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Hydrostatic pressure effects on donor-related absorption spectra in GaAs–Ga1−xAlxAs quantum wells
ARTICLE IN PRESS
Physica B 362 (2005) 41–49 www.elsevier.com/locate/physb
Hydrostatic pressure effects on donor-related absorption spectra in GaAs–Ga1xAlxAs quantum wells S.Y. Lo´peza, N. Porras-Montenegrob, C.A. Duquea, a
Instituto de Fı´sica, Universidad de Antioquia, AA 1226, Medellı´n, Colombia Departamento de Fı´sica, Universidad del Valle, AA 25360, Cali, Colombia
b
Received 11 January 2005; accepted 27 January 2005
Abstract In this paper, the effects of hydrostatic stress on the density of donor impurity states and donor-related optical absorption spectra in a GaAs–Ga1xAlxAs quantum wells are obtained. We calculate the shallow-donor binding energy for different well widths and different values of hydrostatic stress. We find that for large well widths the binding energy increases slowly with hydrostatic stress, as opposed to the behavior of the binding energy for wells with small width. One of the results shows that the binding energy does not change appreciably with the impurity position when the width of the well is small and the hydrostatic stress is high. Two structures in both, the density of states and the optical absorption spectra, associated with impurities located close to the center and to the edges of the structure, are obtained. We have also observed that the density of states and the optical absorption spectra depend strongly on the applied hydrostatic stress. r 2005 Elsevier B.V. All rights reserved. PACS: 74.62.Fj; 78.67.De Keywords: Shallow donors; Quantum-well; Hydrostatic stress effects; Binding energy
1. Introduction GaAs–Ga1xAlxAs quantum wells (QWs), multiple quantum wells (MQWs), quantum well-wires (QWWs), and quantum dots (QDs) have been Corresponding author. Tel.: +57 4 210 56 30;
fax: +57 4 233 01 20. E-mail address: cduque@fisica.udea.edu.co (C.A. Duque).
intensively studied in the last two decades due to their potential importance for photonic devices. Various optical techniques, such as Raman scattering, photoluminescence (PL), modulated spectroscopy, etc., have been used to investigate the systems under external applied influences such as electromagnetic fields and atmospheric pressure. There are also some reports of PL, photoreflectance (PR), and photomodulated transmission
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.473
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spectroscopy studies on the systems under high pressure. Having the advantage of high sensitivity and resolution, modulation spectroscopy has been widely used in studying the interband transitions associated with both the ground and excited states with high accuracy because of the derivate feature of the spectra. The pioneering work by Bastard [1] on impurity properties in QWs has been followed by several authors. They have included the influence of effects as effective-mass and dielectric-constant mismatches at the semiconductor interfaces [2], nonparabolicity of the valence and/or conduction bands, the coupling of the top four valence bands, spatially dependent screening, electron–phonon interactions, etc. [3–5]. Experimentally, some work have been reported about the impurity-related PL features attributed to the recombination of n ¼ 1 electrons with neutral acceptors in nonintentionally doped molecular-beam-epitaxy (MBE)-grown GaAs–Ga1xAlxAs QWs [6]. In the line of the impurity-related optical properties, some theoretical work has been reported (see for example Refs. [7,8] in QWs, Refs. [9,10] in QWWs, and Ref. [11] in QDs). As a general feature, this studies have shown that there are two (or three, depending on the symmetry of the quantum system) special structures in the transition probability per unit time WL(o), i.e. an edge associated with transitions involving impurities at the center of the structure and van Hove-like singularities associated with on-edge impurities. Studies of the effect of hydrostatic stress have proven to be invaluable in the context of the optical properties of semiconductors and their heterostructures [12–19]. For a given structure, the difference in energy between the type-I and -II transitions can be tuned with external hydrostatic pressure in a continuous and reversible manner. Using photomodulated transmission spectroscopy, Dai et al. [12] have investigated the interband transitions in GaAs–Ga1xAlxAs multiple QWs as a function of hydrostatic pressure. They find a wide number of spectral structures associated with the G, L, and X critical points in GaAs wells and Ga1xAlxAs barriers. Itskevich et al. [13] have reported the pressure-dependent PL lines in bulk GaAs. For the GaAs exciton line they have found
a value of (11.270.2) meV kbar1 for the pressure coefficient whereas for the acceptor-related line they have obtained a very similar value of (11.470.2) meV kbar1. Due to the strong pressure dependence of the electron effective mass, a significant change should be observed in the pressure coefficient in donorimpurity-related PL and optical absorption experiments. However, the above-mentioned authors do not present results for transitions from donor states to the valence band. The corresponding PL line should appear in the energy region between the conduction–acceptor–impurity transitions and exciton states recombination. Moreover, the pressure coefficients for the different electron and impurity transitions should be modified by the asymmetries in the semiconductors, which are presented for example by the inclusion of potential barriers that appear in the construction of semiconducting heterostructures such as QWs. The hydrostatic pressure and uniaxial stress effects have been reported for shallow-donor impurities in GaAs–(Ga,Al)As QWs [14–18] and rectangular-shaped QDs [19]. Several authors have shown that in general in the direct-gap regime the binding energy depends linearly with the pressure and/or the stress [14–19]. It has also been shown that in the indirect-gap regime for the barrierregion the energy grows until reaching a maximum and then it falls as an effect of the decrease of the potential barrier that confines the carriers. In the present work, we are concerned with the investigation of the optical-absorption spectra associated with shallow-donors in GaAs– Ga1xAlxAs QWs under the effects of hydrostatic pressure. In particular, the optical-absorption spectra have been calculated from the valence band to the donor-impurity band considering that the donors are evenly distributed in the region of the QW. In order to obtain the wave functions and the donor energy the variational method and the effective mass approach have been used. The pressure effects have been considered in the direct-gap regime (pressure values between 0 and 13.5 kbar) and indirect-gap regime (pressure values between 13.5 and 35 kbar) for the Ga1xAlxAs barrier material. In our model we have considered the change of dielectric constant and effective mass
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in the well and barrier interfaces. Due to the well sizes that we take into account, our model does not consider the image-charge effects because they generate changes smaller than 10% in the electron binding energy. This paper is organized as follows: in Section 2 we present our theoretical model, in Section 3 our results and discussion, and finally in Section 4 some conclusions are presented.
