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AlGaAs superlattices
Vibrational Spectroscopy 54 (2010) 174–178
Contents lists available at ScienceDirect
Vibrational Spectroscopy journal homepage: www.elsevier.com/locate/vibspec
Raman investigation of plasmon localization in GaAs/AlGaAs superlattices A.D. Rodrigues a,∗ , Yu.A. Pusep b , J.C. Galzerani a a b
Departamento de Física, Universidade Federal de São Carlos, CP676, 13565-905, São Carlos, SP, Brazil Instituto de Física de São Carlos, Universidade de São Paulo, 13560-970, São Carlos, SP, Brazil
a r t i c l e
i n f o
Article history: Received 28 January 2010 Received in revised form 30 September 2010 Accepted 4 October 2010 Available online 12 October 2010 PACS: 71.45.Gm 63.20.kd 71.55.Jv 72.15.Rn
a b s t r a c t This work presents a Raman analysis of the influence of disorder in the plasmon localization. The disorder strength control was achieved by growing intentionally disordered GaAs/Al0.3 Ga0.7 As superlattices. The calculation of the localization extents was made by determining the coherency lengths of the plasmonLO coupled modes in the frequency range of the AlAs-like optical vibration, that presents a plasmon character. The plasmon damping was obtained from the Raman scattering related to the overdamped plasmon. These calculations allowed us to establish the relationship between the plasmon localization effect and the disorder strength, and to determine the limit of the transition from the delocalization to the strong localization regimes of the plasmon. © 2010 Elsevier B.V. All rights reserved.
Keywords: Plasmon Disorder Localization Raman spectroscopy
1. Introduction The transport properties of the electrons are directly influenced by the presence of disorder potentials. In weak disorder conditions we can consider that the electronic wave functions extend for the whole material. Otherwise, in the limit of strong disordered system, the propagating carriers undergo successive scatterings that restrict their trajectories, resulting in the localization of the electronic wave functions. Both the coherence length of the localization and the transition to a delocalizated state have been studied considering single-particle excitations, as in references [1,2]. The free-carriers can interact giving rise to a collective excitation, the so called plasmon. The plasmon wave function can be related to its Raman scattering intensity [3]. These facts point to the possibility to study the plasmon localization using Raman spectroscopy. Besides to contribute to the understanding of the physical principles of the plasmon propagation and localization, the understanding of the dynamical properties of the collective carriers excitations is essential to the development of new types of photonic devices [4]. In this work we present a way to use the Raman techniques in the investi-
∗ Corresponding author. Tel.: +55 16 3351 9727; fax: +55 16 3351 8464. E-mail addresses: [email protected], [email protected] (A.D. Rodrigues). 0924-2031/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.vibspec.2010.10.003
gations of the influence of the disorder in the plasmon localization. To proceed our analysis, a system in which the disorder strength could be quantitatively controlled was extremely necessary. It was achieved by growing intentionally disordered GaAs/Al0.3 Ga0.7 As superlattices. 2. Materials and methods The samples prepared to perform our studies consist of intentionally disordered (GaAs)m (Al0.3 Ga0.7 As)6 superlattices (SL’s) grown by molecular beam epitaxy (MBE) techniques. In order to produce a carrier excess with density N = 1.2 × 1018 cm−3 , the samples were homogeneously (well and barriers) doped with Si. The desired disorder control was achieved by the random fluctuations of the GaAs well thicknesses around the nominal value of 17 monolayers (ML’s), during the growth process. In that case, the energies of the non interacting electrons in the wells depend on the well thicknesses. By calculating the lowest energy of the electrons in each isolated quantum well, we obtain a distribution of the energy levels that describes a Gaussian shape. The more it is varied the thicknesses of the wells (i.e., the more disordered the superlattice), the larger is the full width at half maximum of the Gaussian distribution . In order to establish a disorder magnitude we have calculated the period dependent width W of the lowest energetic miniband for a pattern (in the absence of disorder) superlattice,
A.D. Rodrigues et al. / Vibrational Spectroscopy 54 (2010) 174–178
175
with well (barrier) width 17 ML’s (6 ML’s). The obtained value was W = 52 meV. So, the disorder strength of each superlattice can be described by a disorder parameter defined as ı = /W. In order to check the relation between the miniband formation and the barrier thickness, (GaAs)m (Al0.3 Ga0.7 As)n SL’s with N = 5 × 1017 cm−3 were prepared. Every sample in that set presents the same well thickness m = 10 ML’s, meanwhile the barrier width was varied assuming values 8 ≤ n ≤ 100 ML’s for each sample. The Raman scattering were collected from the samples surfaces in the backscattering geometry. The spectra were analysed by a Jobin Yvon T64000 triple monochromator equipped with a liquid nitrogen cooled CCD. The 514.5 nm line of an Ar+ laser was used as the excitation source. Since this energy lies far away from any electrical transition