic pressure values is given in
( and in Fig. 1(b)
Fig. 1(a) for L ¼ Lx ¼ Lz ¼ 50 A
(
for L ¼ Lx ¼ Lz ¼ 200 A. The maximum binding
energy is obtained for impurity located at the
center of QWW and P ¼ 30 kbar pressure value.
As the pressure increases, the well width and
dielectric constant decrease, the effective mass of
electron increases, leading to more confinement in
the well in the z-direction of the impurity electron
and so the impurity binding energy increases for
all impurity positions. The binding energy for the
impurity positions closed to the barriers is lower
than for on-center impurity since, Coulomb
interaction between the electron and impurity
decreases. When the electric field is applied, since
the electron shifts to the left side of the well in the
z-direction with the effect of electric field the
probability of finding the electron in the left side of
the well increases, while it decreases in the right
side of the well so, we observe that for impurity
position close to the left side of the well the
binding energy is higher than in right side. As seen
in Fig. 1(b), for quantum-well wires with large
L x = L z = L = 50
F= 0
----F = 50 kV/cm
30
Eb ( meV)
For the incident light polarized along the axis of
the wire, y-direction, the transition to the first
subbands nx ¼ 1, nz ¼ 1 is dipole allowed. Thus in
that case, wnx ðxÞ and wnz ðzÞare taken to be the same
as in the initial ground state, i.e., the first subbands
wave functions relative to the x and z-directions of
the wire, respectively. The photoionization crosssection for the incident light polarized along the
axis of the wire is given by
" #
xeff 2 nr aFS l5 pm E S 3=2
sð_oÞ ¼
x0
ðPÞ
3
_2
pffiffiffiffiffiffiffiffiffiffiffi
x x 1 exp½m ðl=_Þ2 E S ðx 1Þ
, ð17Þ
79
P = 30 kbar L x = L z = L = 200
F=0
------ F = 50 kV/cm
16
12
P=0
8
4
-0.5
(b)
-0.3
0.0
zi / L
0.3
0.5
Fig. 1. The variation of the binding energy as a function of the
normalized impurity position for different hydrostatic pressure
( and (b)
values (a) wire dimension L ¼ Lx ¼ Lz ¼ 50 A
( F ¼ 0 (solid line) and F ¼ 50 kV=cm
L ¼ Lx ¼ Lz ¼ 200 A.
(dashed line).
dimensions the impurity binding energy decreases,
since the probability of finding the electron and
impurity in the same plane decreases, while the
localization of the electron in the well-wire
increases. The field dependence of the binding
ARTICLE IN PRESS
E. Kasapoglu et al. / Physica B 368 (2005) 76–81
2.5
L x = L z = 50
F=0
------- F = 50 kV/cm
zi= 0
P=0
2.0
σx10-15 (cm2)
energy in narrow well-wire dimensions is very
weak, since the geometric confinement is predominant. But in the wider QWWs, the binding
energy is more sensitive to the external electric
field. Since, when the wire dimension increases, the
energy of electron becomes weaker and the
electron that approaches the well bottom begins
to become sensitive to the electric field strength.
Finally, in Fig. 2, we present the photoionization cross-section as a function of the normalized
photon energy of donor impurity placed at the
center of QWW for several hydrostatic pressure
( and (b)
values for (a) L ¼ Lx ¼ Lz ¼ 50 A
(
L ¼ Lx ¼ Lz ¼ 200 A,
respectively. Solid line
shows
F ¼0
and
dashed
line
shows
F ¼ 50 kV=cm. As the pressure increases, the
magnitude of the cross-section becomes much
smaller, the value of _o associated with the peak
of the cross-section moves the higher photon
energies and the photoionization cross-section
decreases for higher photon energies. Hydrostatic
pressure leads to increment of the binding energy
and hence to the optical photoionization threshold
energy. As a consequence the excitation of an
electron bound to a donor impurity to the first
conduction subband by absorption of a photon
requires higher photon energies in order for the
transition to occur. The electric field on the
photoionization cross-section is not effective in
norrow wire dimensions. Photoionization crosssection increases since the binding energy decreases with the electric field effect for Pa0
pressure values but this increment is not very
important. As known, the binding energy is very
sensitive to the electric field in large wire dimensions. In large wire dimensions, the binding energy
becomes much smaller and thus photoionization
cross-section increases evidently for P ¼ 0 pressure
value in Fig. 2(b). For Pa0 pressure values,
photoionization cross-section shows the same
behavior with in Fig. 2(a). Photoionization crosssection is not very affected from pressure since the
geometric confinement is predominant.
In summary, we have calculated the hydrostatic
pressure and electric field effects on the donorimpurity related photoionization cross-section and
impurity binding energy in GaAs/GaAlAs QWWs.
The electric field on the photoionization cross-
1.5
P = 10 kba r
1.0
0.5
P = 30 kba r
0.0
1.0
1.5
(a)
2.0
hω / Eb
14
2.5
3.0
L x = L z = 200
F=0
------- F = 50 kV/cm
zi= 0
12
10
σx10-15 (cm2)
80
8
P=0
6
4
P = 10 kbar
2
P = 30 kbar
0
1.0
(b)
1.5
2.0
hω / Eb
2.5
3.0
Fig. 2. The photoionization cross-section as a function of the
normalized photon energy of donor impurity placed at the
center of QWW for several hydrostatic pressure values for (a)
( and (b) L ¼ L ¼ L ¼ 200 A.
