Here, we demonstrate that high energy electrons can be used to explore the collective oscillation of s, p, and d orbital electrons at the nanometer length scale. Using epitaxial AlGaN/AlN quantum wells as a test system, we observe the emergence of additional features in the loss spectrum with the increasing Ga content. A comparison of the observed spectra with ab-initio theory reveals that the origin of these spectral features lies in excitations of 3d-electrons contributed by Ga. We find that these modes differ in energy from the valence electron plasmons in Al1–xGaxN due to the different polarizabilities of the d electrons. Finally, we study the dependence of observed spectral features on the Ga content, lending insights into the origin of these spectral features, and their coupling with electron-hole excitations.

The collective oscillation of free electrons (plasmons) is well described by the Drude model, which is based on the classical equations of motion for a free electron gas.1,2 In response to an externally applied electromagnetic field, the polarization of the electron gas generates a restoring force, creating longitudinal oscillatory modes in the Fermi sea. These modes can be sustained in any material at a characteristic frequency where the real part of the dielectric function goes to zero,3 known as the plasma frequency, ωp. For a free electron gas, ωp=ne2ϵm, where n is the density of electrons, e and m are the charge and mass of an electron, respectively, and ϵ is the permittivity. In spectroscopic studies of simple metals, this energy is also referred to as the “ultraviolet transmission limit” since metals reflect photons with energy less than ωp and transmit above this energy. This energy is very sensitive to the density of electrons in the Fermi sea undergoing oscillation, spanning a spectral range from the deep UV to far infrared. In particular, as the density of electrons varies from 1022 cm–3 (in simple metals) to 1010 cm–3 in (in dielectrics), this plasma energy varies from ∼4 eV to ∼4μeV. In a typical semiconducting material such as silicon, the valence band is filled at room temperatures, with each atom contributing four electrons to the Fermi sea. This gives rise to a plasmon of energy of ∼17 eV. Despite spanning this large spectroscopic range, the fundamental physics of these excitations is described with considerable success using the simple free electron model.

Within the free electron model, several important assumptions are made: all electrons in the system are treated as identical (or, equally “free”), and the interactions between these electrons, as well as the influence of the periodic crystal potential generated by the “ion cores” in the solid, are ignored.3 Consequently, any interaction between “collective modes,” i.e., plasmons, and the spectrum of single electron states (interband transitions) are not considered in the Drude model. In most metals, ignoring the periodic crystal potential is justified since typical plasma frequencies (5–50 eV) are significantly higher than the energy of the single particle states (electron–hole pairs), and there is minimal admixing of plasmons and electrons. In wide bandgap semiconductors, such as GaN (Eg = 3.4 eV) or AlN (Eg = 6 eV), however, solid-state effects are pronounced. The problem is further complicated in AlGaN due to the presence of delocalized core states (referred to as “semi-core”).

Furthermore, in semiconductors, the plasmon coupling with the single particle electron–hole pair excitations (Landau damping) is an important decay channel through which plasmons dissipate energy, thus reducing the plasmon lifetime. Several applications of plasmons have been proposed, such as the creation of a perfect lens,4 invisibility cloaking,5 and as conduits of on-chip signals which can interface naturally with optics.6 The realization of these exciting applications, however, requires knowledge of the various scattering mechanisms that affect plasmons and their relative importance. By mapping such plasmon modes at the nanometer scale, the role of interfaces, surfaces, and compositional fluctuations can be connected to plasmon scattering mechanisms in inhomogeneous media.

Features associated with plasmon creation are observed using spectroscopic techniques such as X-Ray absorption spectroscopy (XAS) and electron energy loss spectroscopy (EELS). While XAS typically offers better spectral resolution, EELS is capable of providing information at the atomic scale by scanning a finely focused high energy electron probe across the sample.7 For plasmon imaging, the spatial resolution is limited to nanometer length scales by delocalization of the plasmon excitation.8 In the case of metal nanoparticles, this technique allows experimentalists to probe the dependence of the plasmon energy on a particle shape,9,10 the coupling and energy splitting of plasmonic modes of particles separated by a nanoscale gap,11,12 the mapping of surface plasmons,13 and the quantization of the plasmon modes.14 A collection of such results is presented in the review by Colliex et al.15 

In this letter, we analyze the plasmon loss signal in AlN/Al1–xGaxN quantum wells using scanning transmission electron microscopy (STEM) and EELS. We show that in addition to the primary plasmon peak (due to the collective motion of the valence band electrons), two additional spectral features (δ1 and δ2) appear when Ga is present. We study the energies of these modes as a function of the Ga composition and find that while the primary valence electron mode downshifts in energy, the additional modes shift to higher energies with the increasing Ga fraction. Comparison with ab-initio calculations shows that these additional spectral features arise from the polarization of Ga d orbital electrons. Through this insight, we explain the dependence of plasmon energy on the Ga composition, as well as changes in the peak linewidths, which sheds light on the relative importance of different plasmon scattering mechanisms.