in that materials, no resonance effect is expected. The measurements were carried out at 8 K. 3. Theory Before proceeding any quantitative Raman analysis in the studied systems, it is essential to know the nature of the vibrational excitations and the influence of the density of free-carriers to their energy spectrum. So, we present bellow the calculation of the vibrational dispersion relation for a (GaAs)17 (Al0.3 Ga0.7 As)6 doped superlattice. In polar highly doped semiconductor a quantum of the freecarriers oscillation (plasmon) can interact with the longitudinaloptical (LO) phonons. The resulting coupled LO-plasmon mode (CLPM) is generated when the density fluctuation of the freecarriers acts screening the macroscopic electric field associated with the LO phonon. Its dispersion relation gives rise to two branches, one with energy lower (ω− ) and another with energy higher (ω+ ) than the unscreened LO phonon. For vanishing wave vectors q the energies of the two branches of the CLPM strongly depend on the plasma frequency (or the carrier concentration) ωp = 1/2
(4e2 N/m) , where e, N and m are the electron charge, concentration and mass, respectively. If the plasma frequency is smaller than the LO frequency, the ω− mode presents a plasmon-like character and the ω+ , a phonon-like one. For very high free-carriers density (plasmon frequency larger than the LO frequency), the ω− mode shows a phonon-like behavior and its frequency approaches that of the TO phonon; meanwhile, ω+ presents a plasmon-like character, indicating that the electric field associated with the LO phonon is almost completely screened by the free-carriers. By using the continuum dielectric model, the corresponding diagonal component of the dielectric function tensor along the growth direction z can describe the ionic part of the superlattice dielectric function. Considering the electric contribution due to the collective carriers excitations, the z component of the total superlattice dielectric function can be written in the long wavelength approximation as
εz (ω, q) = (d1 + d2 ) 2 (q) ωpz − ω2
d1 (ω2 − ωT21 (q)) 2 (q)) ε∞1 (ω2 − ωL1
+
d2 (ω2 − ωT22 (q))
−1
Fig. 1. Dispersion relations of the coupled modes calculated for a periodic (GaAs)17 (Al0.3 Ga0.7 As)6 superlattice with carrier density N = 1.2 × 1018 cm−3 .
sion relations for the doped (GaAs)17 (Al0.3 Ga0.7 As)6 superlattice, shown in Fig. 1 4. Results and discussion Fig. 2 shows the optical phonons range of the Raman spectra of the (GaAs)10 (Al0.3 Ga0.7 As)n SL’s with barrier widths 8 ≤ n ≤ 100 ML’s. The spectra were taken in the z(x, y)¯z cross-polarized scattering geometry. In this configuration only the vibrational modes with longitudinal character are expected to contribute to the Raman scattering. At the interval between 275 and 300 cm−1 it can be noticed a mixing of Raman peaks related to the longitudinal optical GaAs-like (LO1 ) mode both from the GaAs wells and the Al0.3 Ga0.7 As barriers. At around 380 cm−1 the superlattices spectra show pronounced peaks with frequency close to the longitudinal optical AlAs-like modes. In fact, for thicker barrier superlattices (n ≥ 25 ML’s) this spectral feature can be attributed to the longitudinal optical AlAs phonon, labeled LO2 . Its asymmetric shape is due to the contribution of a density of phonon states to the Raman signal. If the vibrations have a finite coherence length, an enhancement in the momentum uncertainty of the oscillations promotes a breaking in the selection rule for the momentum conservation, allowing excitations with q > 0 to contribute to the Raman intensity, with energies given by their dispersion relations. Thus, the Raman efficiency of localized vibrations can be given by [8]
2 (q)) ε∞2 (ω2 − ωL2
I(ω) ∝ (1)
where d1 and d2 are the thicknesses of the wells and of the barriers, respectively; ωpz is the q dependent frequency of the vertically polarized plasmon, obtained by the random phase approximation in the limit of long wave lengths [5]. ε∞1 and ε∞2 are, respectively, the GaAs and Al0.3 Ga0.7 As high frequency dielectric constants and ωL1(2) and ωT1(2) are the LO and TO dispersion relations for the GaAs (AlAs) taken from reference [6]. The effective electronic mass, that is necessary in order to determine the vertical plasmon frequency ωpz , was obtained using the envelope function approximation [7]. By calculating the zeros of Eq. (1) we have found the CLPM disper-
exp
−
q2 Lc2 4
d3 q [ω − ωexc (q)]2 + (/2)
2
(2)
where Lc is the collective excitation correlation length, is the damping constant and ωexc (q) is the dispersion relation of the respective excitation. Since the AlAs longitudinal optical phonons present a negative dispersion relation, a contribution of states with wave numbers bellow the center of the Brillouin zone ( point) is expected in the Raman spectrum, provoking downward asymmetric shape in the LO2 lines. On the other hand, for the superlattice with barrier thinner than 10 ML’s the same Raman line acquires an opposite asymmetric shape and its frequency shifts upward. This effect is caused by a miniband formation in the electron energy spectrum. By decreasing the barriers thicknesses the electrons can
176
A.D. Rodrigues et al. / Vibrational Spectroscopy 54 (2010) 174–178
Fig. 3. Raman spectra of the(GaAs)m (Al0.3 Ga0.7 As)6 superlattices with several disorder parameters ı, taken in the z(x, y)¯z scattering geometry.