( Solid line
L ¼ Lx ¼ Lz ¼ 50 A
x
z
corresponds to F ¼ 0, dashed line corresponds to
F ¼ 50 kV=cm.
section is not effective in norrow wire dimensions.
In large wire dimensions, the binding energy
becomes much smaller and thus photoionization
cross-section increases evidently for P ¼ 0 pressure
value. Photoionization cross-section is not very
ARTICLE IN PRESS
E. Kasapoglu et al. / Physica B 368 (2005) 76–81
affected from pressure where the geometric confinement is predominant. Both for the results of
impurity binding energy as a function of the
impurity position and photoionization cross-section for a hydrogenic donor impurity placed at the
center of the QWW as a function of the normalized photon energy in the QWW under the
hydrostatic pressure and electric field for two
different wire dimensions are presented. We have
found that the photoionization cross-section decreases with the applied hydrostatic pressure.
Additionally, photoionization cross-section increases or decreases depending on size of the wire
and the impurity positions. The measurement of
photoionization in low-dimensional systems would
be of great interest for understanding the optical
properties of carriers in QWWs for which the
photoionization can give information about the
impurity position and the distribution inside the
heterostructure. To the best of our knowledge,
there are no experimental reports relating to the
photoionization cross-section in QW, QWWs, and
QDs.
References
[1] M. Lax, In the Proceeding of the 1954 Atlantic City
Conference on Photoconductivity, Wiley, New York, 1956,
p. 111.
[2] M. Takikawa, K. Kelting, G. Burunthaler, M. Takeshi,
J. Komena, J. Appl. Phys. 65 (1989) 3937.
[3] M. El-Said, M. Tomak, J. Phys. Chem. Solids 52 (1991)
603.
[4] M. El-Said, M. Tomak, Solid State Commun. 82 (1993)
721.
[5] K.F. Ilaiwi, M. El-Said, Phys. Stat. Sol. B 187 (1995) 93.
[6] G. Jayam Sr., K. Navaneethakrishnan, Solid State
Commun. 122 (2002) 433.
81
[7] A. Sali, M. Fliyou, L. Roubi, H. Loumrhari, J. Phys.:
Condens. Matter 11 (1999) 2427;
A. Sali, M. Fliyou, H. Satori, H. Loumrhari, Phys. Stat.
Sol. B 211 (1999) 661;
A. Sali, M. Fliyou, H. Loumrhari, Physica B 233 (1997)
196.
[8] A. Sali, H. Satori, M. Fliyou, H. Loumrhari, Phys. Stat.
Sol. B 200 (1997) 145;
A. Sali, M. Fliyou, H. Loumrhari, Phys. Stat. Sol. B 232
(2002) 209.
[9] A. Sali, M. Fliyou, H. Satori, H. Loumrhari, J. Phys.
Chem. Solids 64 (2003) 31.
[10] H. Heon, J.L. Cheol, J. Korean Phys. Soc 42 (2003) S289.
[11] A.M. Elabsy, J. Phys. Conds. Matter 6 (1994) 10025.
[12] A.L. Morales, A. Montes, S.Y. Lopez, C.A. Duque,
J. Phys.: Condens. Matter 14 (2002) 987.
[13] S.Y. Lopez, N. Porras-Montenegro, C.A. Duque, Phys.
Stat. Sol. C (2003) 648.
[14] A.L. Morales, A. Montes, S.Y. Lopez, N. Raigoza, C.A.
Duque, Phys. Stat. Sol. C (2003) 652.
[15] S.Y. Lopez, N. Porras-Montenegro, C.A. Duque, Semicond. Sci. Technol. 18 (2003) 718.
[16] H.O. Oyoko, C.A. Duque, N. Porras-Montenegro,
J. Appl. Phys. 90 (2001) 819.
[17] J.D. Correa, N. Porras-Montenegro, C.A. Duque, Phys.
Stat. Sol. B 241 (2004) 2440.
[18] B.K. Ridley, Quantum Process in Semiconductors, Clarendon Press, Oxford, 1982.
[19] H.J. Ehrenreich, J. Appl. Phys. 32 (1961) 2155.
[20] D.E. Aspens, Phys. Rev. B 14 (1976) 5331.
[21] S. Adachi, J. Appl. Phys. 58 (1985) R1.
[22] B.B. Welber, M. Cardona, C.K. Kim, S. Rodriguez, Phys.
Rev. B 12 (1975) 5729;
A. Jayaraman, Rev. Mod. Phys. 55 (1983) 65.
[23] G.A. Samara, Phys. Rev. B 27 (1983) 3494.
[24] R.F. Kopf, M.H. Herman, M.L. Schnoes, A.P. Perley,
G. Livescu, M. Ohring, J. Appl. Phys. 71 (1992) 5004.
[25] A. Montes, A.L. Morales, C.A. Duque, Surf. Rev. Lett. 9
(2002) 1753.
[26] R.O. Klepher, F.L. Madarasz, F. Szmulowichz, Phys. Rev.
B 51 (1995) 4633.
[27] T. Fukui, H. Saito, Appl. Phys. Lett. 50 (1987) 824.
[28] E. Kasapoglu, H. Sari, M. Bursal, I. Sökmen, Physica E 16
(2003) 237.