2. Theoretical framework In the effective-mass approximation, the Hamiltonian for a hydrogenic shallow-donor impurity in a single GaAs–Ga1xAlxAs QW (where x is the aluminum concentration in the barrier region) under the effect of a hydrostatic pressure (P) and the temperature (T) is given by _2 1 H¼ r r mwc;bc ðP; TÞ 2
!
e2 þ V ðz; PÞ, w;b ðP; TÞr
ð1Þ
where r ¼ ½x2 þ y2 þ ðz zi Þ2 1=2 is the carrierimpurity distance and subscripts w and b stand for the quantum well (WL) and the barrier layer (BL) materials, respectively. The WL and BL materials parabolic conduction effective-masses are given by [20] 2 mwc ðP; TÞ ¼ 1 þ 7:51 E g ðP; TÞ 1 1 þ m0 ð2Þ E g ðP; TÞ þ 0:341 and mbc ðP; T; xÞ ¼ mwc ðP; TÞ þ 0:083xm0 ,
(3)
where m0 is the free-electron mass. In Eq. (2) E g ðP; TÞ is the band gap for the GaAs semiconductor at the G point and at low temperatures. This is given by [21] E g ðP; TÞ ¼ b1:519 þ 10:7 103 P 5:405 104 T 2 =ðT þ 204Þc eV. ð4Þ
43
We want to emphasize that for single QWs larger than 50 A˚, the nonparabolic effective mass effects are lower than 5% [22]. In Eq. (1) the parameters w;b ðP; TÞ are the WL and BL materials static dielectric constants. At T ¼ 4 K the stressdependent GaAs static dielectric constant is given by [23] w ðP; 4 KÞ ¼ 12:83 expð1:67 103 PÞ.
(5)
Single GaAs–(Ga,Al)As QW dielectric constant mismatch effects on the shallow-donor impurity binding energy have been reported. They show that the binding energy changes occur for small well widths and high aluminum concentration [24]. ( the For example for x ¼ 0:3 and L ¼ 100 A binding energy change is 2%, decreasing with the increase of the well-size. Strictly speaking, the image potential in QWs cannot be neglected when considering electronic and impurity states, especially if the dimensions of the wells are small [25] (well sizes of 50 A˚ implies a 9% change on the donor binding energy). In our calculations we use x ¼ 0:3 [26,27] and due to the fact that in the present work we focus our attention on the stress effects the charge image effects have not been considered. This means that in the Hamiltonian in Eq. (1) we take b ðP; TÞ ¼ w ðP; TÞ [28]. In Eq. (1) V(z, P) is the potential which confines the donor electron in the QW region, given by ( V 0 ðPÞ for jzj4L=2; V ðz; PÞ ¼ (6) 0 for jzjoL=2; where L is the stress-dependent width of the WL and V0(P) is the stress-dependent barrier height [21]. L can be obtained by the fractional change in volume, which for the zinc-blende crystal of volume V is given by [29] dV ¼ 3PðS 11 þ 2S 12 Þ, (7) V where S 11 ð¼ 1:16 103 kbar1 Þ and S 12 ð¼ 3:7 104 kbar1 Þ are the compliance constants of GaAs [21]. The trial wave function for the ground state is chosen to be CðrÞ ¼ Nf f ðzÞgðrÞ,
(8)
ARTICLE IN PRESS S.Y. Lo´pez et al. / Physica B 362 (2005) 41–49
44
where N is the normalization constant, and gðrÞ ¼ e
ð1=lÞr
with (9)
for the hydrogenic part and f f ðzÞ is the eigenfunction of the Hamiltonian in Eq. (1) without the impurity potential term. This is given by 8 k1c ðzþL=2Þ ; zp L=2; > < þAe þ cosðk zÞ; jzjo þ L=2; (10) f f ðzÞ ¼ 2c > : þAek1c ðzL=2Þ ; zX þ L=2; where A is obtained from the matching conditions at the interfaces, k1c ¼
2mbc ðP; TÞ ½V 0 ðP; TÞ E 0 ðP; TÞ _2
1=2 (11)
and k2c ¼
2mwc ðP; TÞ E 0 ðP; TÞ _2
1=2 .
(12)
The compressive stress dependence of the donor binding energy is calculated from the definition E b ðP; TÞ ¼ E 0 ðP; TÞ E min ðP; TÞ
(13)
with Emin(P,T) the eigenvalue with the impurity potential term, minimized with respect to the variational parameter l. The transition probability per unit time for valence-to-donor transitions (associated with a single impurity located at z ¼ zi ) is proportional to the square of the matrix element of the electron–photon interaction H int between the wave functions of the initial (valence) and final (impurity) states, i.e. 2p X W¼ jhf jH int jiij2 dðE f E i _oÞ (14) _ i with H int ¼ Ce p; where e is the polarization vector in the direction of the electric field of the radiation, p is the momentum operator, and C is a prefactor which contains the photon vector potential. The above matrix element may be written as jhf jH int jiij ffi Ce Pfi Sfi
(15)
1 Pfi ¼ O
Z O
dr uf ðrÞ pui ðrÞ
(16)
and Z Sfi ¼
dr F f ðrÞF i ðrÞ,
(17)
where O denotes the volume of the unit cell, uf ðui Þ is the periodic part of the Bloch state for the final (initial) state, and F f ðF i Þ is the envelope function for the final (initial) state. The f i ðzÞ wave function is obtained by using the GaAs and Ga1xAlxAs valence-band material parameters in Eqs. (10)–(12). Since the hydrostatic pressure breaks the degeneration in the G-point between the heavyand light-hole bands with an smaller energy heavyhole related effective gap, in this work we will be interested in absorption processes from the heavy hole states to the donor-impurity band. For simplicity, we will consider a hydrostatic pressure independent [30] spherical GaAs effective mass of mGaAshh ¼ 0:4m0 : For the Ga1xAlxAs barrier region the corresponding heavy-hole effective mass is obtained from a linear interpolation between the GaAs and AlAs compounds, i.e. mbhh ðxÞ ¼ mGaAshh 0:1xm0 .