Fig. 2. Raman spectra of (GaAs)10 (Al0.3 Ga0.7 As)n superlattices with8 ≤ n ≤ 100 ML’s, measured in z(x, y)¯z configuration.
tunnel between adjacent wells. The plasmon propagating parallel to the heterostructure growth axis interacts with the AlAs LO phonon of the barriers, giving rise to the coupled LO2 -plasmon mode ω2+ . Since these vibrations present positive dispersion relation due to their plasmon-like character (see Fig. 1), in presence of localization an upward asymmetry is expected for the related Raman peak. The opposition between the LO phonon and the coupled mode dispersion relations is also responsible for the blue shift in the peak maximum. Additionally, in the referred spectra we can notice the low-frequency coupled mode ω− and the mode resulting from the coupling between the vertical plasmon and the LO GaAs-like phonon, labeled ω1+ . Our analysis previously made showed that, for superlattices with thin enough barriers, the vertically polarized plasmons can couple with the LO phonons, provoking clear signatures in the Raman spectra. Thus, the Raman measurements present an efficient technique to determine how thin the superlattice barriers must be in order to promote the miniband formation. Furthermore, the quantitative analysis in the line shape modifications can be successfully used to investigate the dynamic properties of the plasmons and how they can be influenced by structural parameters such as disorder potentials. Fig. 3 shows the Raman spectra of random (GaAs)m (Al0.3 Ga0.7 As)6 SL’s prepared with the disorder parameters ı = 0.18, 0.35, 0.59, 0.82 and 1.13, taken in the same geometric configuration as in the first analysis. Sample with ı = 0.18 is a nominal ordered superlattice. The monolayer fluctuation in the pattern superlattice period produces a small disorder parameter even for the periodic superlattice. In fact, the obtained Raman
Fig. 4. Experimental (thin line) and calculated (dark gray line) Raman spectra of (GaAs)m (Al0.3 Ga0.7 As)6 superlattice with disorder parameterı = 0.59. The dotted and the dashed lines are components of the calculated spectrum due to the ω− and ω2+ , respectively. For simplicity, the Lorentzian curves related to TO1 , LO1 and TO2 used in the fitting are omitted in this figure.
A.D. Rodrigues et al. / Vibrational Spectroscopy 54 (2010) 174–178
Fig. 5. (a) Plasmon spatial extent Lc (full circles) and damping constant p (open circles) as function of disorder parameters. The dashed lines are guide to the eyes. (b) Ratio between the integral intensity of ω− (Iω− ) and ω2+ (Iω+ ) peaks, as function 2
of disorder parameter. (c) Temperature dependence of the plasmon localization for the strongest disordered superlattice.
