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GaAs heterostructures
Superlattices and Microstructures 52 (2012) 738–749
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Optical spectra in the region of exciton resonances in quantum wells and quantum dots of In0.3Ga0.7As/GaAs heterostructures N.N. Syrbu a,⇑, V. Dorogan a, A. Dorogan a, T. Vieru a, V.V. Ursaki b, V.V. Zalamai b a b
Technical University of Moldova, 168 Stefan cel Mare Avenue, 2004 Chisinau, Republic of Moldova Institute of Applied Physics, Academy of Sciences of Moldova, 5 Academy Street, 2028 Chisinau, Republic of Moldova
a r t i c l e
i n f o
Article history: Received 30 May 2012 Accepted 20 June 2012 Available online 28 June 2012 Subject classification: 71.36.+c 78.40.Fy 78.20.Ci 78.55.Cr 78.67.De 78.67.Hc Keywords: Heterostructures Quantum wells Quantum dots Reflection Absorption Luminescence Excitons Optical functions
a b s t r a c t The transparency, reflection and luminescence spectra of In0.3Ga0.7As structures with 8 nm thickness and quantum wells limited by the barrier layer GaAs of a 9 nm (upper layer) and 100 nm (bottom layer) thickness had been studied in the region of photon energy 0.5–1.6 eV. Lines associated with the transitions hh,lh1-e1(1s,2s,3s), hh2,lh2-e2(1s,2s,3s), hh1,lh1-e2(1s) and hh3,lh3-e3(1s) had been revealed in reflection spectra. The shapes of the reflection and transparency lines had been calculated using a single oscillator model of dispersion relations and the Kramers– Kronig integrals. The binding energy of hh,lh1-e1 excitons, the effective mass mhh⁄ and mlh⁄ and the damping factor for the optical transitions to QW and QD had been determined. The lifetime of charge carriers on quantum dots varies in the range of 0.04–0.1 ps, while the radiative lifetime of excitons in quantum wells in the considered structure is around 2 ps. Ó 2012 Elsevier Ltd. All rights reserved.
⇑ Corresponding author. Tel.: +373 22 237508. E-mail address: [email protected] (N.N. Syrbu). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.06.022
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1. Introduction In0.3Ga0.7As/GaAs heterostructures with quantum wells and quantum dots are used as working media in different optoelectronic devices [1–3]. Injection lasers based on layers with QW and QD, which demonstrate a higher temperature stability of the threshold current density Jth, a low value of Jth [4–6], and generation in continuous mode at room temperature are created nowadays, that assure a 3 W output power [7]. The devices based on heterostructures with QW are used in many domains, including the optical communication systems [7–12]. The study showed that the optical reflection and transparency spectra with QW positioned periodically are complex, they being composed of excitonic resonances in QW. The manufacturing of In0.3Ga0.7As/GaAs heterostructures with quantum wells is accompanied by the formation of quantum dots (QD) at the boundary of layer with QW and the barrier. The emission properties of semiconductor heterostructures with QW and QD are important from the point of view of identification of quantum layers’ parameters, which are determined by the electronic processes in QW and QD. In this paper we investigate the reflection, transparency and luminescence spectra of In0.3Ga0.7As heterostructures with QW and QD. The shapes of the reflection and transparency lines had been calculated in a single oscillator model of dispersion relations and Kramers–Kronig integrals. 2. Experimental method The reflection, transparency and luminescence spectra had been measured using the spectrometers MDR-2, JASCO-670 and a double Raman diffraction spectrometer of high resolution CDL-1 at 10, 20 and 300 K temperature. InGaAsP photodiode was used as a photodetector. The cooing of samples was done using a Workhorse-type optical cryogenic system LTS-22-S-330. The backside surface of nanostructure was polished to a mirror state to study the transparency spectra TðxÞ ¼ jtðxÞj2 . 3. Experimental results and discussions The transitions in quantum wells between different size-quantization subbands from V-band to Cband, induced by the light with hx > Eg , can generate a whole family of electronic transitions and, consequently, spectral bands of the interband reflection, absorption and luminescence [13–15]. The reflection spectra RðxÞ ¼ jrðxÞj2 , absorption spectra TðxÞ ¼ jtðxÞj2 , and luminescence spectra are the easiest way for the study of quantum wells in heterostructures. The amplitude of light reflection and transparency coefficients are:
r ¼ Er =E0 ;
t ¼ Et =E0
ð1Þ
The law of energy conservation limits these coefficients in the absence of energy dissipation inside the quantum well:
jrj2 þ jtj2 ¼ 1:
ð2Þ
The exciton with a null two-dimensional wave vector is excited at a normal incident light, i.e. with K x ¼ K y ¼ 0. The analysis of reflection and transparency coefficients of heterostructures with quantum wells was performed in previous works [13–15], where the reflection rQW was easily determined:
rQW ¼
iC0
x0 x iðC þ C0 Þ x00 ¼ x0 þ C10 C0 sin2u;
ð3Þ
C0 ¼ C0 ð1 þ r10 cos 2uÞ:
ð4Þ
In this case x00 and U0 are the resonant frequency and the radiative decay parameter of the exciton, renormalized taking into consideration the exciton interaction with the light wave induced by this exciton and reflected from the external surface. After several transformations [13] of the reflection and absorption coefficients, the dependence of the reflection coefficient was obtained:
RðxÞ ¼ jrðxÞj2 ¼ R0 þ ðA þ BxÞ=ð1 þ x2 Þ;
ð5Þ
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where x ¼ ðx x0 Þ=C; R0 ¼ r201 .
A ¼ t 01 t 10 S½t01 t 10 S 2r01 ð1 þ SÞ cos 2u
ð6Þ
B ¼ 2r 01 t 01 t 10 S sin 2u;
ð7Þ
S ¼ C0 =C; S ¼ C0 =C
According to the Fresnel formulas, in the case of normal light incidence on the crystal’s surface we have:
r 10 ¼ r01 ¼ ðnb 1Þ=ðnb þ 1Þ;
t01 t 10 ¼ 4nb =ðnb þ 1Þ2
ð8Þ
Depending on the distance between quantum well center and the external surface, the A and B coefficients can take values of different sign, and, particularly, values equal to zero. If A = 0, B < 0 the resonant shape consists of a maximum at x0 < x00 and a minimum at x > x00 . If B = 0 the spectrum contains one maximum (A > 0) or one minimum (A < 0) [13–15]. The reflection and transparency spectra of the studied heterostructure In0.3Ga0.7As/GaAs (Fig. 1) correspond to the case when B = 0 and A < 0. The maxima of transparency spectra and minima of reflection spectra practically coincide. It is necessary to measure R and T in order to determine the accurate value of A(x):
AðxÞ ¼ 1 RðxÞ RðTÞ
ð9Þ
The imperfection of the structure influences the optical reflection and absorption spectra, leading to inhomogeneous broadening of resonant frequency of the exciton. The inhomogeneity can lead to a smooth dependence of x0 on the coordinate in the plane of the quantum well or in the volume of supperlattice, which leads to the broadening of the reflection and absorption lines. From an experimental point of view, the narrow lines of absorption and reflection are indicative of a high quality of the structure with quantum wells. The simplest and most effective way of accounting for inhomogeneous broadening in the calculation of the reflection coefficient is the formal change in the corresponding formulas of nonradiative decay U parameter by an effective nonradiative decay parameter Ueff = U + U0, where U0 is the parameter of broadening. Fig. 1 shows the transparency spectra of the structure consisting of two In0.3Ga0.7As layers of 8 nm thickness with quantum wells that are divided by the GaAs barrier layers with 9 nm thickness. The transparency spectra are measured at a normal incident light on the surface of In0.3Ga0.7As/GaAs heterostructure with quantum wells. Transparency peaks associated with electronic transitions in quantum dots and between levels in quantum well are observed in the spectrum. The quantization of the charge carriers (electrons or holes) is characteristic for a thin quantum well, that create quasitwo-dimensional excitons [13]. The exciton properties in a wide quantum well remain basically the same as in the respective bulk semiconductor, but the exiton movement is quantized [13–15]. In this
Fig. 1. Optical transmittance spectra of In0.3Ga0.7As/GaAs heterostructures. The inset shows the structure consisting of two In0.3Ga0.7As layers with 8 nm thickness with quantum wells divided by a GaAs barrier layer with 9 nm thickness.