AlN/Al1–xGaxN quantum well structures were grown by metal-organic chemical vapor deposition (MOCVD) on vicinal c-plane AlN and sapphire substrates as described previously.16,17 The quantum well width was varied between 2 nm and 50 nm, and the composition range was varied between AlN (x = 0) and GaN (x = 1). Electron microscopy cross-sectional samples were wedge polished to electron transparency using an Allied Multiprep system.18 For final thinning, the samples were argon ion milled (Fischione Model 1050) at energies decreasing from 2 keV to 200 eV. A probe-corrected FEI Titan 60–300 kV STEM/TEM equipped with an X-FEG source was used for imaging and spectroscopy. Energy dispersive X-Ray spectroscopy was used to determine the composition of each quantum well and was found to be in agreement with the expected values from the MOCVD conditions. The electron beam acceleration voltage was 80 keV to increase the cross-section of plasmon creation. Electron energy loss spectroscopy (EELS) was performed with a Gatan Enfinium ER spectrometer and with an energy spread of approximately 0.9 eV. As measured with the EELS log-ratio method, the sample thickness varied between 40 and 80 nm.19 For plasmon energy and linewidth analysis, the spectra were first aligned relative to the zero-loss signal. The effects of plural scattering were then removed using Fourier-log deconvolution.8 The plasmon loss peak was fit to the Drude model, where the permittivity of an electron gas with density n is given by

ϵ(ω)=1ωp2ω2+iω/τ,
(1)

and the loss function is given by Im {1/ϵ}, which corresponds to a Lorentzian peak centered around the plasmon energy, ωp, having a linewidth given by /τ, where τ is the plasmon lifetime.

The EEL spectra shown in Fig. 1(a) are acquired from quantum well structures with various Al1–xGaxN compositions. The different quantum wells, as in Fig. 1(b), are otherwise similar in terms of sample thickness. The image intensity variation in Fig. 1(b) results from the atomic number sensitivity of HAADF STEM and scales with the Ga content. The variation of the valence plasmon energy across this quantum well structure is shown in Fig. 1(c). Two trends are established from these data. First, increasing Ga incorporation leads to a downshift of the “primary” plasmon peak (p1) to lower energy. Second, additional energy loss signatures emerge at ∼24 and ∼28 eV. The downshift in the energy of the primary plasmon could result from the decreasing bandgap and the increasing unit cell volume with the Ga content. Since both Al and Ga are group III elements, they both contribute 3 valence electrons per atom to the crystal. Hence, this increase in the unit cell size leads to a decrease in valence electron density and consequently a downshift in the plasma frequency (given by ωP=ne2ϵ0me). A similar argument was put forth for aluminum undergoing thermal expansion.20 A third mechanism for this downshift is electron-plasmon coupling, which is discussed further below. By mapping the position, width, and amplitude of these plasmon modes across the quantum well structures, one can spatially map the Ga fraction of these Al1–xGaxN alloys. This offers a complementary technique to methods such as EDS or quantitative STEM imaging for identifying the alloy composition.21,22

FIG. 1.

(a) Energy loss spectra at varying compositions of Al1–xGaxN. (b) HAADF STEM overview of the quantum well structures used to obtain the spectra in (a). (c) The corresponding plasmon energy shift across the region in (b), revealing a downshift in the plasmon energy as the Ga fraction increases.

FIG. 1.

(a) Energy loss spectra at varying compositions of Al1–xGaxN. (b) HAADF STEM overview of the quantum well structures used to obtain the spectra in (a). (c) The corresponding plasmon energy shift across the region in (b), revealing a downshift in the plasmon energy as the Ga fraction increases.

Close modal

The emergence of two additional excitations at 24 eV and 28 eV can be understood by considering the difference in orbital configurations of Ga and Al. Ga contributes d electrons to the Fermi sea due to its occupied 3–d shell, while Al does not. To explore this, we examine results from ab-initio calculations of the Al1–xGaxN dielectric response in the random phase approximation. Many body effects are taken into account using the GW approximation, and the resulting spectral function is calculated. To understand the resulting spectra, rather than using Drude's classical description of a “free-electron” gas, we use the Lindhard dielectric response theory3 to provide a rigorous framework for exploring plasmons in semiconducting materials. The Lindhard model describes a many-electron system in the influence of an external perturbation. This perturbation modifies the electron wavefunctions in this system, leading to a redistribution of charges and hence the creation of an electrostatic potential. By ensuring self-consistency between this generated response field and the original perturbation, we arrive at the following expression for permittivity

ϵ(q,ω)=1+4πe2q2k[f(k)f(k+q)ε(k+q)ε(k)ω+iα],
(2)

where f(k) is the probability of occupation of an electronic state with momentum k and energy ε(k), while ω is the frequency of the original perturbation, q is the scattering wavevector, and α is introduced to avoid singularities in the summation. It should be noted that this summation extends over all electronic states (occupied as well as unoccupied), denoted by their momentum quantum number k. Although we have omitted oscillator strengths from this equation, those matrix elements are included in the full calculation of the spectral function. At energies much higher than the Fermi energy, states do not contribute to the summation as the numerator goes to zero.