177
excitation for each superlattice could be determined by fitting its experimental spectrum and the calculated one, as shown in Fig. 4 for the sample with ı = 0.59. There the solid thin and the dark gray lines are the experimental and the calculated spectra, respectively. In order to obtain the Lc values, the Raman intensity in the range of AlAs optical vibrations was calculated using Eq. (2) (dashed line). In these calculations we have used the dispersion relation for the coupled mode ω2+ , determined as shown in Section 3. The damping constants were determined considering the spectral contribution of an overdamped plasmon (dotted line) using the theory presented in reference [10]. The obtained plasmon length (Lc ) and damping ( p ) are presented in Fig. 5(a) as full and opened circles, respectively, as a function of the disorder strength. With the increasing of disorder the spacial extent of the plasmon continuously diminishes, indicating a tendency of the collective carriers excitations to localize due to the disorder potential in random superlattices. At the same time the plasmon damping constant rapidly increases. For ı values larger than 0.6 the damping does not significantly change, indicating that the overdamped character of the plasmon in the highly disordered superlattices leads to a strong localization in the carriers excitation. The transition from the weak to a strong localization regime in the plasmon, for disorder parameter values upper than that, can be faced as a changing in the collective carriers oscillations behavior from a conducting regime to a insulator one. Another information about the plasmon localization can be taken from the analysis of the relation between the integral intensity of the overdamped plasmon Raman line in the GaAs energy range and the ω2+ plasmon-like mode with frequency in the AlAs range. This ratio presents a substantial increase with increasing disorder, as can be seen in Fig. 5(b). This suggests that the plasmons exhibit a tendency to localize in the GaAs wells of the random superlattices. In addition, the dependence of the plasmon localization length with temperature was also studied. Fig. 5(c) shows the obtained Lc for the superlattice with ı = 1.13 at temperatures varying between 8 and 250 K. The localization length increase reveals that the plasmon presents a tendency to delocalize as the temperature increases, as expected for any localized quasiparticule. Even for the most disordered superlattices, where the plasmon undergoes a strong localized regime at low temperatures, the obtained spacial extent of the plasmon is comparable to that found for the periodic superlattice at 8 K (Fig. 5(a)).
5. Conclusions spectrum for this sample is quite similar to that found in Fig. 2 for the periodic superlattices with barrier thicknesses shorter than 10 ML’s. As can be seen in Fig. 3, the ω2+ plasmon-like Raman line presents different line broadenings for distinct disorder parameters, as a manifestation of different non conservation strengths in the momentum, probably due to distinct localization lengths. Another significantly change in the Raman spectra of the disordered superlattices can be observed; as the disorder strength increases, the ω− line shape changes, from a down to and upward asymmetry. This feature is attributed to an overdamped behavior of the plasmon, that occurs when the plasmon damping p is of the same order or higher than the plasmon frequency ωp . In the presence of an overdamped plasmon in a specific polar semiconductor, the low and high-frequency coupled modes vanish and only one peak presenting upward broadening shows up in the Raman spectrum in the vicinity of the TO mode frequency [9]. In our case, the potential produced by the most disordered superlattices is strong enough to enhance the plasmon damping until its magnitude becomes comparable to (or higher) the plasma frequency. Since the disorder dependent modifications in the spectra are influenced by the dynamic properties of the vertically polarized plasmons, both the localization extent Lc and the damping of the collective carriers
We have shown how the Raman analysis can be used to study the dynamic properties of plasmons, specially in presence of disorder. The chosen system was the GaAs/Al0.3 Ga0.7 As superlattice due to the possibility of controlling the disorder strength. Firstly, we have probed the existence of a vertically polarized plasmon due to the miniband formation, confirmed for superlattices with barrier thickness shorter than 10 ML’s. The calculation of both the plasmon localization and damping could be studied as a function of disorder using random superlattices with different disorder parameters. The results allow us to conclude that for ı > 0.6 the plasmon presents a strong localization character. So, as the disorder increases, the plasmon undergoes a transition from the delocalization to localization regimes, similar to that found in the metal-insulator transition. This assumption is complemented by the relative increasing in the integral intensity of the Raman line related to the overdamped plasmon, with the increasing of the disorder strength. Furthermore, even for the most disordered superlattice, the plasmon localization diminishes with the temperature increase. The localization length calculated for the superlattice with ı = 1.13 at room temperature is very close to that found in the periodic superlattice (ı = 0.18), where the plasmon is delocalized.
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Acknowledgements The authors are grateful to financial support from Brazilian agencies FAPESP and CNPq. References [1] P.H. Song, F. Oppen, Phys. Rev. B 59 (1999) 46–49. [2] S.V. Kravchenko, D. Simonian, M.P. Sarachik, W. Mason, J.E. Furneaux, Phys. Rev. Lett. 77 (1996) 4938–4941.