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Fig. 2. Transmittance spectra caused by the ground (1s) and excited (2s,3s) states of excitonic transitions hh1-e1, lh1-e1 in a quantum well of the In0.3Ga0.7As/GaAs heterostructure.
case, especially important are the spatial dispersion effects that represent the presence of additional light waves and the necessity of imposing additional boundary conditions considering the exitonic polarization [13]. The valence band of the A3B5 compounds is four-time degenerated in the Brillouin zone center (k = 0). At k = 0 there are zones of heavy and light holes. The application of uniaxial strains leads to the appearance of two maxima in the valence band with a small distance in energy (light and heavy holes). In the case of uniaxial strain applied in a random direction, different maxima (minima) are differently shifted. These extremums are also shifted by changing temperature. It is difficult to determine the temperature shifting coefficient of the valence bands of the light and heavy holes. These difficulties are partially overcome in semiconductors by investigating the excitonic absorption spectra and the interband magneto-optical effect. The presence of heavy and light holes in quantum wells leads to the appearance of a series of electronic transitions from their quantum levels. The valence bands of the light and heavy holes in In0.3Ga0.7As/GaAs heterostructures cause the appearance of the electronic transitions hh1-e1 and lh1-e1, hh2-e2 and lh2-e2, etc. In addition, the electrons and holes are bound in excitonic states forming their ground and excited states hh1-e1 (1s, 2s, 3s. . .) and lh1-e1(1s, 2s, 3s. . .), hh2-e2 (1s, 2s, 3s. . .) and lh2-e2 (1s, 2s, 3s. . .). Fig. 2 shows transparency spectra of In0.3Ga0.7As/GaAs heterostructure with quantum wells at 20 K. Narrow maxima are revealed in transparency spectra, which are caused by the ground (1s) and excited (2s, 3s) states of the excitonic transitions hh1-e1, lh1-e1 in the quantum well. The energetic position of the observed transitions in QW is indicated with numbers on this figure. The energy of the excitonic line in QW is determined by the expression (9) [13–15]:
Elex ¼ Eg þ E0e þ E0h
le4 2 2h 2 ðl
e
þ 0:5Þ2
ð10Þ
where l is the quantum number, l = mh⁄me⁄/(mh⁄ + me⁄) is the exciton binding energy.
R¼
le4 : 2 2h e2
ð11Þ
The exciton binding energy (R) was determined from the energetic position of the lines lh1e1(1s,2s,3s. . .) caused by the light holes (lh) and from the heavy holes’ energies hh1-e1(1s,2s,3s. . .). Table 1 shows the obtained results. The binding energy R = 11.5 meV of the hh1-e1 excitons
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Table 1 Ground and excited excitonic states in the heterostructure’s quantum well. Quantum state
Electronic transitions
Energy of transition (eV)
Binding energy, (R meV)
Quantum number
Electronic transitions
Energy of transition (eV)
Binding energy (R meV)
n=1
hh1-e1(1s) hh1-e1(2s) hh1-e1(3s) lh1-e1(1s) lh-e1(2s) lh1-e1(3s)
1.1631 1.1711 1.1741 1.1778 1.1860 1.1889
11.5–12.4
n=2
hh2-e2(1s) hh2-e2(2s)
1.2964 1.3064
16.5
lh2-e2(1s) lh2-e2(2s) lh2-e2(3s)
1.340 1.357 1.362
22.0– 23.5
12.4–13.2
determined from the energy positions of 1s–2s lines equals R = 11.5 meV, while that determined from the energy positions of 2s–3s lines equals 12.4 meV. For the lh1-e1 excitons, the values of binding energy are R = 12.4 meV and 13.2 meV, respectively. Doublet structures A, B of the hh1-e1(1s) excitons and C, D of the lh1-e1(1s) excitons are revealed in transparency spectra. The value of splitting is shown in the figure. The A, B and C, D splitting is caused by the exchange interaction of excitons. The exchange interaction of excitons, which arises according to the theory of Wannier–Mott taking into account corrections to the effective mass approximation, leads to a partial lifting of the degeneracy of the ground excitonic state and splits it to corresponding irreducible representations. The ground state of the U6 U8 exciton is eightfold degenerated in a bulk semiconductor with zinc-blend structure [13–15]. The spin indices of the envelope wave function take the values of s = 1/2 and m = 3/2, ±1/2. The exchange interaction splits this state into three terms: U6 U8 = U12 + U15 + U25. In the scheme of angular momentum s = 1/2 and j = 3/2 summation, the triplet level U15 corresponds to the total angular moment j = 1 with projections M = 1, 0, 1. The U12 and U25 terms correspond to the angular moment j = 2 and are shifted by a value of D0 relative to the U15 term. The splitting between U12 and U25 terms differs from zero due to the asphericity of the valence band U8. The splitting value caused by the exchange interaction of the singlet–triplet excitonic states in bulk crystals varies from around 0.1 meV up to 10 meV. The ground state of the 1s-exciton in the GaAs/AlAs (0 0 1) quantum well is fourfold degenerated. U6 U6 = A1 + A2 + E in the notation of irreducible representations of point group D2d. Consequently, taking into account the exchange interaction, this state is split into a radiative doublet E with the projections M = s + m = ± 1 of the angular momentum on the Z axis and the terms A1, A2 (s = ± 1/2, t = ± 3/2) [13–15]. The last ones are symmetrized and antisymmetrized linear combinations of states with momentum projection ±2. The slitting between them is small, usually it is neglected. The states ±1 are dipole active at r+ and r polarizations, respectively. In an anisotropic quantum well, the symmetry of the system decreases and the radiative doublet should be split into two sublevels, which orientation is determined by the form of the localizing potential. Indeed, the electron–hole exchange interaction partially lifts the degeneracy of the exciton states and leads to splitting of the exciton 1s level into sublevels, corresponding to the irreducible representations. The exchange splitting into two components of e1-hh1(1s) doublet was observed while studying luminescence spectra of localized excitons in GaAs/AlAs (0 0 1) quantum wells in the optical near-field 0, i.e. the phase difference regime. The two components are polarized along the axes [1 1 0] and ½1 1 equals ±90°. An analogical splitting was observed in GaAs/A1As (0 0 1) superlattices of type II, which is associated with the exciton localization on a specific interface and with the low C2V symmetry of the single interface. The photoluminescence of GaAs/AlGaAs structures with quantum wells was discussed in previous works [16–18], in which a fine structure of ground and excited states of the localized exciton was revealed. Each of the observed exciton levels consists of two sublevels linearly polarized in orthogonal directions [1 1 0] and [1 1 0]. The splitting of the sublevels depends on the number of levels and varies within ±20–± 50 leV. The splitting values for the ground and four excited levels observed in Refs. [16–18], vary depending on the parameters of the quantum well and quantum dot. Emission maxima caused by the ground (1s) and excited (2s,3s) states of the exitonic transitions e1-hh1 and e1-lh1 as well as peaks of e1-hh2, e1- lh2 transitions are revealed in luminescence spectra
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Fig. 3. Luminescence spectra caused by the ground (1s) and excited (2s,3s) states of excitonic transitions hh1-e1, lh1-e1 in a quantum well of the In0.3Ga0.7As/GaAs heterostructure.
measured at 10 K under the excitation by the 542 nm laser line (Fig. 3). The doublet A, B and C, D is faintly observable in luminescence spectra. Fig. 4 shows the transparency spectra caused by the ground (1s) and excited (2s, 3s) states of the excitonic transitions hh2-e2, lh2-e2 in quantum well of the In0.3Ga0.7As/GaAs heterostructure. The measurements were done at S–P and P–P polarizations of light waves at 20 K. The binding energy of the hh2-e2 excitons determined according to the energetic position of 1s–2s lines equals R = 16.5 meV. The binding energy of the lh2-e2 exciton equals 22 meV, or 23.5 meV when determined according to the energetic position of 1s–2s, or 2s–3s lines, respectively. The obtained values of exciton binding energy for light and heavy holes allow one to determine the value of effective masses mh⁄ and ml⁄ by using the calculated or experimental values of the background dielectric constant eb. In the exciton resonances of light and heavy holes, the electron effective mass me⁄ is the same.