(18)
For a GaAs–Ga1xAlxAs QW of width L, the transition probability per unit time for valence-todonor transitions (associated with a single impurity located at z ¼ zi ) is therefore given by [7] 1 a0 2 mv W L ðzi ; oÞ ¼ W 0 ðN 2b a0 Þ 2 a0 m0 N 2 J 2 ð19Þ 3 Y ðDÞ, a0 k ? where the overlapping between the initial and final envelope functions is given by Z 2p þ1 f i ðzÞf f ðzÞ J½zi ; l; k? ðoÞ ¼ lb2 1 1 bjzzi j dz ð20Þ jz zi j þ e b with b ¼ ðk2? þ l2 Þ1=2 .
(21)
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and Z
e
þ1
l f 2f ðzÞ jz zi j þ 2
1 ð2=lÞjzzi j
dz,
14 L0 = 50 Å
12
L0 = 100 Å
ð23Þ
where k1v and k2v are obtained by using Eqs. (11) and (12) but taking into account the valence effective-masses for the well and barrier layers and the appropriate percentage of the band-offset for the potential barriers. For a homogeneous distribution of impurities and assuming that the QW thickness is much larger than the lattice spacing, one has for the total transition probability per unit of time Z 1 þL=2 W L ðoÞ ¼ dzi W L ðzi ; oÞ. (24) L L=2 Additionally, if we consider the case in which the zi impurity position is a random variable, the density of impurity states per unit binding energy is given by 2 dzi gL ð E i Þ ¼ , (25) L dE i
10
10
0
20
30
Pressure (kbar)
(a)
14
12 4
E b ( meV)
N 2 ¼ pl
16
E b ( meV)
The normalization constants for the initial and final envelope wave functions are given by L 2 1 2 þ Nb ¼ 2 k1v L cos2 ðk2v L=2Þ sinðk2v LÞ 1þ ð22Þ k2v L
45
1
10
2
3
8
where E i ¼ E b ðP; T; L; zi Þ with zi X0: L = 100 Å
6 -0.5
3. Results and discussion (b)
In Fig. 1(a) we present the binding energy of a shallow-donor impurity in a GaAs–Ga0.7Al0.3As QW as a function on the hydrostatic pressure considering two values of the well width. In Fig. 1(b) the results are shown as a function of the impurity position along the growth direction in a QW with 100 A˚ in width. Fig. 1(a) shows a linear behavior of the binding energy with the hydrostatic pressure up to 13.5 kbar (direct gap regime). In this pressure regime the G–X mixing effects are not considered and as a consequence the barrier
0.0
0.5
zi / L0
Fig. 1. Binding energies of a donor impurity in a GaAs–Ga0.7Al0.3As QW. In (a) results are presented as a function of the hydrostatic pressure for two well widths. In (b) our results are depicted as a function of the impurity position along the growth direction of the QW for different values of the hydrostatic pressures 0, 10, 20, and 30 kbar, denoted by the numbers 1, 2, 3, and 4, respectively.
height that confines the electrons in the GaAs layer remains constant. For higher pressures the rate at which the binding energy increases is lower and
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8 L = 50 Å
g (E) (1/au*)
finally the curves bend down to smaller values. This last behavior is mainly due to the linear decrease of the barrier height which is associated with the G–X crossover in the Ga1xAlxAs. From this behavior of the binding energy, we can infer that the density of impurity states will present two peaks: the first one related with on-center (with infinite weight due to the observed maximum in the corresponding binding energy curve) and the second one related with on-edge (which weight depends on the QW length) impurities, both separated by approximately 4.0 meV [7,8]. As it can be observed in Fig. 1(b), the hydrostatic pressure raises the binding energy mainly for oncenter impurities than for on-edge ones. The combined effects of hydrostatic pressure and the impurity position are really not so simple. For example, for on-center impurities the largest changes happen mainly for pressures values between 0 and 20 kbar, while for on-edge impurities the main changes on the donor binding energy are given for pressures near to 30 kbar. The density of impurity states as a function of the impurity position-dependent binding energy in a GaAs–Ga0.7Al0.3As QW for different values of the hydrostatic pressure is presented in Fig. 2. As expected, we can observe that the center of gravity of the density of states is closed to energies corresponding to on-center impurities, while the structure at lowest energy values is related with onedge impurities. The effect of the hydrostatic pressure on the density of impurity states is to shift the structure related to on-edge impurities to higher energies, while for on-center impurityrelated structure the shift to higher energies due to the hydrostatic pressure is observed only up to 26 kbar approximately. For pressures higher than 26 kbar a red shift is observed in this maximum of infinite weight. In Fig. 3 we display the valence to donorimpurity-related optical-absorption spectra. They are given as a function of the difference between the photon energy and the gap energy in GaAs–Ga0.7Al0.3As QWs. The results we present are for zero-pressure well widths of 50 and 100 A˚ and for different values of the hydrostatic pressure. The main effect of the hydrostatic pressure is to shift the absorption spectra to lower energies. For a
4
1 0
2
3
10
4
12
(a)
14
16
Eb ( meV) 2 L = 100 Å
g (E) (1/au*)
46
1 1
2
3 4 0 (b)
6
8
10
12
14
Eb ( meV)
Fig. 2. Density of impurity states as a function of the impurity position-dependent binding energy in a GaAs–Ga0.7Al0.3As QW for different values of the hydrostatic pressure: 0, 10, 20, and 30 kbar, denoted by the numbers 1, 2, 3 and 4, respectively. In (a) our results are for a QW with 50 A˚ in width, whereas in (b) the well width is 100 A˚.
well width of 50 A˚ there is no clear evidence of the separate contribution to the optical-absorption spectra due to on-edge and on-center impurities because the width of the well is too small. In this
ARTICLE IN PRESS S.Y. Lo´pez et al. / Physica B 362 (2005) 41–49
6
0.2 L = 50 Å
4
1
4
2 3 g(E) (1 / au*)
WL ( ω ) / 104
47
2
0
50
52
54
75
80
85
0.1
90
(E Ph-E g) (meV)
(a)
L = 100 Å
WL ( ω ) / 104
2
1 4
0.0 0
2
2
1
8
4 6 Eb ( meV)
(a)
3
10
0.4 20
(b)
16
20 24 28 (E Ph-E g) (meV)
32
Fig. 3. Donor-related optical-absorption spectra as a function of the difference between the photon energy and the gap energy. The well widths and hydrostatic pressure values are the same than those in Fig. 2.