[3] R. Merlin, M. Cardona, G. Güntherodt, Light Scattering in Solids V, SpringerVerlag, Berlin, 1989, pp. 214–232 (Chapter 5). [4] W.L. Barnes, A. Dereux, T.W. Ebbesen, Nature 424 (2003) 824–830. [5] G.D. Mahan, Many-Particle Physics, 2nd ed., Plenum Press, New York, 1990. [6] Yu.A. Pusep, S.W. da Silva, J.C. Galzerani, Phys. Rev. B 52 (1995) 2610– 2618. [7] G. Bastard, Phys. Rev. B 24 (1981) 5693–5697. [8] H. Richter, Z.P. Wang, L. Ley, Solid State Commun. 39 (1981) 625–629. [9] G. Irmer, M. Wenzel, J. Monecke, Phys. Rev. B 56 (1997) 9524–9538. ˜ [10] A. Mlayah, R. Carles, G. Landa, E. Bedel, A. Munoz-Yagüe, J. Appl. Phys. 69 (1991) 4064–4070.
Contents lists available at ScienceDirect
Vibrational Spectroscopy journal homepage: www.elsevier.com/locate/vibspec
Raman investigation of plasmon localization in GaAs/AlGaAs superlattices A.D. Rodrigues a,∗ , Yu.A. Pusep b , J.C. Galzerani a a b
Departamento de Física, Universidade Federal de São Carlos, CP676, 13565-905, São Carlos, SP, Brazil Instituto de Física de São Carlos, Universidade de São Paulo, 13560-970, São Carlos, SP, Brazil
a r t i c l e
i n f o
Article history: Received 28 January 2010 Received in revised form 30 September 2010 Accepted 4 October 2010 Available online 12 October 2010 PACS: 71.45.Gm 63.20.kd 71.55.Jv 72.15.Rn
a b s t r a c t This work presents a Raman analysis of the influence of disorder in the plasmon localization. The disorder strength control was achieved by growing intentionally disordered GaAs/Al0.3 Ga0.7 As superlattices. The calculation of the localization extents was made by determining the coherency lengths of the plasmonLO coupled modes in the frequency range of the AlAs-like optical vibration, that presents a plasmon character. The plasmon damping was obtained from the Raman scattering related to the overdamped plasmon. These calculations allowed us to establish the relationship between the plasmon localization effect and the disorder strength, and to determine the limit of the transition from the delocalization to the strong localization regimes of the plasmon. © 2010 Elsevier B.V. All rights reserved.
Keywords: Plasmon Disorder Localization Raman spectroscopy
1. Introduction The transport properties of the electrons are directly influenced by the presence of disorder potentials. In weak disorder conditions we can consider that the electronic wave functions extend for the whole material. Otherwise, in the limit of strong disordered system, the propagating carriers undergo successive scatterings that restrict their trajectories, resulting in the localization of the electronic wave functions. Both the coherence length of the localization and the transition to a delocalizated state have been studied considering single-particle excitations, as in references [1,2]. The free-carriers can interact giving rise to a collective excitation, the so called plasmon. The plasmon wave function can be related to its Raman scattering intensity [3]. These facts point to the possibility to study the plasmon localization using Raman spectroscopy. Besides to contribute to the understanding of the physical principles of the plasmon propagation and localization, the understanding of the dynamical properties of the collective carriers excitations is essential to the development of new types of photonic devices [4]. In this work we present a way to use the Raman techniques in the investi-
∗ Corresponding author. Tel.: +55 16 3351 9727; fax: +55 16 3351 8464. E-mail addresses: [email protected], [email protected] (A.D. Rodrigues). 0924-2031/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.vibspec.2010.10.003