Fig. 4. Transmittance spectra caused by the ground (1s) and excited (2s, 3s) states of excitonic transitions hh2-e2, lh2-e2 in a quantum well of the In0.3Ga0.7As/GaAs heterostructure.
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Due to compressive or tensile strains of the crystal lattice during the growth of heterostructures, the lattice symmetry in strained semiconductor layers changes. In this case, the valence band is shifted as a whole, the subbands of light and heavy holes are split in the center of the Brillouin zone and the effective masses of electrons and holes change. For small deformations, the splitting energy of heavy and light excitons (De) is determined mainly by the splitting of hole subbands [19–21]. Since the relative position of subbands of heavy and light holes depends on the sign of the strain [19], in general, De can have any sign: the crystal lattice of one component of the heterostructure is compressed in the direction of growth, while that of another component is stretched. The deformation effects lead to the change of the De splitting, the parameters of heavy and light excitons, and the value of the background dielectric constant near the exciton resonance. The presence of two close exciton resonances in a quantum well is taken into account by using the following dielectric function of the quantum well material:
0
eðx; KÞ ¼ eb B @1 þ
2 X j¼1
ðjÞ LT
1
ðlÞ 0
2x x ðx
ðjÞ 2 0 Þ
x þ 2
ðlÞ
hK 2 x0 M
ðjÞ
ixC
0
C A
ð12Þ
where x is frequency, K is the wave vector value, eb is the background dielectric constant. The index of the exciton resonance j takes value j = 1 for the heavy hole exciton (h) and j = 2 for the light hole exciton (l). Usually, only one resonance mode is considered [2–7]. For the excitons of j-type, x0(j) is resonant frequency, xLT(j) is frequency of the longitudinal transversal splitting, U(j) is the damping factor , M(j) is translational mass of exciton in a quantum well.In [22–25] there are analyzed The dispersion pffiffiffi pffiffiffiffiffi branches ReK p =ðk0 = e0 Þ and their damping ImK p =ðk 0 = e0 Þ were analyzed in Refs. [23–25] for the case ðlÞ ðkÞ jDej (j) when the splitting due to deformation x0 x0 ¼ h=2 p is comparable with xLT . The influence of layers’ parameters on the nature (normal/anomalous) of dispersion branches was taken into account in Refs. [24,25]. One should mention that the ratio between xLT(j) and the splitting of the bands of light and heavy holes is especially important in the case of two near exciton resonances, such as the resonances associated with light and heavy holes. The value of longitudinal transversal splitting xLT(j) can be higher or lower than the value of the splitting of the bands of heavy and light holes. This imposes certain requirements for the analysis of optical reflection and absorption spectra of excitonic resonances of light and heavy holes. To calculate the effective mass of holes, it is necessary to know the background dielectric constant eb of the quantum well material in the region of excitonic resonance. The contour of reflection spectra was calculated to determine the background dielectric constant at 20 K and 300 K (Fig. 5). At 20 K, the values of DxLT = 69 meV, U = 18 meV and ed = 7 were obtained for the resonance frequency x1 = 0.98 eV, while the values of DxLT = 75 meV, U = 7.8 meV and ed = 7 where obtained for the resonance frequency x2 = 1.086 eV. The change of the value of effective mass M has a little effect on the shapes of the calculated curves. At 300 K, the best agreement between experimental and calculated shapes is realized with the following parameters: DxLT = 70 meV, U = 12.8 meV and ed = 6 for x1 = 0.905 eV; DxLT = 70 meV, U = 15.8 meV and ed = 5.5 for x2 = 1.016 eV; and DxLT = 55 meV, U = 27 meV and ed = 4.6 for x3 = 1.12 eV. The obtained values of DxLT = 70 meV characterize two intensive minima in the reflection spectra, which are due to QD. A method which is usually used in calculations of exciton polaritons spectra in bulk crystals [26] was used for the determination of the background dielectric constant eb in the region of electronic transitions region in QW. The maxima of reflection spectra in points ‘‘a’’ and ‘‘b’’ correspond to transversal frequency of excitons (x0), while the minima (indicated by arrows) correspond to longitudinal frequency of excitons (xL). Actually, the real values of the transverse frequencies of the oscillators in QDs correspond to the minima of the reflection spectra as indicated by arrows in the left side of Fig. 5. The right part of Fig. 5 shows the experimental and calculated reflection spectra (1-R) in the model of excitonic polaritons in QW [13–15]. The barrier layers have a finite thickness. One of these layers is neighbored by vacuum, and another one neighbors on a thick substrate. The fundamental absorption edge of the substrate lies below the resonance frequency of the exciton in quantum well, and the refractive indices of substrate material
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Fig. 5. Measured (exp.) and calculated by dispersion relations (cal.) reflection spectra contours at temperatures of 300 K and 20 K.
and barriers are close to each other. The internal barrier can be considered semi-infinite in thickness, and the reflectivity coefficient of the entire structure is written as [13–15]:
t 01 t 10 e2iu r QW ; 1 r10 rQW e2iu x a u ¼ nb bþ ; 2 c r ¼ r01 þ
ð13Þ ð14Þ
here rjl and tjl are the amplitudes of reflection and transmission coefficients at the boundary between the media j and l (j, l = 0 in vacuum and 1 in the barrier). These coefficients are calculated according to
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the Fresnel formulas (15). rQW is the reflection coefficient of the quantum well defined according to (14). u is the phase shift acquired when the light passes through the thickness of the outer barrier b and a half of the thickness of the well a. c is the speed of light. The denominator in the second term in (13) takes into account multiple reflections of the light from the quantum well and from the border between the outer barrier and the vacuum
rðxÞ ¼
iC0 ~ 0 x iðC þ C0 Þ x
ð15Þ
;
~ 0 is the resonant frequency renormalized taking into account the interaction of the exciton with Here x the light wave, induced by this exciton and reflected from the outer surface. Note that s0 = (2U0)1 is the radiative lifetime, U is the nonradiative decay parameter of the exciton
r 10 ¼ r01 ¼
nb 1 ; nb þ 1
t01 t10 ¼
4nb ðnb þ 1Þ2
;
ð16Þ
where nb is the refractive index of the external barrier. Calculation of the reflection coefficient for the entire structure was carried out according to the formula:
RðxÞ ¼ jrðxÞj2 ;
ð17Þ
In a case of several periodic quantum wells, rQW in the expression (12) is replaced by rN calculated by the formula:
rN ¼
iNC0 ~ 0 x iðC þ N C0 Þ x
;
ð18Þ
i.e. the dependence on the number of wells N is determined by replacing the radiation damping parameter U0 by NU0. The formula for calculating the phase shift takes the form:
u¼
x c
na Nd;
ð19Þ
where na nb is the refraction index of quantum well material. N is the number of quantum wells, d is the distance between neighboring quantum wells. In spectra with several oscillators, the calculation was performed for each oscillator on a single-oscillator basis with a consequent adjustment of spectra on the boundaries. The calculations are made in the region of respective excitonic transitions in quantum wells by using the value of ed = 6.0. The resonant frequencies of transitions and the damping parameters are given in the right side of Fig. 5. Calculations of the reflection spectra of In0.3Ga0.7As/GaAs heterostructure using the Kramers– Kronig relations were made to determine the optical constants. Fig. 6 shows the spectral dependence of optical constants n, k, e1 and e2 at room temperature in a wide range of energies. The data obtained show that the strongest changes occur for the dielectric functions e1 and e2 at the resonance frequency of QD. Measurements and calculations were performed for S–P and P–P polarized light waves to estimate the value of eb in the region of resonances in QW, i.e. in the region of hh1-e1(lh1-e1) and hh2-e2(lh2-e2) transitions. The reflection coefficient R is larger in S–P polarization than in the P–P polarization, while for the transmission coefficient there is an inverse relationship). In S–P polarization the hh2-e2(1s, 2s) transitions occur, and therefore a value of eb(e1) = 6 is characteristic for these transitions. The dielectric constant for this polarization in the region of 1.16 eV, i.e. in the region of hh1-e1 transitions, equals eb(e1) = 6.9. In the P–P polarization, the dielectric constant equals eb(e1) = 5.5 and eb(e1) = 4.3 for the energy of 1.16 eV and 1.3 eV, respectively. One can see that the value of the background dielectric constant obtained from the calculation of the reflection spectra using the Kramers– Kronig relations are consistent with the calculations of the dispersion relation. Thus, in the region of the exciton transitions hh1-e1 (lh1-e1) the difference between the dielectric constants is De = ebhh–eblh = 6.0–4.3 = 1.7, and in the region of hh2-e2 (lh2-e2) transitions we have De = ebhh–eblh = 6.9–5.5 = 1.4. Therefore, these values coincide in the limits of experimental errors. The effective mass of electrons me⁄ = 0.05m0 [15] was used to estimate the effective mass of the heavy and light holes, and the effective mass of excitons l was calculated on the basis of binding energies R defined as
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Fig. 6. Spectral dependence of optical functions n, k, e1 and e2 obtained from the calculation of the reflection spectra by means of Kramers–Kronig relations. The inset shows the spectra of the real part of dielectric constant e1 for S–P and P–P polarizations of the light waves in the energy region of hh1-e1(lh1-e1) and hh2-e2(lh2-e2) transitions.