30 kbar
0.3
36 WL (ω) / 104
0
15 10 5
0.2
0 116
120
124
0.1
case, for every position along the growth direction, the impurity feels the same effects of the presence of the barriers. However, when the well width increases, as it is shown in Fig. 3(b), the absorption spectra present clearly the contribution due to oncenter (on-edge) impurities at lower (higher) energies. In order to illustrate the behavior in the bulk limit, in Fig. 4 we present the density of impurity states as a function of the impurity positiondependent binding energy (a). We also give the donor-related optical-absorption spectra as a function of the difference between the photon energy and the gap energy (b) for a donor impurity in a GaAs–Ga0.7Al0.3As QW for two values of the hydrostatic pressure, 0 and 20 kbar, respectively. The zero-pressure width of the well is 1000 A˚. One of the roles of the hydrostatic pressure is to
0.0 -8 (b)
-6
-4 (E Ph-E g) (meV)
-2
0
Fig. 4. Density of impurity states as a function of the impurity position-dependent binding energy (a) and donor-related optical-absorption spectra as a function of the difference between the photon energy and the gap energy (b) for a donor ( GaAs–Ga Al As QW for two impurity in a L0 ¼ 1000 A 0.7 0.3 values of the hydrostatic pressure: 0 (solid lines) and 20 kbar (dotted lines). In Fig. (b), the inset shows the absorption spectra ( GaAs–Ga Al As QW with a hydrostatic in a L0 ¼ 20 A 0.7 0.3 pressure of 30 kbar.
increase the intensity of the structure related with on-center impurities and to shift the entire spectra to lower energy values. As expected, the structure
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S.Y. Lo´pez et al. / Physica B 362 (2005) 41–49
associated with on-center impurities is dominant over the one related with on-edge impurities. However, for large widths of the well the absorption spectra reveal (by means of the lower peak) the presence of the barrier. This is due to the effect of a homogeneous distribution of impurities in which case almost 20 percent of the impurities are located close to the well edges. In the inset, the absorption spectrum is shown for a well width of 20 A˚ with a hydrostatic pressure of 30 kbar. We observe that only one structure is present which corresponds practically to the 30 kbar bulk limit of the GaAs. This 3D limit is related to the barrier model that we use to describe the conduction G–X mixing associated with the G–X crossover.
4. Conclusions Theoretical calculations related with the influence of a hydrostatic pressure on the donor binding energy, density of impurity states, and valence-to-donor-related optical-absorption spectra in GaAs–Ga0.7Al0.3As QWs are presented. As a general feature, we observe that the binding energy increases with the pressure and with the decrease of the well width. We have shown that with the increasing of the well width two special structures appear in the donor-related optical-absorption spectra, an edge associated with transitions involving impurities at the center of the well and a peak associated with transitions related to impurities at the edges of the QW. We have observed a shift to higher energies of the density of impurity states as a function of the binding energy and a change in the intensity with a red shift of the absorption spectra with the hydrostatic pressure. The difficulties that should be had to verify experimentally the results that we present in our work summary in aspects such as: (i) the low energy separation between the both structures due to the impurity position in the well (essentially 4.0 meV), (ii) the necessity of doping the structure with a sufficiently large number of donor impurities, that would allow to observe very defined structures, could introduce a large broadening of the peak of the optical transition, in such a way that the energy that separates the two theoretical
well-defined structures related to on-center and on-edge impurities, could not be easily observed, and finally (iii) the large broadening of the excitonic absorption edge could mask the transitions to donor impurities. Taking in mind that nowadays the experimental skills are more accurate, we hope this work motivates future investigations related with the hydrostatic pressure effects on the optical absorption and photoluminescence processes in GaAs– Ga1xAlxAs QWs and other low dimensional heterostructures such as QW wires and QDs.
Acknowledgements We are grateful to the Universidad de Antioquia (CODI) for the financial support. This work was partially financed by Colciencias, the Colombian Scientific Agencies, under the Grants nos. 1115-0511502 and 1106-05-13828. References [1] G. Bastard, Phys. Rev. B 24 (1981) 4714. [2] C. Mailhiot, Y.-C. Chang, T.C. McGill, Phys. Rev. B 26 (1982) 4449. [3] S. Chaudhuri, K.K. Bajaj, Phys. Rev. B 29 (1984) 1803. [4] F.A.P. Oso´rio, M.H. Degani, O. Hipo´lito, Phys. Rev. B 37 (1988) 1402. [5] Z.Y. Deng, T.R. Lai, J.K. Guo, S.W. Gu, J. Appl. Phys. 75 (1994) 7389. [6] R.C. Miller, A.C. Gossard, W.T. Tsang, O. Munteanu, Phys. Rev. B 25 (1982) 3871. [7] L.E. Oliveira, R. Pe´rez-Alvarez, Phys. Rev. B 40 (1989) 10460. [8] R.B. Santiago, L.E. Oliveira, J. d’Albuquerque e Castro, Phys. Rev. B 46 (1992) 4041. [9] N. Porras-Montenegro, L.E. Oliveira, Solid State Commun. 76 (1990) 275. [10] N. Porras-Montenegro, A. Latge´, L.E. Oliveira, J. Appl. Phys. 70 (1991) 5555. [11] J. Silva-Valencia, N. Porras-Montenegro, J. Appl. Phys. 81 (1997) 901. [12] N. Dai, D. Huang, X.Q. Liu, Y.M. Mu, W. Lu, S.C. Shen, J. Appl. Phys. 82 (1997) 6359; N. Dai, D. Huang, X.Q. Liu, Y.M. Mu, W. Lu, S.C. Shen, Phys. Rev. B 57 (1998) 6566. [13] I.E. Itskevich, M. Henini, H.A. Carmona, L. Eaves, P.C. Main, D.K. Maude, J.C. Portal, Appl. Phys. Lett. 70 (1997) 505; I.E. Itskevich, S.G. Lyapin, I.A. Troyan, P.C. Klipstein,
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Physica B 362 (2005) 41–49 www.elsevier.com/locate/physb
Hydrostatic pressure effects on donor-related absorption spectra in GaAs–Ga1xAlxAs quantum wells S.Y. Lo´peza, N. Porras-Montenegrob, C.A. Duquea, a
Instituto de Fı´sica, Universidad de Antioquia, AA 1226, Medellı´n, Colombia Departamento de Fı´sica, Universidad del Valle, AA 25360, Cali, Colombia
b
Received 11 January 2005; accepted 27 January 2005