gations of the influence of the disorder in the plasmon localization. To proceed our analysis, a system in which the disorder strength could be quantitatively controlled was extremely necessary. It was achieved by growing intentionally disordered GaAs/Al0.3 Ga0.7 As superlattices. 2. Materials and methods The samples prepared to perform our studies consist of intentionally disordered (GaAs)m (Al0.3 Ga0.7 As)6 superlattices (SL’s) grown by molecular beam epitaxy (MBE) techniques. In order to produce a carrier excess with density N = 1.2 × 1018 cm−3 , the samples were homogeneously (well and barriers) doped with Si. The desired disorder control was achieved by the random fluctuations of the GaAs well thicknesses around the nominal value of 17 monolayers (ML’s), during the growth process. In that case, the energies of the non interacting electrons in the wells depend on the well thicknesses. By calculating the lowest energy of the electrons in each isolated quantum well, we obtain a distribution of the energy levels that describes a Gaussian shape. The more it is varied the thicknesses of the wells (i.e., the more disordered the superlattice), the larger is the full width at half maximum of the Gaussian distribution . In order to establish a disorder magnitude we have calculated the period dependent width W of the lowest energetic miniband for a pattern (in the absence of disorder) superlattice,
A.D. Rodrigues et al. / Vibrational Spectroscopy 54 (2010) 174–178
175
with well (barrier) width 17 ML’s (6 ML’s). The obtained value was W = 52 meV. So, the disorder strength of each superlattice can be described by a disorder parameter defined as ı = /W. In order to check the relation between the miniband formation and the barrier thickness, (GaAs)m (Al0.3 Ga0.7 As)n SL’s with N = 5 × 1017 cm−3 were prepared. Every sample in that set presents the same well thickness m = 10 ML’s, meanwhile the barrier width was varied assuming values 8 ≤ n ≤ 100 ML’s for each sample. The Raman scattering were collected from the samples surfaces in the backscattering geometry. The spectra were analysed by a Jobin Yvon T64000 triple monochromator equipped with a liquid nitrogen cooled CCD. The 514.5 nm line of an Ar+ laser was used as the excitation source. Since this energy lies far away from any electrical transition in that materials, no resonance effect is expected. The measurements were carried out at 8 K. 3. Theory Before proceeding any quantitative Raman analysis in the studied systems, it is essential to know the nature of the vibrational excitations and the influence of the density of free-carriers to their energy spectrum. So, we present bellow the calculation of the vibrational dispersion relation for a (GaAs)17 (Al0.3 Ga0.7 As)6 doped superlattice. In polar highly doped semiconductor a quantum of the freecarriers oscillation (plasmon) can interact with the longitudinaloptical (LO) phonons. The resulting coupled LO-plasmon mode (CLPM) is generated when the density fluctuation of the freecarriers acts screening the macroscopic electric field associated with the LO phonon. Its dispersion relation gives rise to two branches, one with energy lower (ω− ) and another with energy higher (ω+ ) than the unscreened LO phonon. For vanishing wave vectors q the energies of the two branches of the CLPM strongly depend on the plasma frequency (or the carrier concentration) ωp = 1/2
(4e2 N/m) , where e, N and m are the electron charge, concentration and mass, respectively. If the plasma frequency is smaller than the LO frequency, the ω− mode presents a plasmon-like character and the ω+ , a phonon-like one. For very high free-carriers density (plasmon frequency larger than the LO frequency), the ω− mode shows a phonon-like behavior and its frequency approaches that of the TO phonon; meanwhile, ω+ presents a plasmon-like character, indicating that the electric field associated with the LO phonon is almost completely screened by the free-carriers. By using the continuum dielectric model, the corresponding diagonal component of the dielectric function tensor along the growth direction z can describe the ionic part of the superlattice dielectric function. Considering the electric contribution due to the collective carriers excitations, the z component of the total superlattice dielectric function can be written in the long wavelength approximation as
εz (ω, q) = (d1 + d2 ) 2 (q) ωpz − ω2
d1 (ω2 − ωT21 (q)) 2 (q)) ε∞1 (ω2 − ωL1
+
d2 (ω2 − ωT22 (q))
−1
Fig. 1. Dispersion relations of the coupled modes calculated for a periodic (GaAs)17 (Al0.3 Ga0.7 As)6 superlattice with carrier density N = 1.2 × 1018 cm−3 .