described above. The value of exciton binding energy R = 0.0124 eV and the effective mass m = 0.0447m0 were obtained for the hh1-e1 transitions on the basis of data presented in Table 1. A value of the effective mass equal to mhh⁄ = 0.32m0 was obtained for the background dielectric constant eb = 6.9. The values of the exciton binding energy and the effective mass equal R = 0.0132 eV and m = 0.0293m0, respectively, for transitions lh1-e1. The effective mass of mlh⁄ = 0.07m0 is obtained for the background dielectric constant eb = 5.5. The obtained values of effective masses of light and heavy holes are consistent with those obtained previously for heterostructures [27–29]. The reflection spectra measured at 20 K and their contours calculated by using dispersion relations in the region of exciton transitions hh1-e1(1s) and hh1-e1(2s) are presented in Fig. 7 for structures which transmission spectra are shown in Fig. 2. The best agreement between experimental and calculated data was obtained for the oscillator x0 = 1.1629 eV, with the following parameters: DxLT = 1.1 meV, U = 0.3 meV and ed = 6.7. Fig. 7 shows also the scheme of optical transitions in the ground and excited states of QW excitons (right panel of the figure). The longitudinal-transverse splitting DxLT obtained from the calculation of reflection spectra of the hh1-e1 contour is than that obtained for the hh1-e1(1s) contour. This is because the reflection contour hh1-e1 contains contribution of both the ground and the excited excitonic states of heavy and light holes. The exciton binding energy and the effective mass equal R = 0.0165 eV and m = 0.0436m0, respectively, for hh2-e2 transitions. The effective mass calculated for a background dielectric constant of
Fig. 7. Measured (exp.) and calculated by the dispersion relations (cal.) contours of the reflection spectra of exciton transitions hh1-e1(1s) and hh1-e1(2s) at the temperature of 20 K.
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Fig. 8. Comparison of measured and calculated optical density spectra in the spectral range of 0.7–1.1 eV.
eb = 6.0 equals mhh⁄ = 0.30m0. For the lh2-e2 transitions, the exciton binding energy and the effective mass are R = 0.022 eV and m = 0.0293m0, respectively. The effective mass is mlh⁄ = 0.07m0 for the background dielectric constant of eb = 4.5. The parameter M (M = mhh⁄ + me⁄) obtained from the calculation of the reflection spectra contour of exciton transition hh1-e1(1s) is equal to 0.4m0. Consequently, the effective mass is mhh⁄ = M–me⁄ = 0.4–0.07 = 0.33m0. The possibility to achieve a high quantum yield (g) was demonstrated in laser structures based on quantum dots (QD). The maximum values of g, higher than 80%, for short wavelength (k 0.94– 0.98 lm) lasers was reported in Refs. [4–9]. In the long wavelength region (k 1.3 lm), for the lasers used in fiber–optic communication systems, the highest value of gD constitute only 55–57% [10–12]. The optical gain of QD lasers can be increased by depositing a larger number of QDs in the active region. It was previously shown [6–12] that for the arrays of QDs emitting near 1.3 lm it is necessary to deposit a larger number of InAs monolayers as compared to QDs emitting in the short-wavelength region for which 5–6 monolayers is enough. Apart from that, the increase of the ratio of QD height to its lateral size leads to increasing photosensitivity and decreasing the dark current of infrared detectors [12]. Narrow and intense emission peaks at wavelengths of 1550 nm 1240 nm, 1100 nm, 1000 nm, 930 nm and 911 nm, which are due to QD, were observed in photoluminescence spectra at 300 K [30]. These features are explained by the population of small and large QD. A 0.975 lm line at 77 K [30] indicates on the appearance of a population of small QDs. We suggest that the QD1 line situated near the Eg in the investigated spectra is due to small QDs. Similarly to bulk materials, the intensity of the absorption lines and the lifetime of carriers on centers (QD, QW) in nanostructures influence the photosensitivity of the photodetectors. Fig. 8 shows a comparison of optical density spectra experimentally measured and calculated for 1200–1800 nm. The table in Fig. 8 presents the values of the oscillator strength, which determine the absorption coefficient at the resonance frequency. The value of the damping parameter is associated with the lifetime of carriers on a given localized QD center or on QW levels. The damping parameters for QD1–QD4 and for excitonic levels of QW are not very different from each other and vary from 20 meV to 70 meV. Thus, the lifetime of charge carriers on quantum dots varies in the range of 0.04–0.1 ps. 4. Conclusions A series of lines caused by the transitions hh, lh1-e1 (1s, 2s, 3s), hh2, lh2-e2 (1s, 2s, 3s), hh1, lh1-e2 (1s), and hh3, lh3, -e3 (1s) have been detected in the reflection, transmission and luminescence spectra. The calculations of the reflection and transmission contours in a single-oscillator model of
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the dispersion relations and the Kramers–Kronig integrals show that the values of the background dielectric constant in the region of transitions in the ground and excited states of excitons in QW are different. The energy of the hh-e1, lh1-e1 excitons, the effective masses mhh⁄ and mlh⁄, and the damping parameters for optical transitions in QW and QD were determined. The value of the damping parameter is associated with the lifetime of carriers on localized centers of QD or QW levels. The damping parameters for QD1–QD4 quantum dots and exciton levels of QW do not differ very much. The lifetime of charge carriers on quantum dots varies in the range of 0.04–0.1 ps, while the radiative lifetime of excitons in quantum wells in the considered structure is around 2 ps. Acknowledgment Financial support from the SCOPES Program (Project IZ73Z0-128019) is acknowledged. References [1] [2] [3] [4]
[5] [6] [7] [8]
[9] [10] [11]
[12]
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Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Optical spectra in the region of exciton resonances in quantum wells and quantum dots of In0.3Ga0.7As/GaAs heterostructures N.N. Syrbu a,⇑, V. Dorogan a, A. Dorogan a, T. Vieru a, V.V. Ursaki b, V.V. Zalamai b a b
Technical University of Moldova, 168 Stefan cel Mare Avenue, 2004 Chisinau, Republic of Moldova Institute of Applied Physics, Academy of Sciences of Moldova, 5 Academy Street, 2028 Chisinau, Republic of Moldova
a r t i c l e
i n f o
Article history: Received 30 May 2012 Accepted 20 June 2012 Available online 28 June 2012 Subject classification: 71.36.+c 78.40.Fy 78.20.Ci 78.55.Cr 78.67.De 78.67.Hc Keywords: Heterostructures Quantum wells Quantum dots Reflection Absorption Luminescence Excitons Optical functions
a b s t r a c t The transparency, reflection and luminescence spectra of In0.3Ga0.7As structures with 8 nm thickness and quantum wells limited by the barrier layer GaAs of a 9 nm (upper layer) and 100 nm (bottom layer) thickness had been studied in the region of photon energy 0.5–1.6 eV. Lines associated with the transitions hh,lh1-e1(1s,2s,3s), hh2,lh2-e2(1s,2s,3s), hh1,lh1-e2(1s) and hh3,lh3-e3(1s) had been revealed in reflection spectra. The shapes of the reflection and transparency lines had been calculated using a single oscillator model of dispersion relations and the Kramers– Kronig integrals. The binding energy of hh,lh1-e1 excitons, the effective mass mhh⁄ and mlh⁄ and the damping factor for the optical transitions to QW and QD had been determined. The lifetime of charge carriers on quantum dots varies in the range of 0.04–0.1 ps, while the radiative lifetime of excitons in quantum wells in the considered structure is around 2 ps. Ó 2012 Elsevier Ltd. All rights reserved.