Abstract In this paper, the effects of hydrostatic stress on the density of donor impurity states and donor-related optical absorption spectra in a GaAs–Ga1xAlxAs quantum wells are obtained. We calculate the shallow-donor binding energy for different well widths and different values of hydrostatic stress. We find that for large well widths the binding energy increases slowly with hydrostatic stress, as opposed to the behavior of the binding energy for wells with small width. One of the results shows that the binding energy does not change appreciably with the impurity position when the width of the well is small and the hydrostatic stress is high. Two structures in both, the density of states and the optical absorption spectra, associated with impurities located close to the center and to the edges of the structure, are obtained. We have also observed that the density of states and the optical absorption spectra depend strongly on the applied hydrostatic stress. r 2005 Elsevier B.V. All rights reserved. PACS: 74.62.Fj; 78.67.De Keywords: Shallow donors; Quantum-well; Hydrostatic stress effects; Binding energy
1. Introduction GaAs–Ga1xAlxAs quantum wells (QWs), multiple quantum wells (MQWs), quantum well-wires (QWWs), and quantum dots (QDs) have been Corresponding author. Tel.: +57 4 210 56 30;
fax: +57 4 233 01 20. E-mail address: cduque@fisica.udea.edu.co (C.A. Duque).
intensively studied in the last two decades due to their potential importance for photonic devices. Various optical techniques, such as Raman scattering, photoluminescence (PL), modulated spectroscopy, etc., have been used to investigate the systems under external applied influences such as electromagnetic fields and atmospheric pressure. There are also some reports of PL, photoreflectance (PR), and photomodulated transmission
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.473
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spectroscopy studies on the systems under high pressure. Having the advantage of high sensitivity and resolution, modulation spectroscopy has been widely used in studying the interband transitions associated with both the ground and excited states with high accuracy because of the derivate feature of the spectra. The pioneering work by Bastard [1] on impurity properties in QWs has been followed by several authors. They have included the influence of effects as effective-mass and dielectric-constant mismatches at the semiconductor interfaces [2], nonparabolicity of the valence and/or conduction bands, the coupling of the top four valence bands, spatially dependent screening, electron–phonon interactions, etc. [3–5]. Experimentally, some work have been reported about the impurity-related PL features attributed to the recombination of n ¼ 1 electrons with neutral acceptors in nonintentionally doped molecular-beam-epitaxy (MBE)-grown GaAs–Ga1xAlxAs QWs [6]. In the line of the impurity-related optical properties, some theoretical work has been reported (see for example Refs. [7,8] in QWs, Refs. [9,10] in QWWs, and Ref. [11] in QDs). As a general feature, this studies have shown that there are two (or three, depending on the symmetry of the quantum system) special structures in the transition probability per unit time WL(o), i.e. an edge associated with transitions involving impurities at the center of the structure and van Hove-like singularities associated with on-edge impurities. Studies of the effect of hydrostatic stress have proven to be invaluable in the context of the optical properties of semiconductors and their heterostructures [12–19]. For a given structure, the difference in energy between the type-I and -II transitions can be tuned with external hydrostatic pressure in a continuous and reversible manner. Using photomodulated transmission spectroscopy, Dai et al. [12] have investigated the interband transitions in GaAs–Ga1xAlxAs multiple QWs as a function of hydrostatic pressure. They find a wide number of spectral structures associated with the G, L, and X critical points in GaAs wells and Ga1xAlxAs barriers. Itskevich et al. [13] have reported the pressure-dependent PL lines in bulk GaAs. For the GaAs exciton line they have found
a value of (11.270.2) meV kbar1 for the pressure coefficient whereas for the acceptor-related line they have obtained a very similar value of (11.470.2) meV kbar1. Due to the strong pressure dependence of the electron effective mass, a significant change should be observed in the pressure coefficient in donorimpurity-related PL and optical absorption experiments. However, the above-mentioned authors do not present results for transitions from donor states to the valence band. The corresponding PL line should appear in the energy region between the conduction–acceptor–impurity transitions and exciton states recombination. Moreover, the pressure coefficients for the different electron and impurity transitions should be modified by the asymmetries in the semiconductors, which are presented for example by the inclusion of potential barriers that appear in the construction of semiconducting heterostructures such as QWs. The hydrostatic pressure and uniaxial stress effects have been reported for shallow-donor impurities in GaAs–(Ga,Al)As QWs [14–18] and rectangular-shaped QDs [19]. Several authors have shown that in general in the direct-gap regime the binding energy depends linearly with the pressure and/or the stress [14–19]. It has also been shown that in the indirect-gap regime for the barrierregion the energy grows until reaching a maximum and then it falls as an effect of the decrease of the potential barrier that confines the carriers. In the present work, we are concerned with the investigation of the optical-absorption spectra associated with shallow-donors in GaAs– Ga1xAlxAs QWs under the effects of hydrostatic pressure. In particular, the optical-absorption spectra have been calculated from the valence band to the donor-impurity band considering that the donors are evenly distributed in the region of the QW. In order to obtain the wave functions and the donor energy the variational method and the effective mass approach have been used. The pressure effects have been considered in the direct-gap regime (pressure values between 0 and 13.5 kbar) and indirect-gap regime (pressure values between 13.5 and 35 kbar) for the Ga1xAlxAs barrier material. In our model we have considered the change of dielectric constant and effective mass
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in the well and barrier interfaces. Due to the well sizes that we take into account, our model does not consider the image-charge effects because they generate changes smaller than 10% in the electron binding energy. This paper is organized as follows: in Section 2 we present our theoretical model, in Section 3 our results and discussion, and finally in Section 4 some conclusions are presented.