sion relations for the doped (GaAs)17 (Al0.3 Ga0.7 As)6 superlattice, shown in Fig. 1 4. Results and discussion Fig. 2 shows the optical phonons range of the Raman spectra of the (GaAs)10 (Al0.3 Ga0.7 As)n SL’s with barrier widths 8 ≤ n ≤ 100 ML’s. The spectra were taken in the z(x, y)¯z cross-polarized scattering geometry. In this configuration only the vibrational modes with longitudinal character are expected to contribute to the Raman scattering. At the interval between 275 and 300 cm−1 it can be noticed a mixing of Raman peaks related to the longitudinal optical GaAs-like (LO1 ) mode both from the GaAs wells and the Al0.3 Ga0.7 As barriers. At around 380 cm−1 the superlattices spectra show pronounced peaks with frequency close to the longitudinal optical AlAs-like modes. In fact, for thicker barrier superlattices (n ≥ 25 ML’s) this spectral feature can be attributed to the longitudinal optical AlAs phonon, labeled LO2 . Its asymmetric shape is due to the contribution of a density of phonon states to the Raman signal. If the vibrations have a finite coherence length, an enhancement in the momentum uncertainty of the oscillations promotes a breaking in the selection rule for the momentum conservation, allowing excitations with q > 0 to contribute to the Raman intensity, with energies given by their dispersion relations. Thus, the Raman efficiency of localized vibrations can be given by [8]
2 (q)) ε∞2 (ω2 − ωL2
I(ω) ∝ (1)
where d1 and d2 are the thicknesses of the wells and of the barriers, respectively; ωpz is the q dependent frequency of the vertically polarized plasmon, obtained by the random phase approximation in the limit of long wave lengths [5]. ε∞1 and ε∞2 are, respectively, the GaAs and Al0.3 Ga0.7 As high frequency dielectric constants and ωL1(2) and ωT1(2) are the LO and TO dispersion relations for the GaAs (AlAs) taken from reference [6]. The effective electronic mass, that is necessary in order to determine the vertical plasmon frequency ωpz , was obtained using the envelope function approximation [7]. By calculating the zeros of Eq. (1) we have found the CLPM disper-
exp
−
q2 Lc2 4
d3 q [ω − ωexc (q)]2 + (/2)
2
(2)
where Lc is the collective excitation correlation length, is the damping constant and ωexc (q) is the dispersion relation of the respective excitation. Since the AlAs longitudinal optical phonons present a negative dispersion relation, a contribution of states with wave numbers bellow the center of the Brillouin zone ( point) is expected in the Raman spectrum, provoking downward asymmetric shape in the LO2 lines. On the other hand, for the superlattice with barrier thinner than 10 ML’s the same Raman line acquires an opposite asymmetric shape and its frequency shifts upward. This effect is caused by a miniband formation in the electron energy spectrum. By decreasing the barriers thicknesses the electrons can
176
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Fig. 3. Raman spectra of the(GaAs)m (Al0.3 Ga0.7 As)6 superlattices with several disorder parameters ı, taken in the z(x, y)¯z scattering geometry.
Fig. 2. Raman spectra of (GaAs)10 (Al0.3 Ga0.7 As)n superlattices with8 ≤ n ≤ 100 ML’s, measured in z(x, y)¯z configuration.
tunnel between adjacent wells. The plasmon propagating parallel to the heterostructure growth axis interacts with the AlAs LO phonon of the barriers, giving rise to the coupled LO2 -plasmon mode ω2+ . Since these vibrations present positive dispersion relation due to their plasmon-like character (see Fig. 1), in presence of localization an upward asymmetry is expected for the related Raman peak. The opposition between the LO phonon and the coupled mode dispersion relations is also responsible for the blue shift in the peak maximum. Additionally, in the referred spectra we can notice the low-frequency coupled mode ω− and the mode resulting from the coupling between the vertical plasmon and the LO GaAs-like phonon, labeled ω1+ . Our analysis previously made showed that, for superlattices with thin enough barriers, the vertically polarized plasmons can couple with the LO phonons, provoking clear signatures in the Raman spectra. Thus, the Raman measurements present an efficient technique to determine how thin the superlattice barriers must be in order to promote the miniband formation. Furthermore, the quantitative analysis in the line shape modifications can be successfully used to investigate the dynamic properties of the plasmons and how they can be influenced by structural parameters such as disorder potentials. Fig. 3 shows the Raman spectra of random (GaAs)m (Al0.3 Ga0.7 As)6 SL’s prepared with the disorder parameters ı = 0.18, 0.35, 0.59, 0.82 and 1.13, taken in the same geometric configuration as in the first analysis. Sample with ı = 0.18 is a nominal ordered superlattice. The monolayer fluctuation in the pattern superlattice period produces a small disorder parameter even for the periodic superlattice. In fact, the obtained Raman
Fig. 4. Experimental (thin line) and calculated (dark gray line) Raman spectra of (GaAs)m (Al0.3 Ga0.7 As)6 superlattice with disorder parameterı = 0.59. The dotted and the dashed lines are components of the calculated spectrum due to the ω− and ω2+ , respectively. For simplicity, the Lorentzian curves related to TO1 , LO1 and TO2 used in the fitting are omitted in this figure.
A.D. Rodrigues et al. / Vibrational Spectroscopy 54 (2010) 174–178
Fig. 5. (a) Plasmon spatial extent Lc (full circles) and damping constant p (open circles) as function of disorder parameters. The dashed lines are guide to the eyes. (b) Ratio between the integral intensity of ω− (Iω− ) and ω2+ (Iω+ ) peaks, as function 2
of disorder parameter. (c) Temperature dependence of the plasmon localization for the strongest disordered superlattice.