⇑ Corresponding author. Tel.: +373 22 237508. E-mail address: [email protected] (N.N. Syrbu). 0749-6036/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2012.06.022
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1. Introduction In0.3Ga0.7As/GaAs heterostructures with quantum wells and quantum dots are used as working media in different optoelectronic devices [1–3]. Injection lasers based on layers with QW and QD, which demonstrate a higher temperature stability of the threshold current density Jth, a low value of Jth [4–6], and generation in continuous mode at room temperature are created nowadays, that assure a 3 W output power [7]. The devices based on heterostructures with QW are used in many domains, including the optical communication systems [7–12]. The study showed that the optical reflection and transparency spectra with QW positioned periodically are complex, they being composed of excitonic resonances in QW. The manufacturing of In0.3Ga0.7As/GaAs heterostructures with quantum wells is accompanied by the formation of quantum dots (QD) at the boundary of layer with QW and the barrier. The emission properties of semiconductor heterostructures with QW and QD are important from the point of view of identification of quantum layers’ parameters, which are determined by the electronic processes in QW and QD. In this paper we investigate the reflection, transparency and luminescence spectra of In0.3Ga0.7As heterostructures with QW and QD. The shapes of the reflection and transparency lines had been calculated in a single oscillator model of dispersion relations and Kramers–Kronig integrals. 2. Experimental method The reflection, transparency and luminescence spectra had been measured using the spectrometers MDR-2, JASCO-670 and a double Raman diffraction spectrometer of high resolution CDL-1 at 10, 20 and 300 K temperature. InGaAsP photodiode was used as a photodetector. The cooing of samples was done using a Workhorse-type optical cryogenic system LTS-22-S-330. The backside surface of nanostructure was polished to a mirror state to study the transparency spectra TðxÞ ¼ jtðxÞj2 . 3. Experimental results and discussions The transitions in quantum wells between different size-quantization subbands from V-band to Cband, induced by the light with hx > Eg , can generate a whole family of electronic transitions and, consequently, spectral bands of the interband reflection, absorption and luminescence [13–15]. The reflection spectra RðxÞ ¼ jrðxÞj2 , absorption spectra TðxÞ ¼ jtðxÞj2 , and luminescence spectra are the easiest way for the study of quantum wells in heterostructures. The amplitude of light reflection and transparency coefficients are:
r ¼ Er =E0 ;
t ¼ Et =E0
ð1Þ
The law of energy conservation limits these coefficients in the absence of energy dissipation inside the quantum well:
jrj2 þ jtj2 ¼ 1:
ð2Þ
The exciton with a null two-dimensional wave vector is excited at a normal incident light, i.e. with K x ¼ K y ¼ 0. The analysis of reflection and transparency coefficients of heterostructures with quantum wells was performed in previous works [13–15], where the reflection rQW was easily determined:
rQW ¼
iC0
x0 x iðC þ C0 Þ x00 ¼ x0 þ C10 C0 sin2u;
ð3Þ
C0 ¼ C0 ð1 þ r10 cos 2uÞ:
ð4Þ
In this case x00 and U0 are the resonant frequency and the radiative decay parameter of the exciton, renormalized taking into consideration the exciton interaction with the light wave induced by this exciton and reflected from the external surface. After several transformations [13] of the reflection and absorption coefficients, the dependence of the reflection coefficient was obtained:
RðxÞ ¼ jrðxÞj2 ¼ R0 þ ðA þ BxÞ=ð1 þ x2 Þ;
ð5Þ
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where x ¼ ðx x0 Þ=C; R0 ¼ r201 .
A ¼ t 01 t 10 S½t01 t 10 S 2r01 ð1 þ SÞ cos 2u
ð6Þ
B ¼ 2r 01 t 01 t 10 S sin 2u;
ð7Þ
S ¼ C0 =C; S ¼ C0 =C
According to the Fresnel formulas, in the case of normal light incidence on the crystal’s surface we have:
r 10 ¼ r01 ¼ ðnb 1Þ=ðnb þ 1Þ;
t01 t 10 ¼ 4nb =ðnb þ 1Þ2
ð8Þ
Depending on the distance between quantum well center and the external surface, the A and B coefficients can take values of different sign, and, particularly, values equal to zero. If A = 0, B < 0 the resonant shape consists of a maximum at x0 < x00 and a minimum at x > x00 . If B = 0 the spectrum contains one maximum (A > 0) or one minimum (A < 0) [13–15]. The reflection and transparency spectra of the studied heterostructure In0.3Ga0.7As/GaAs (Fig. 1) correspond to the case when B = 0 and A < 0. The maxima of transparency spectra and minima of reflection spectra practically coincide. It is necessary to measure R and T in order to determine the accurate value of A(x):
AðxÞ ¼ 1 RðxÞ RðTÞ
ð9Þ
The imperfection of the structure influences the optical reflection and absorption spectra, leading to inhomogeneous broadening of resonant frequency of the exciton. The inhomogeneity can lead to a smooth dependence of x0 on the coordinate in the plane of the quantum well or in the volume of supperlattice, which leads to the broadening of the reflection and absorption lines. From an experimental point of view, the narrow lines of absorption and reflection are indicative of a high quality of the structure with quantum wells. The simplest and most effective way of accounting for inhomogeneous broadening in the calculation of the reflection coefficient is the formal change in the corresponding formulas of nonradiative decay U parameter by an effective nonradiative decay parameter Ueff = U + U0, where U0 is the parameter of broadening. Fig. 1 shows the transparency spectra of the structure consisting of two In0.3Ga0.7As layers of 8 nm thickness with quantum wells that are divided by the GaAs barrier layers with 9 nm thickness. The transparency spectra are measured at a normal incident light on the surface of In0.3Ga0.7As/GaAs heterostructure with quantum wells. Transparency peaks associated with electronic transitions in quantum dots and between levels in quantum well are observed in the spectrum. The quantization of the charge carriers (electrons or holes) is characteristic for a thin quantum well, that create quasitwo-dimensional excitons [13]. The exciton properties in a wide quantum well remain basically the same as in the respective bulk semiconductor, but the exiton movement is quantized [13–15]. In this
Fig. 1. Optical transmittance spectra of In0.3Ga0.7As/GaAs heterostructures. The inset shows the structure consisting of two In0.3Ga0.7As layers with 8 nm thickness with quantum wells divided by a GaAs barrier layer with 9 nm thickness.