We calculate the spectral function of GaN, systematically increasing the number of electronic bands included in the summation (supplementary material). The results of the calculations are shown in Figs. 2(a) and 2(b), where the calculated real part of the dielectric function and the loss function [i.e., 1/ϵ(ω)] are provided along with experimental data. As the electronic states of the Ga 3–d shell are included in the calculation, we observe the emergence of additional loss signatures at approximately 24 eV and 28 eV, in excellent agreement with the experiment. When these states are excluded from the calculation, the loss function of GaN very closely resembles that of AlN. This leads to the conclusion that these additional features arise from the polarization of d orbital electrons contributed to the crystal as Ga is incorporated in the lattice. While these states resemble excitations of core electrons from the 3–d states, we argue that they show characteristic signs of delocalization and are referred to these as “semi-core.”

FIG. 2.

Evolution of the (a) calculated real component of the dielectric function and (b) loss function of GaN with the varying number of Ga d states taken into account. The experimentally recorded EEL spectrum of GaN is also shown.

FIG. 2.

Evolution of the (a) calculated real component of the dielectric function and (b) loss function of GaN with the varying number of Ga d states taken into account. The experimentally recorded EEL spectrum of GaN is also shown.

Close modal

We also observe that in our calculations for the real part of the dielectric function (Fig. 2), as Ga-3d states are included in the calculation, the zero crossing downshifts by about 1 eV. This tracks very closely with the observed downshift as the composition varies from AlN to GaN. Raether had shown that in the case of electron-plasmon coupling,23 the plasmon mode is expected to downshift if the energy of electronic transitions (Ei) is greater than the energy of the plasmon (EP). Since Ei= 24 eV and EP= 20 eV here, the combination of electron-plasmon coupling, increasing lattice constant, and decreasing bandgap agree with the observed downshifting plasmon energy.

We can also calculate the expected magnitude of the p1 shift due to each of the three aforementioned mechanisms. The shift due to the presence of a bandgap is expected to add in quadrature with the plasmon energy, E=EP2+Eg2, where E is the observed peak in the loss spectrum, EP is the plasmon energy, and Eg is the bandgap. Considering that the bandgap changes from 3.4 eV for GaN to 6.05 eV for AlN, the maximum shift due to the change in bandgap is expected to be ∼0.6 eV. Similarly, the unit cell volume for GaN is about 6%. The resulting change in plasmon energy is again ∼0.5 eV. The p1 energy, however, shifts by nearly a factor of two more (∼2 eV) as the composition is varied from GaN to AlN. The shift is therefore not accounted for by the change in bandgap and unit cell volume but is instead likely a consequence of electron-plasmon coupling, as discussed above.

Several different Al1–xGaxN compositions are also used to study the evolution of the energy-loss features as a function of the Ga content, as in Fig. 3. As explained above, the main bulk plasmon (p1), originating from the s and p valence electrons, downshifts as the Ga composition is increased. In contrast, both δ1 and δ2 modes associated with d-electrons upshift as Ga is added. This provides additional evidence that the origin of these modes is the oscillation of d orbital electrons. The dispersion of these states with the composition also sheds light on the nature of these states. Solid state excitations are typically either described as “free”/“delocalized” (such as valence electrons and plasmons) or as “bound”/“localized” (such as core electrons and defect states).

FIG. 3.

(a) Energy loss spectrum for GaN with the three spectral features. Each mode is fit to a Lorentzian, which is used to extract [(b) and (c)] energies and [(d) and (e)] linewidths. The fits from δ2 suffer from fitting uncertainty and are, hence, shown only in Fig. S2 (supplementary material).

FIG. 3.

(a) Energy loss spectrum for GaN with the three spectral features. Each mode is fit to a Lorentzian, which is used to extract [(b) and (c)] energies and [(d) and (e)] linewidths. The fits from δ2 suffer from fitting uncertainty and are, hence, shown only in Fig. S2 (supplementary material).