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excitation for each superlattice could be determined by fitting its experimental spectrum and the calculated one, as shown in Fig. 4 for the sample with ı = 0.59. There the solid thin and the dark gray lines are the experimental and the calculated spectra, respectively. In order to obtain the Lc values, the Raman intensity in the range of AlAs optical vibrations was calculated using Eq. (2) (dashed line). In these calculations we have used the dispersion relation for the coupled mode ω2+ , determined as shown in Section 3. The damping constants were determined considering the spectral contribution of an overdamped plasmon (dotted line) using the theory presented in reference [10]. The obtained plasmon length (Lc ) and damping ( p ) are presented in Fig. 5(a) as full and opened circles, respectively, as a function of the disorder strength. With the increasing of disorder the spacial extent of the plasmon continuously diminishes, indicating a tendency of the collective carriers excitations to localize due to the disorder potential in random superlattices. At the same time the plasmon damping constant rapidly increases. For ı values larger than 0.6 the damping does not significantly change, indicating that the overdamped character of the plasmon in the highly disordered superlattices leads to a strong localization in the carriers excitation. The transition from the weak to a strong localization regime in the plasmon, for disorder parameter values upper than that, can be faced as a changing in the collective carriers oscillations behavior from a conducting regime to a insulator one. Another information about the plasmon localization can be taken from the analysis of the relation between the integral intensity of the overdamped plasmon Raman line in the GaAs energy range and the ω2+ plasmon-like mode with frequency in the AlAs range. This ratio presents a substantial increase with increasing disorder, as can be seen in Fig. 5(b). This suggests that the plasmons exhibit a tendency to localize in the GaAs wells of the random superlattices. In addition, the dependence of the plasmon localization length with temperature was also studied. Fig. 5(c) shows the obtained Lc for the superlattice with ı = 1.13 at temperatures varying between 8 and 250 K. The localization length increase reveals that the plasmon presents a tendency to delocalize as the temperature increases, as expected for any localized quasiparticule. Even for the most disordered superlattices, where the plasmon undergoes a strong localized regime at low temperatures, the obtained spacial extent of the plasmon is comparable to that found for the periodic superlattice at 8 K (Fig. 5(a)).
5. Conclusions spectrum for this sample is quite similar to that found in Fig. 2 for the periodic superlattices with barrier thicknesses shorter than 10 ML’s. As can be seen in Fig. 3, the ω2+ plasmon-like Raman line presents different line broadenings for distinct disorder parameters, as a manifestation of different non conservation strengths in the momentum, probably due to distinct localization lengths. Another significantly change in the Raman spectra of the disordered superlattices can be observed; as the disorder strength increases, the ω− line shape changes, from a down to and upward asymmetry. This feature is attributed to an overdamped behavior of the plasmon, that occurs when the plasmon damping p is of the same order or higher than the plasmon frequency ωp . In the presence of an overdamped plasmon in a specific polar semiconductor, the low and high-frequency coupled modes vanish and only one peak presenting upward broadening shows up in the Raman spectrum in the vicinity of the TO mode frequency [9]. In our case, the potential produced by the most disordered superlattices is strong enough to enhance the plasmon damping until its magnitude becomes comparable to (or higher) the plasma frequency. Since the disorder dependent modifications in the spectra are influenced by the dynamic properties of the vertically polarized plasmons, both the localization extent Lc and the damping of the collective carriers
We have shown how the Raman analysis can be used to study the dynamic properties of plasmons, specially in presence of disorder. The chosen system was the GaAs/Al0.3 Ga0.7 As superlattice due to the possibility of controlling the disorder strength. Firstly, we have probed the existence of a vertically polarized plasmon due to the miniband formation, confirmed for superlattices with barrier thickness shorter than 10 ML’s. The calculation of both the plasmon localization and damping could be studied as a function of disorder using random superlattices with different disorder parameters. The results allow us to conclude that for ı > 0.6 the plasmon presents a strong localization character. So, as the disorder increases, the plasmon undergoes a transition from the delocalization to localization regimes, similar to that found in the metal-insulator transition. This assumption is complemented by the relative increasing in the integral intensity of the Raman line related to the overdamped plasmon, with the increasing of the disorder strength. Furthermore, even for the most disordered superlattice, the plasmon localization diminishes with the temperature increase. The localization length calculated for the superlattice with ı = 1.13 at room temperature is very close to that found in the periodic superlattice (ı = 0.18), where the plasmon is delocalized.
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