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Fig. 2. Transmittance spectra caused by the ground (1s) and excited (2s,3s) states of excitonic transitions hh1-e1, lh1-e1 in a quantum well of the In0.3Ga0.7As/GaAs heterostructure.
case, especially important are the spatial dispersion effects that represent the presence of additional light waves and the necessity of imposing additional boundary conditions considering the exitonic polarization [13]. The valence band of the A3B5 compounds is four-time degenerated in the Brillouin zone center (k = 0). At k = 0 there are zones of heavy and light holes. The application of uniaxial strains leads to the appearance of two maxima in the valence band with a small distance in energy (light and heavy holes). In the case of uniaxial strain applied in a random direction, different maxima (minima) are differently shifted. These extremums are also shifted by changing temperature. It is difficult to determine the temperature shifting coefficient of the valence bands of the light and heavy holes. These difficulties are partially overcome in semiconductors by investigating the excitonic absorption spectra and the interband magneto-optical effect. The presence of heavy and light holes in quantum wells leads to the appearance of a series of electronic transitions from their quantum levels. The valence bands of the light and heavy holes in In0.3Ga0.7As/GaAs heterostructures cause the appearance of the electronic transitions hh1-e1 and lh1-e1, hh2-e2 and lh2-e2, etc. In addition, the electrons and holes are bound in excitonic states forming their ground and excited states hh1-e1 (1s, 2s, 3s. . .) and lh1-e1(1s, 2s, 3s. . .), hh2-e2 (1s, 2s, 3s. . .) and lh2-e2 (1s, 2s, 3s. . .). Fig. 2 shows transparency spectra of In0.3Ga0.7As/GaAs heterostructure with quantum wells at 20 K. Narrow maxima are revealed in transparency spectra, which are caused by the ground (1s) and excited (2s, 3s) states of the excitonic transitions hh1-e1, lh1-e1 in the quantum well. The energetic position of the observed transitions in QW is indicated with numbers on this figure. The energy of the excitonic line in QW is determined by the expression (9) [13–15]:
Elex ¼ Eg þ E0e þ E0h
le4 2 2h 2 ðl
e
þ 0:5Þ2
ð10Þ
where l is the quantum number, l = mh⁄me⁄/(mh⁄ + me⁄) is the exciton binding energy.
R¼
le4 : 2 2h e2
ð11Þ
The exciton binding energy (R) was determined from the energetic position of the lines lh1e1(1s,2s,3s. . .) caused by the light holes (lh) and from the heavy holes’ energies hh1-e1(1s,2s,3s. . .). Table 1 shows the obtained results. The binding energy R = 11.5 meV of the hh1-e1 excitons
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Table 1 Ground and excited excitonic states in the heterostructure’s quantum well. Quantum state
Electronic transitions
Energy of transition (eV)
Binding energy, (R meV)
Quantum number
Electronic transitions
Energy of transition (eV)
Binding energy (R meV)
n=1
hh1-e1(1s) hh1-e1(2s) hh1-e1(3s) lh1-e1(1s) lh-e1(2s) lh1-e1(3s)
1.1631 1.1711 1.1741 1.1778 1.1860 1.1889
11.5–12.4
n=2
hh2-e2(1s) hh2-e2(2s)
1.2964 1.3064
16.5
lh2-e2(1s) lh2-e2(2s) lh2-e2(3s)
1.340 1.357 1.362
22.0– 23.5
12.4–13.2
determined from the energy positions of 1s–2s lines equals R = 11.5 meV, while that determined from the energy positions of 2s–3s lines equals 12.4 meV. For the lh1-e1 excitons, the values of binding energy are R = 12.4 meV and 13.2 meV, respectively. Doublet structures A, B of the hh1-e1(1s) excitons and C, D of the lh1-e1(1s) excitons are revealed in transparency spectra. The value of splitting is shown in the figure. The A, B and C, D splitting is caused by the exchange interaction of excitons. The exchange interaction of excitons, which arises according to the theory of Wannier–Mott taking into account corrections to the effective mass approximation, leads to a partial lifting of the degeneracy of the ground excitonic state and splits it to corresponding irreducible representations. The ground state of the U6 U8 exciton is eightfold degenerated in a bulk semiconductor with zinc-blend structure [13–15]. The spin indices of the envelope wave function take the values of s = 1/2 and m = 3/2, ±1/2. The exchange interaction splits this state into three terms: U6 U8 = U12 + U15 + U25. In the scheme of angular momentum s = 1/2 and j = 3/2 summation, the triplet level U15 corresponds to the total angular moment j = 1 with projections M = 1, 0, 1. The U12 and U25 terms correspond to the angular moment j = 2 and are shifted by a value of D0 relative to the U15 term. The splitting between U12 and U25 terms differs from zero due to the asphericity of the valence band U8. The splitting value caused by the exchange interaction of the singlet–triplet excitonic states in bulk crystals varies from around 0.1 meV up to 10 meV. The ground state of the 1s-exciton in the GaAs/AlAs (0 0 1) quantum well is fourfold degenerated. U6 U6 = A1 + A2 + E in the notation of irreducible representations of point group D2d. Consequently, taking into account the exchange interaction, this state is split into a radiative doublet E with the projections M = s + m = ± 1 of the angular momentum on the Z axis and the terms A1, A2 (s = ± 1/2, t = ± 3/2) [13–15]. The last ones are symmetrized and antisymmetrized linear combinations of states with momentum projection ±2. The slitting between them is small, usually it is neglected. The states ±1 are dipole active at r+ and r polarizations, respectively. In an anisotropic quantum well, the symmetry of the system decreases and the radiative doublet should be split into two sublevels, which orientation is determined by the form of the localizing potential. Indeed, the electron–hole exchange interaction partially lifts the degeneracy of the exciton states and leads to splitting of the exciton 1s level into sublevels, corresponding to the irreducible representations. The exchange splitting into two components of e1-hh1(1s) doublet was observed while studying luminescence spectra of localized excitons in GaAs/AlAs (0 0 1) quantum wells in the optical near-field 0, i.e. the phase difference regime. The two components are polarized along the axes [1 1 0] and ½1 1 equals ±90°. An analogical splitting was observed in GaAs/A1As (0 0 1) superlattices of type II, which is associated with the exciton localization on a specific interface and with the low C2V symmetry of the single interface. The photoluminescence of GaAs/AlGaAs structures with quantum wells was discussed in previous works [16–18], in which a fine structure of ground and excited states of the localized exciton was revealed. Each of the observed exciton levels consists of two sublevels linearly polarized in orthogonal directions [1 1 0] and [1 1 0]. The splitting of the sublevels depends on the number of levels and varies within ±20–± 50 leV. The splitting values for the ground and four excited levels observed in Refs. [16–18], vary depending on the parameters of the quantum well and quantum dot. Emission maxima caused by the ground (1s) and excited (2s,3s) states of the exitonic transitions e1-hh1 and e1-lh1 as well as peaks of e1-hh2, e1- lh2 transitions are revealed in luminescence spectra
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Fig. 3. Luminescence spectra caused by the ground (1s) and excited (2s,3s) states of excitonic transitions hh1-e1, lh1-e1 in a quantum well of the In0.3Ga0.7As/GaAs heterostructure.
measured at 10 K under the excitation by the 542 nm laser line (Fig. 3). The doublet A, B and C, D is faintly observable in luminescence spectra. Fig. 4 shows the transparency spectra caused by the ground (1s) and excited (2s, 3s) states of the excitonic transitions hh2-e2, lh2-e2 in quantum well of the In0.3Ga0.7As/GaAs heterostructure. The measurements were done at S–P and P–P polarizations of light waves at 20 K. The binding energy of the hh2-e2 excitons determined according to the energetic position of 1s–2s lines equals R = 16.5 meV. The binding energy of the lh2-e2 exciton equals 22 meV, or 23.5 meV when determined according to the energetic position of 1s–2s, or 2s–3s lines, respectively. The obtained values of exciton binding energy for light and heavy holes allow one to determine the value of effective masses mh⁄ and ml⁄ by using the calculated or experimental values of the background dielectric constant eb. In the exciton resonances of light and heavy holes, the electron effective mass me⁄ is the same.