2. Theoretical framework In the effective-mass approximation, the Hamiltonian for a hydrogenic shallow-donor impurity in a single GaAs–Ga1xAlxAs QW (where x is the aluminum concentration in the barrier region) under the effect of a hydrostatic pressure (P) and the temperature (T) is given by _2 1 H¼ r r mwc;bc ðP; TÞ 2
!
e2 þ V ðz; PÞ, w;b ðP; TÞr
ð1Þ
where r ¼ ½x2 þ y2 þ ðz zi Þ2 1=2 is the carrierimpurity distance and subscripts w and b stand for the quantum well (WL) and the barrier layer (BL) materials, respectively. The WL and BL materials parabolic conduction effective-masses are given by [20] 2 mwc ðP; TÞ ¼ 1 þ 7:51 E g ðP; TÞ 1 1 þ m0 ð2Þ E g ðP; TÞ þ 0:341 and mbc ðP; T; xÞ ¼ mwc ðP; TÞ þ 0:083xm0 ,
(3)
where m0 is the free-electron mass. In Eq. (2) E g ðP; TÞ is the band gap for the GaAs semiconductor at the G point and at low temperatures. This is given by [21] E g ðP; TÞ ¼ b1:519 þ 10:7 103 P 5:405 104 T 2 =ðT þ 204Þc eV. ð4Þ
43
We want to emphasize that for single QWs larger than 50 A˚, the nonparabolic effective mass effects are lower than 5% [22]. In Eq. (1) the parameters w;b ðP; TÞ are the WL and BL materials static dielectric constants. At T ¼ 4 K the stressdependent GaAs static dielectric constant is given by [23] w ðP; 4 KÞ ¼ 12:83 expð1:67 103 PÞ.
(5)
Single GaAs–(Ga,Al)As QW dielectric constant mismatch effects on the shallow-donor impurity binding energy have been reported. They show that the binding energy changes occur for small well widths and high aluminum concentration [24]. ( the For example for x ¼ 0:3 and L ¼ 100 A binding energy change is 2%, decreasing with the increase of the well-size. Strictly speaking, the image potential in QWs cannot be neglected when considering electronic and impurity states, especially if the dimensions of the wells are small [25] (well sizes of 50 A˚ implies a 9% change on the donor binding energy). In our calculations we use x ¼ 0:3 [26,27] and due to the fact that in the present work we focus our attention on the stress effects the charge image effects have not been considered. This means that in the Hamiltonian in Eq. (1) we take b ðP; TÞ ¼ w ðP; TÞ [28]. In Eq. (1) V(z, P) is the potential which confines the donor electron in the QW region, given by ( V 0 ðPÞ for jzj4L=2; V ðz; PÞ ¼ (6) 0 for jzjoL=2; where L is the stress-dependent width of the WL and V0(P) is the stress-dependent barrier height [21]. L can be obtained by the fractional change in volume, which for the zinc-blende crystal of volume V is given by [29] dV ¼ 3PðS 11 þ 2S 12 Þ, (7) V where S 11 ð¼ 1:16 103 kbar1 Þ and S 12 ð¼ 3:7 104 kbar1 Þ are the compliance constants of GaAs [21]. The trial wave function for the ground state is chosen to be CðrÞ ¼ Nf f ðzÞgðrÞ,
(8)
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44
where N is the normalization constant, and gðrÞ ¼ e
ð1=lÞr
with (9)
for the hydrogenic part and f f ðzÞ is the eigenfunction of the Hamiltonian in Eq. (1) without the impurity potential term. This is given by 8 k1c ðzþL=2Þ ; zp L=2; > < þAe þ cosðk zÞ; jzjo þ L=2; (10) f f ðzÞ ¼ 2c > : þAek1c ðzL=2Þ ; zX þ L=2; where A is obtained from the matching conditions at the interfaces, k1c ¼
2mbc ðP; TÞ ½V 0 ðP; TÞ E 0 ðP; TÞ _2
1=2 (11)
and k2c ¼
2mwc ðP; TÞ E 0 ðP; TÞ _2
1=2 .
(12)
The compressive stress dependence of the donor binding energy is calculated from the definition E b ðP; TÞ ¼ E 0 ðP; TÞ E min ðP; TÞ
(13)
with Emin(P,T) the eigenvalue with the impurity potential term, minimized with respect to the variational parameter l. The transition probability per unit time for valence-to-donor transitions (associated with a single impurity located at z ¼ zi ) is proportional to the square of the matrix element of the electron–photon interaction H int between the wave functions of the initial (valence) and final (impurity) states, i.e. 2p X W¼ jhf jH int jiij2 dðE f E i _oÞ (14) _ i with H int ¼ Ce p; where e is the polarization vector in the direction of the electric field of the radiation, p is the momentum operator, and C is a prefactor which contains the photon vector potential. The above matrix element may be written as jhf jH int jiij ffi Ce Pfi Sfi
(15)
1 Pfi ¼ O
Z O
dr uf ðrÞ pui ðrÞ
(16)
and Z Sfi ¼
dr F f ðrÞF i ðrÞ,
(17)
where O denotes the volume of the unit cell, uf ðui Þ is the periodic part of the Bloch state for the final (initial) state, and F f ðF i Þ is the envelope function for the final (initial) state. The f i ðzÞ wave function is obtained by using the GaAs and Ga1xAlxAs valence-band material parameters in Eqs. (10)–(12). Since the hydrostatic pressure breaks the degeneration in the G-point between the heavyand light-hole bands with an smaller energy heavyhole related effective gap, in this work we will be interested in absorption processes from the heavy hole states to the donor-impurity band. For simplicity, we will consider a hydrostatic pressure independent [30] spherical GaAs effective mass of mGaAshh ¼ 0:4m0 : For the Ga1xAlxAs barrier region the corresponding heavy-hole effective mass is obtained from a linear interpolation between the GaAs and AlAs compounds, i.e. mbhh ðxÞ ¼ mGaAshh 0:1xm0 .