Close modal

The observed losses are reminiscent of photoemission spectroscopy, where such features have been associated with excitation of localized core d electrons into the empty conduction band. The observed energy splitting is associated with the crystal field splitting of d orbitals, which hybridizes during bonding to introduce a further energy split.24,25 In the EELS data reported here, however, three findings point to an alternative picture. First, the energy of truly localized core states is not expected to vary with the Ga composition since these states do not interact with each other. In that case, the reduced bandgap GaN would lead to a downward shift in both δ1 and δ2 spectral features with increasing Ga. Second, the binding energy of the 3d electrons of Ga is ∼18.7 eV, significantly smaller than the observed 24–28 eV. Lastly, although the binding energy of core states is expected to vary somewhat with changes in Ga coordination, since the Ga-N bond is less covalent than Al-N, the binding energy of Ga 3–d states would be expected to increase in Al rich Al1–xGaxN.24 In contrast to our observations, all three factors indicate that the δ1 and δ2 energies would downshift if they resulted from purely localized core state transitions.

Next, we consider the other extreme, i.e., a purely delocalized picture. To a naive approximation, we treat the system of d electrons as a polarizable electronic sea. In that case, the corresponding energy loss varies as n. An upshift in energy loss is then expected from the increasing number density of available d electrons, in agreement with the experiment. Although significant simplifications are being made in this picture, this n dependence yields the energy for a polarizable sea of d electrons to be approximately 23 eV. In ab-initio theory calculations of the electronic properties of GaN, the importance of this delocalized nature of the Ga-3d states has already been recognized. This delocalization is enabled by the 2s states of N, which are almost degenerate with the Ga-3d states. In the absence of this degeneracy, d-shell electrons in Ga would be well localized core states, owing to the large atomic radius of Ga, which reduces the overlap integral. The resonance with N-2s states enables this delocalization.26 Hence, what should be localized core-states originating from the Ga-3d shells turn into somewhat delocalized “semi-core” states, and wavefunctions of all the d-electrons in the system hybridize significantly. Neglecting this delocalization was the source of several inaccuracies in the very first band-structure calculations of AlGaN, as highlighted by Fiorentini.26 

We also note the trends for the linewidth of these spectral features. As the Ga fraction increases, we observe a sharpening of the true plasmon peak p1. In principle, these linewidths yield information about the plasmon excitation lifetime.27 Due to the broad energy spread of the electron beam, however, these linewidths provide only a qualitative measure of the various scattering rates. The excitation lifetime is determined by various scattering mechanisms such as the interaction with phonons, or decay into electron–hole pairs, as well as disorder scattering due to alloying. If disorder related scattering due to compositional fluctuation were the dominant scattering mechanism, one expects the highest scattering rates at compositions around Al0.5Ga0.5N. This is not supported by trends presented in Fig. 3. The results are, however, consistent with what can be expected if electron–plasmon coupling is dominant. Typically, the scattering rate between two quantum states is of the form 1τ=|VAB|εAεB, where |VAB| is the coupling matrix element between states A and B and εA and εB are the energies of these states. Hence, this scattering rate is inversely proportional to the difference in energies of the states being coupled. In the case here, εA is the energy of the valence electron plasmon (p1) and εB is the energy of an electron–hole pair. Since the bandgap of GaN is much smaller than AlN, it is expected that the higher energy single electron states in AlN are more strongly coupled to plasmons than those in the lower bandgap GaN. The plasmon linewidth is thus expected to broaden more for AlN than GaN, in agreement with our observation. Recent theoretical studies have predicted similar phenomena in semiconducting systems including emergence of new collective excitations and broadening due to electron–plasmon coupling in semiconductors.28–30 

In conclusion, we find that low-loss EEL spectra of AlGaN alloys show three dominant spectral features (p1 from the oscillation of valence electrons and δ1 and δ2 from the delocalized oscillation of d-electrons of Ga). The nitride materials used in this study offer the ideal platform to observe such effects. As the alloy composition is varied from AlN to GaN, the increased unit cell volume, reduced bandgap, and coupling of Ga-3d states to the valence electron plasmon lead to changes in the loss spectra of AlGaN. By quantifying changes in the plasmon energies and linewidths, the influence of solid-state effects, such as electron–plasmon coupling, is revealed in these wide bandgap semiconductors. These results further demonstrate the applicability of low-loss EELS to probe subtle solid-state effects. Coupled with simultaneous atomic resolution STEM imaging, the approach can lead to spatial mapping of electron scattering pathways.

See supplementary material for additional details on the dielectric response calculations and additional spectral feature measurements.

The authors gratefully acknowledge the support for this research from the Air Force Office of Scientific Research (FA9550-14-1-0182). J.H.D. acknowledges the support by the National Science Foundation Graduate Research Fellowship (DGE-1252376). This work was performed in part at the Analytical Instrumentation Facility (AIF) at North Carolina State University, which was supported by the State of North Carolina and the National Science Foundation (ECCS-1542015). The AIF is a member of the North Carolina Research Triangle Nanotechnology Network (RTNN), a site in the National Nanotechnology Coordinated Infrastructure (NNCI).

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Supplementary Material