Fig. 4. Transmittance spectra caused by the ground (1s) and excited (2s, 3s) states of excitonic transitions hh2-e2, lh2-e2 in a quantum well of the In0.3Ga0.7As/GaAs heterostructure.
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Due to compressive or tensile strains of the crystal lattice during the growth of heterostructures, the lattice symmetry in strained semiconductor layers changes. In this case, the valence band is shifted as a whole, the subbands of light and heavy holes are split in the center of the Brillouin zone and the effective masses of electrons and holes change. For small deformations, the splitting energy of heavy and light excitons (De) is determined mainly by the splitting of hole subbands [19–21]. Since the relative position of subbands of heavy and light holes depends on the sign of the strain [19], in general, De can have any sign: the crystal lattice of one component of the heterostructure is compressed in the direction of growth, while that of another component is stretched. The deformation effects lead to the change of the De splitting, the parameters of heavy and light excitons, and the value of the background dielectric constant near the exciton resonance. The presence of two close exciton resonances in a quantum well is taken into account by using the following dielectric function of the quantum well material:
0
eðx; KÞ ¼ eb B @1 þ
2 X j¼1
ðjÞ LT
1
ðlÞ 0
2x x ðx
ðjÞ 2 0 Þ
x þ 2
ðlÞ
hK 2 x0 M
ðjÞ
ixC
0
C A
ð12Þ
where x is frequency, K is the wave vector value, eb is the background dielectric constant. The index of the exciton resonance j takes value j = 1 for the heavy hole exciton (h) and j = 2 for the light hole exciton (l). Usually, only one resonance mode is considered [2–7]. For the excitons of j-type, x0(j) is resonant frequency, xLT(j) is frequency of the longitudinal transversal splitting, U(j) is the damping factor , M(j) is translational mass of exciton in a quantum well.In [22–25] there are analyzed The dispersion pffiffiffi pffiffiffiffiffi branches ReK p =ðk0 = e0 Þ and their damping ImK p =ðk 0 = e0 Þ were analyzed in Refs. [23–25] for the case ðlÞ ðkÞ jDej (j) when the splitting due to deformation x0 x0 ¼ h=2 p is comparable with xLT . The influence of layers’ parameters on the nature (normal/anomalous) of dispersion branches was taken into account in Refs. [24,25]. One should mention that the ratio between xLT(j) and the splitting of the bands of light and heavy holes is especially important in the case of two near exciton resonances, such as the resonances associated with light and heavy holes. The value of longitudinal transversal splitting xLT(j) can be higher or lower than the value of the splitting of the bands of heavy and light holes. This imposes certain requirements for the analysis of optical reflection and absorption spectra of excitonic resonances of light and heavy holes. To calculate the effective mass of holes, it is necessary to know the background dielectric constant eb of the quantum well material in the region of excitonic resonance. The contour of reflection spectra was calculated to determine the background dielectric constant at 20 K and 300 K (Fig. 5). At 20 K, the values of DxLT = 69 meV, U = 18 meV and ed = 7 were obtained for the resonance frequency x1 = 0.98 eV, while the values of DxLT = 75 meV, U = 7.8 meV and ed = 7 where obtained for the resonance frequency x2 = 1.086 eV. The change of the value of effective mass M has a little effect on the shapes of the calculated curves. At 300 K, the best agreement between experimental and calculated shapes is realized with the following parameters: DxLT = 70 meV, U = 12.8 meV and ed = 6 for x1 = 0.905 eV; DxLT = 70 meV, U = 15.8 meV and ed = 5.5 for x2 = 1.016 eV; and DxLT = 55 meV, U = 27 meV and ed = 4.6 for x3 = 1.12 eV. The obtained values of DxLT = 70 meV characterize two intensive minima in the reflection spectra, which are due to QD. A method which is usually used in calculations of exciton polaritons spectra in bulk crystals [26] was used for the determination of the background dielectric constant eb in the region of electronic transitions region in QW. The maxima of reflection spectra in points ‘‘a’’ and ‘‘b’’ correspond to transversal frequency of excitons (x0), while the minima (indicated by arrows) correspond to longitudinal frequency of excitons (xL). Actually, the real values of the transverse frequencies of the oscillators in QDs correspond to the minima of the reflection spectra as indicated by arrows in the left side of Fig. 5. The right part of Fig. 5 shows the experimental and calculated reflection spectra (1-R) in the model of excitonic polaritons in QW [13–15]. The barrier layers have a finite thickness. One of these layers is neighbored by vacuum, and another one neighbors on a thick substrate. The fundamental absorption edge of the substrate lies below the resonance frequency of the exciton in quantum well, and the refractive indices of substrate material
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745
Fig. 5. Measured (exp.) and calculated by dispersion relations (cal.) reflection spectra contours at temperatures of 300 K and 20 K.
and barriers are close to each other. The internal barrier can be considered semi-infinite in thickness, and the reflectivity coefficient of the entire structure is written as [13–15]:
t 01 t 10 e2iu r QW ; 1 r10 rQW e2iu x a u ¼ nb bþ ; 2 c r ¼ r01 þ
ð13Þ ð14Þ
here rjl and tjl are the amplitudes of reflection and transmission coefficients at the boundary between the media j and l (j, l = 0 in vacuum and 1 in the barrier). These coefficients are calculated according to
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the Fresnel formulas (15). rQW is the reflection coefficient of the quantum well defined according to (14). u is the phase shift acquired when the light passes through the thickness of the outer barrier b and a half of the thickness of the well a. c is the speed of light. The denominator in the second term in (13) takes into account multiple reflections of the light from the quantum well and from the border between the outer barrier and the vacuum
rðxÞ ¼
iC0 ~ 0 x iðC þ C0 Þ x
ð15Þ
;
~ 0 is the resonant frequency renormalized taking into account the interaction of the exciton with Here x the light wave, induced by this exciton and reflected from the outer surface. Note that s0 = (2U0)1 is the radiative lifetime, U is the nonradiative decay parameter of the exciton
r 10 ¼ r01 ¼
nb 1 ; nb þ 1
t01 t10 ¼
4nb ðnb þ 1Þ2
;
ð16Þ
where nb is the refractive index of the external barrier. Calculation of the reflection coefficient for the entire structure was carried out according to the formula:
RðxÞ ¼ jrðxÞj2 ;
ð17Þ
In a case of several periodic quantum wells, rQW in the expression (12) is replaced by rN calculated by the formula:
rN ¼
iNC0 ~ 0 x iðC þ N C0 Þ x
;
ð18Þ
i.e. the dependence on the number of wells N is determined by replacing the radiation damping parameter U0 by NU0. The formula for calculating the phase shift takes the form:
u¼
x c
na Nd;
ð19Þ
where na nb is the refraction index of quantum well material. N is the number of quantum wells, d is the distance between neighboring quantum wells. In spectra with several oscillators, the calculation was performed for each oscillator on a single-oscillator basis with a consequent adjustment of spectra on the boundaries. The calculations are made in the region of respective excitonic transitions in quantum wells by using the value of ed = 6.0. The resonant frequencies of transitions and the damping parameters are given in the right side of Fig. 5. Calculations of the reflection spectra of In0.3Ga0.7As/GaAs heterostructure using the Kramers– Kronig relations were made to determine the optical constants. Fig. 6 shows the spectral dependence of optical constants n, k, e1 and e2 at room temperature in a wide range of energies. The data obtained show that the strongest changes occur for the dielectric functions e1 and e2 at the resonance frequency of QD. Measurements and calculations were performed for S–P and P–P polarized light waves to estimate the value of eb in the region of resonances in QW, i.e. in the region of hh1-e1(lh1-e1) and hh2-e2(lh2-e2) transitions. The reflection coefficient R is larger in S–P polarization than in the P–P polarization, while for the transmission coefficient there is an inverse relationship). In S–P polarization the hh2-e2(1s, 2s) transitions occur, and therefore a value of eb(e1) = 6 is characteristic for these transitions. The dielectric constant for this polarization in the region of 1.16 eV, i.e. in the region of hh1-e1 transitions, equals eb(e1) = 6.9. In the P–P polarization, the dielectric constant equals eb(e1) = 5.5 and eb(e1) = 4.3 for the energy of 1.16 eV and 1.3 eV, respectively. One can see that the value of the background dielectric constant obtained from the calculation of the reflection spectra using the Kramers– Kronig relations are consistent with the calculations of the dispersion relation. Thus, in the region of the exciton transitions hh1-e1 (lh1-e1) the difference between the dielectric constants is De = ebhh–eblh = 6.0–4.3 = 1.7, and in the region of hh2-e2 (lh2-e2) transitions we have De = ebhh–eblh = 6.9–5.5 = 1.4. Therefore, these values coincide in the limits of experimental errors. The effective mass of electrons me⁄ = 0.05m0 [15] was used to estimate the effective mass of the heavy and light holes, and the effective mass of excitons l was calculated on the basis of binding energies R defined as
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Fig. 6. Spectral dependence of optical functions n, k, e1 and e2 obtained from the calculation of the reflection spectra by means of Kramers–Kronig relations. The inset shows the spectra of the real part of dielectric constant e1 for S–P and P–P polarizations of the light waves in the energy region of hh1-e1(lh1-e1) and hh2-e2(lh2-e2) transitions.