(18)
For a GaAs–Ga1xAlxAs QW of width L, the transition probability per unit time for valence-todonor transitions (associated with a single impurity located at z ¼ zi ) is therefore given by [7] 1 a0 2 mv W L ðzi ; oÞ ¼ W 0 ðN 2b a0 Þ 2 a0 m0 N 2 J 2 ð19Þ 3 Y ðDÞ, a0 k ? where the overlapping between the initial and final envelope functions is given by Z 2p þ1 f i ðzÞf f ðzÞ J½zi ; l; k? ðoÞ ¼ lb2 1 1 bjzzi j dz ð20Þ jz zi j þ e b with b ¼ ðk2? þ l2 Þ1=2 .
(21)
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and Z
e
þ1
l f 2f ðzÞ jz zi j þ 2
1 ð2=lÞjzzi j
dz,
14 L0 = 50 Å
12
L0 = 100 Å
ð23Þ
where k1v and k2v are obtained by using Eqs. (11) and (12) but taking into account the valence effective-masses for the well and barrier layers and the appropriate percentage of the band-offset for the potential barriers. For a homogeneous distribution of impurities and assuming that the QW thickness is much larger than the lattice spacing, one has for the total transition probability per unit of time Z 1 þL=2 W L ðoÞ ¼ dzi W L ðzi ; oÞ. (24) L L=2 Additionally, if we consider the case in which the zi impurity position is a random variable, the density of impurity states per unit binding energy is given by 2 dzi gL ð E i Þ ¼ , (25) L dE i
10
10
0
20
30
Pressure (kbar)
(a)
14
12 4
E b ( meV)
N 2 ¼ pl
16
E b ( meV)
The normalization constants for the initial and final envelope wave functions are given by L 2 1 2 þ Nb ¼ 2 k1v L cos2 ðk2v L=2Þ sinðk2v LÞ 1þ ð22Þ k2v L
45
1
10
2
3
8
where E i ¼ E b ðP; T; L; zi Þ with zi X0: L = 100 Å
6 -0.5
3. Results and discussion (b)
In Fig. 1(a) we present the binding energy of a shallow-donor impurity in a GaAs–Ga0.7Al0.3As QW as a function on the hydrostatic pressure considering two values of the well width. In Fig. 1(b) the results are shown as a function of the impurity position along the growth direction in a QW with 100 A˚ in width. Fig. 1(a) shows a linear behavior of the binding energy with the hydrostatic pressure up to 13.5 kbar (direct gap regime). In this pressure regime the G–X mixing effects are not considered and as a consequence the barrier
0.0
0.5
zi / L0
Fig. 1. Binding energies of a donor impurity in a GaAs–Ga0.7Al0.3As QW. In (a) results are presented as a function of the hydrostatic pressure for two well widths. In (b) our results are depicted as a function of the impurity position along the growth direction of the QW for different values of the hydrostatic pressures 0, 10, 20, and 30 kbar, denoted by the numbers 1, 2, 3, and 4, respectively.
height that confines the electrons in the GaAs layer remains constant. For higher pressures the rate at which the binding energy increases is lower and
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8 L = 50 Å
g (E) (1/au*)
finally the curves bend down to smaller values. This last behavior is mainly due to the linear decrease of the barrier height which is associated with the G–X crossover in the Ga1xAlxAs. From this behavior of the binding energy, we can infer that the density of impurity states will present two peaks: the first one related with on-center (with infinite weight due to the observed maximum in the corresponding binding energy curve) and the second one related with on-edge (which weight depends on the QW length) impurities, both separated by approximately 4.0 meV [7,8]. As it can be observed in Fig. 1(b), the hydrostatic pressure raises the binding energy mainly for oncenter impurities than for on-edge ones. The combined effects of hydrostatic pressure and the impurity position are really not so simple. For example, for on-center impurities the largest changes happen mainly for pressures values between 0 and 20 kbar, while for on-edge impurities the main changes on the donor binding energy are given for pressures near to 30 kbar. The density of impurity states as a function of the impurity position-dependent binding energy in a GaAs–Ga0.7Al0.3As QW for different values of the hydrostatic pressure is presented in Fig. 2. As expected, we can observe that the center of gravity of the density of states is closed to energies corresponding to on-center impurities, while the structure at lowest energy values is related with onedge impurities. The effect of the hydrostatic pressure on the density of impurity states is to shift the structure related to on-edge impurities to higher energies, while for on-center impurityrelated structure the shift to higher energies due to the hydrostatic pressure is observed only up to 26 kbar approximately. For pressures higher than 26 kbar a red shift is observed in this maximum of infinite weight. In Fig. 3 we display the valence to donorimpurity-related optical-absorption spectra. They are given as a function of the difference between the photon energy and the gap energy in GaAs–Ga0.7Al0.3As QWs. The results we present are for zero-pressure well widths of 50 and 100 A˚ and for different values of the hydrostatic pressure. The main effect of the hydrostatic pressure is to shift the absorption spectra to lower energies. For a
4
1 0
2
3
10
4
12
(a)
14
16
Eb ( meV) 2 L = 100 Å
g (E) (1/au*)
46
1 1
2
3 4 0 (b)
6
8
10
12
14
Eb ( meV)
Fig. 2. Density of impurity states as a function of the impurity position-dependent binding energy in a GaAs–Ga0.7Al0.3As QW for different values of the hydrostatic pressure: 0, 10, 20, and 30 kbar, denoted by the numbers 1, 2, 3 and 4, respectively. In (a) our results are for a QW with 50 A˚ in width, whereas in (b) the well width is 100 A˚.
well width of 50 A˚ there is no clear evidence of the separate contribution to the optical-absorption spectra due to on-edge and on-center impurities because the width of the well is too small. In this
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6
0.2 L = 50 Å
4
1
4
2 3 g(E) (1 / au*)
WL ( ω ) / 104
47
2
0
50
52
54
75
80
85
0.1
90
(E Ph-E g) (meV)
(a)
L = 100 Å
WL ( ω ) / 104
2
1 4
0.0 0
2
2
1
8
4 6 Eb ( meV)
(a)
3
10
0.4 20
(b)
16
20 24 28 (E Ph-E g) (meV)
32
Fig. 3. Donor-related optical-absorption spectra as a function of the difference between the photon energy and the gap energy. The well widths and hydrostatic pressure values are the same than those in Fig. 2.