described above. The value of exciton binding energy R = 0.0124 eV and the effective mass m = 0.0447m0 were obtained for the hh1-e1 transitions on the basis of data presented in Table 1. A value of the effective mass equal to mhh⁄ = 0.32m0 was obtained for the background dielectric constant eb = 6.9. The values of the exciton binding energy and the effective mass equal R = 0.0132 eV and m = 0.0293m0, respectively, for transitions lh1-e1. The effective mass of mlh⁄ = 0.07m0 is obtained for the background dielectric constant eb = 5.5. The obtained values of effective masses of light and heavy holes are consistent with those obtained previously for heterostructures [27–29]. The reflection spectra measured at 20 K and their contours calculated by using dispersion relations in the region of exciton transitions hh1-e1(1s) and hh1-e1(2s) are presented in Fig. 7 for structures which transmission spectra are shown in Fig. 2. The best agreement between experimental and calculated data was obtained for the oscillator x0 = 1.1629 eV, with the following parameters: DxLT = 1.1 meV, U = 0.3 meV and ed = 6.7. Fig. 7 shows also the scheme of optical transitions in the ground and excited states of QW excitons (right panel of the figure). The longitudinal-transverse splitting DxLT obtained from the calculation of reflection spectra of the hh1-e1 contour is than that obtained for the hh1-e1(1s) contour. This is because the reflection contour hh1-e1 contains contribution of both the ground and the excited excitonic states of heavy and light holes. The exciton binding energy and the effective mass equal R = 0.0165 eV and m = 0.0436m0, respectively, for hh2-e2 transitions. The effective mass calculated for a background dielectric constant of
Fig. 7. Measured (exp.) and calculated by the dispersion relations (cal.) contours of the reflection spectra of exciton transitions hh1-e1(1s) and hh1-e1(2s) at the temperature of 20 K.
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Fig. 8. Comparison of measured and calculated optical density spectra in the spectral range of 0.7–1.1 eV.
eb = 6.0 equals mhh⁄ = 0.30m0. For the lh2-e2 transitions, the exciton binding energy and the effective mass are R = 0.022 eV and m = 0.0293m0, respectively. The effective mass is mlh⁄ = 0.07m0 for the background dielectric constant of eb = 4.5. The parameter M (M = mhh⁄ + me⁄) obtained from the calculation of the reflection spectra contour of exciton transition hh1-e1(1s) is equal to 0.4m0. Consequently, the effective mass is mhh⁄ = M–me⁄ = 0.4–0.07 = 0.33m0. The possibility to achieve a high quantum yield (g) was demonstrated in laser structures based on quantum dots (QD). The maximum values of g, higher than 80%, for short wavelength (k 0.94– 0.98 lm) lasers was reported in Refs. [4–9]. In the long wavelength region (k 1.3 lm), for the lasers used in fiber–optic communication systems, the highest value of gD constitute only 55–57% [10–12]. The optical gain of QD lasers can be increased by depositing a larger number of QDs in the active region. It was previously shown [6–12] that for the arrays of QDs emitting near 1.3 lm it is necessary to deposit a larger number of InAs monolayers as compared to QDs emitting in the short-wavelength region for which 5–6 monolayers is enough. Apart from that, the increase of the ratio of QD height to its lateral size leads to increasing photosensitivity and decreasing the dark current of infrared detectors [12]. Narrow and intense emission peaks at wavelengths of 1550 nm 1240 nm, 1100 nm, 1000 nm, 930 nm and 911 nm, which are due to QD, were observed in photoluminescence spectra at 300 K [30]. These features are explained by the population of small and large QD. A 0.975 lm line at 77 K [30] indicates on the appearance of a population of small QDs. We suggest that the QD1 line situated near the Eg in the investigated spectra is due to small QDs. Similarly to bulk materials, the intensity of the absorption lines and the lifetime of carriers on centers (QD, QW) in nanostructures influence the photosensitivity of the photodetectors. Fig. 8 shows a comparison of optical density spectra experimentally measured and calculated for 1200–1800 nm. The table in Fig. 8 presents the values of the oscillator strength, which determine the absorption coefficient at the resonance frequency. The value of the damping parameter is associated with the lifetime of carriers on a given localized QD center or on QW levels. The damping parameters for QD1–QD4 and for excitonic levels of QW are not very different from each other and vary from 20 meV to 70 meV. Thus, the lifetime of charge carriers on quantum dots varies in the range of 0.04–0.1 ps. 4. Conclusions A series of lines caused by the transitions hh, lh1-e1 (1s, 2s, 3s), hh2, lh2-e2 (1s, 2s, 3s), hh1, lh1-e2 (1s), and hh3, lh3, -e3 (1s) have been detected in the reflection, transmission and luminescence spectra. The calculations of the reflection and transmission contours in a single-oscillator model of
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the dispersion relations and the Kramers–Kronig integrals show that the values of the background dielectric constant in the region of transitions in the ground and excited states of excitons in QW are different. The energy of the hh-e1, lh1-e1 excitons, the effective masses mhh⁄ and mlh⁄, and the damping parameters for optical transitions in QW and QD were determined. The value of the damping parameter is associated with the lifetime of carriers on localized centers of QD or QW levels. The damping parameters for QD1–QD4 quantum dots and exciton levels of QW do not differ very much. The lifetime of charge carriers on quantum dots varies in the range of 0.04–0.1 ps, while the radiative lifetime of excitons in quantum wells in the considered structure is around 2 ps. Acknowledgment Financial support from the SCOPES Program (Project IZ73Z0-128019) is acknowledged. References [1] [2] [3] [4]
[5] [6] [7] [8]
[9] [10] [11]
[12]
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