30 kbar
0.3
36 WL (ω) / 104
0
15 10 5
0.2
0 116
120
124
0.1
case, for every position along the growth direction, the impurity feels the same effects of the presence of the barriers. However, when the well width increases, as it is shown in Fig. 3(b), the absorption spectra present clearly the contribution due to oncenter (on-edge) impurities at lower (higher) energies. In order to illustrate the behavior in the bulk limit, in Fig. 4 we present the density of impurity states as a function of the impurity positiondependent binding energy (a). We also give the donor-related optical-absorption spectra as a function of the difference between the photon energy and the gap energy (b) for a donor impurity in a GaAs–Ga0.7Al0.3As QW for two values of the hydrostatic pressure, 0 and 20 kbar, respectively. The zero-pressure width of the well is 1000 A˚. One of the roles of the hydrostatic pressure is to
0.0 -8 (b)
-6
-4 (E Ph-E g) (meV)
-2
0
Fig. 4. Density of impurity states as a function of the impurity position-dependent binding energy (a) and donor-related optical-absorption spectra as a function of the difference between the photon energy and the gap energy (b) for a donor ( GaAs–Ga Al As QW for two impurity in a L0 ¼ 1000 A 0.7 0.3 values of the hydrostatic pressure: 0 (solid lines) and 20 kbar (dotted lines). In Fig. (b), the inset shows the absorption spectra ( GaAs–Ga Al As QW with a hydrostatic in a L0 ¼ 20 A 0.7 0.3 pressure of 30 kbar.
increase the intensity of the structure related with on-center impurities and to shift the entire spectra to lower energy values. As expected, the structure
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associated with on-center impurities is dominant over the one related with on-edge impurities. However, for large widths of the well the absorption spectra reveal (by means of the lower peak) the presence of the barrier. This is due to the effect of a homogeneous distribution of impurities in which case almost 20 percent of the impurities are located close to the well edges. In the inset, the absorption spectrum is shown for a well width of 20 A˚ with a hydrostatic pressure of 30 kbar. We observe that only one structure is present which corresponds practically to the 30 kbar bulk limit of the GaAs. This 3D limit is related to the barrier model that we use to describe the conduction G–X mixing associated with the G–X crossover.
4. Conclusions Theoretical calculations related with the influence of a hydrostatic pressure on the donor binding energy, density of impurity states, and valence-to-donor-related optical-absorption spectra in GaAs–Ga0.7Al0.3As QWs are presented. As a general feature, we observe that the binding energy increases with the pressure and with the decrease of the well width. We have shown that with the increasing of the well width two special structures appear in the donor-related optical-absorption spectra, an edge associated with transitions involving impurities at the center of the well and a peak associated with transitions related to impurities at the edges of the QW. We have observed a shift to higher energies of the density of impurity states as a function of the binding energy and a change in the intensity with a red shift of the absorption spectra with the hydrostatic pressure. The difficulties that should be had to verify experimentally the results that we present in our work summary in aspects such as: (i) the low energy separation between the both structures due to the impurity position in the well (essentially 4.0 meV), (ii) the necessity of doping the structure with a sufficiently large number of donor impurities, that would allow to observe very defined structures, could introduce a large broadening of the peak of the optical transition, in such a way that the energy that separates the two theoretical
well-defined structures related to on-center and on-edge impurities, could not be easily observed, and finally (iii) the large broadening of the excitonic absorption edge could mask the transitions to donor impurities. Taking in mind that nowadays the experimental skills are more accurate, we hope this work motivates future investigations related with the hydrostatic pressure effects on the optical absorption and photoluminescence processes in GaAs– Ga1xAlxAs QWs and other low dimensional heterostructures such as QW wires and QDs.
Acknowledgements We are grateful to the Universidad de Antioquia (CODI) for the financial support. This work was partially financed by Colciencias, the Colombian Scientific Agencies, under the Grants nos. 1115-0511502 and 1106-05-13828. References [1] G. Bastard, Phys. Rev. B 24 (1981) 4714. [2] C. Mailhiot, Y.-C. Chang, T.C. McGill, Phys. Rev. B 26 (1982) 4449. [3] S. Chaudhuri, K.K. Bajaj, Phys. Rev. B 29 (1984) 1803. [4] F.A.P. Oso´rio, M.H. Degani, O. Hipo´lito, Phys. Rev. B 37 (1988) 1402. [5] Z.Y. Deng, T.R. Lai, J.K. Guo, S.W. Gu, J. Appl. Phys. 75 (1994) 7389. [6] R.C. Miller, A.C. Gossard, W.T. Tsang, O. Munteanu, Phys. Rev. B 25 (1982) 3871. [7] L.E. Oliveira, R. Pe´rez-Alvarez, Phys. Rev. B 40 (1989) 10460. [8] R.B. Santiago, L.E. Oliveira, J. d’Albuquerque e Castro, Phys. Rev. B 46 (1992) 4041. [9] N. Porras-Montenegro, L.E. Oliveira, Solid State Commun. 76 (1990) 275. [10] N. Porras-Montenegro, A. Latge´, L.E. Oliveira, J. Appl. Phys. 70 (1991) 5555. [11] J. Silva-Valencia, N. Porras-Montenegro, J. Appl. Phys. 81 (1997) 901. [12] N. Dai, D. Huang, X.Q. Liu, Y.M. Mu, W. Lu, S.C. Shen, J. Appl. Phys. 82 (1997) 6359; N. Dai, D. Huang, X.Q. Liu, Y.M. Mu, W. Lu, S.C. Shen, Phys. Rev. B 57 (1998) 6566. [13] I.E. Itskevich, M. Henini, H.A. Carmona, L. Eaves, P.C. Main, D.K. Maude, J.C. Portal, Appl. Phys. Lett. 70 (1997) 505; I.E. Itskevich, S.G. Lyapin, I.A. Troyan, P.C. Klipstein,
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