We present a detailed calculation of intensities of two-phonon and four-phonon Raman peaks in graphene. Writing the low-energy hamiltonian of the interaction of electrons with the crystal vibrations and the electromagnetic field from pure symmetry considerations, we describe the system in terms of just a few independent coupling constants, considered to be parameters of the theory. The electron scattering rate is introduced phenomenologically as another parameter. The results of the calculation are used to extract information about these parameters from the experimentally measured Raman peak intensities.
We calculate the probabilities of two- and four-phonon Raman scattering in graphene and show how the relative intensities of the overtone peaks encode information about relative rates of different inelastic processes electrons are subject to. If the most important processes are electron-phonon and electron-electron scattering, the rate of the latter can be deduced from the Raman spectra.
We present a comprehensive multiphonon Raman and complementary infrared analysis for bulk and monolayer MoS2.For the bulk the analysis consists of symmetry assignment from which we obtain a broad set of allowed second order transitions at the high symmetry M,K and gamma Brillouin zone points. The attribution of about 80 transitions of up to fifth order Raman processes are proposed in the low temperature(95K)resonant Raman spectrum measured with the excitation energy of 1.96 eV,which is slightly shifted from the A exciton. We propose that the main contributions come from four phonons:A1g(M),E12…
We report a study of long-range MoS2-substrate interaction using resonant Raman imaging. We observed a strong thickness-dependent peak shift of a Raman-forbidden mode that can be used as a new method of determining the thickness of multilayered MoS2 flakes. In addition, dependence of the Raman scattering intensity on thickness is explained by the interference enhancement theory. Finally, the resonant Raman spectra on different substrates are analysed.
Pb(Mg$_{1/3}$Nb$_{2/3}$)O$_{3}$ is one of the simplest representatives of the class of lead relaxors and often serves as a model system for more complicated compounds. In this paper we analyse both polarized and depolarized Raman scattering spectra, measured in the temperature range between 1000 and 100 K, by multiple peak decomposition. Basing on this analysis, we present the picture of transformations in the crystal and the nature of the spectral lines. According to our model, the formation of the Fm3m symmetry in the chemically ordered regions as well as the appearance and freezing of the p…
We present an analysis of deep-UV Raman measurements of graphite, graphene and carbon nanotubes. For excitation energies above the strong optical absorption peak at the $M$ point in the Brillouin zone ($\approx 4.7\,\text{eV}$), we partially suppress double-resonant scattering processes and observe the two-phonon density of states of carbon nanomaterials. The measured peaks are assigned to contributions from LO, TO, and LA phonon branches, supported by calculations of the phonon dispersion. Moreover, we gain access to the infrared-active $E_{1u}$ mode in graphite. By lowering the excitation en…
We have developed a theory of Raman scattering with single and double spin flips of localized resident electrons and holes as well as nonequilibrium localized excitons in semiconductor perovskite crystals under optical excitation in the resonant exciton region. Scattering mechanisms involving localized excitons, biexcitons and exciton polaritons as intermediate states has been examined, the spin-flip Raman scattering by polaritons being a novel mechanism. The derived equations are presented in the invariant form allowing one for the analysis of the dependence of scattering efficiency on the po…
We present a practical method to compute the vibrational resonant Raman spectra in solids with delocalized excitations. We apply this approach to the study of tetrahedrous amorphous carbon. We determine the vibrational eigenmodes and eigenvalues using density functional theory in the local density approximation, and the Raman intensities using a tight binding approximation. The computed spectra are in good agreement with the experimental ones measured with visible and UV lasers. We analyze the Raman spectra in terms of vibrational modes of microscopic units. We show that, at any frequency, the…
The resonant Raman scattering in N-IR - UV range from amorphous hydrogenated carbon (a-C:H) reveal inclusions of trans-polyacetylene (trans-(CH)x) chains with approximate length of up to 120 C=C units and inclusions of poly(p-phenylene vinylene) (PPV) polymer chains. The PPV is evidenced by a strong dispersive mode at ca. 1175 cm-1. It was found that the Raman response from core Ag trans-(CH)x modes incorporated into a-C:H to changing excitation energy is identical to of free-standing chains thus facilitating identification of trans-(CH)x in complex carbonaceous materials spectra.
Knowledge of the electronic band structure of multiferroic oxides, crucial for the understanding and tuning of photo-induced effects, remains very limited even in the model and thoroughly studied BiFeO3. Here, we investigate the electronic band structure of BiFeO3 using Raman scattering with twelve different excitation wavelengths ranging from the blue to the near infrared. We show that resonant Raman signatures can be assigned to direct and indirect electronic transitions, as well as in-gap electronic levels, most likely associated to oxygen vacancies. Their temperature evolution establishes …
Resonant Raman spectroscopies beyond density-functional theory
École polytechnique fédérale de Lausanne（洛桑联邦理工学院）；University of Bremen（不来梅大学）；PSI Center for Scientific Computing, Theory and Data（瑞士保罗谢勒研究所科学计算、理论与数据中心）
We report non-resonant x-ray Raman scattering (XRS) measurements from hexagonal boron nitride for transferred momentum from 2 to 9 $\mathrmÅ^{-1}$ along directions both in and out of the basal plane. A symmetry-based argument, together with real-space full multiple scattering calculations of the projected density of states in the spherical harmonics basis, reveals that a strong pre-edge feature is a dominantly $Y_{10}$-type Frenkel exciton with no other \textit{s}-, \textit{p}-, or \textit{d}- components. This conclusion is supported by a second, independent calculation of the \textbf{q}-depen…
Infrared reflectivity spectra of cubic SrMnO$_{3}$ ceramics reveal 18 % stiffening of the lowest-frequency phonon below the antiferromagnetic phase transition occurring at T$_{N}$ = 233 K. Such a large temperature change of the polar phonon frequency is extraordinary and we attribute it to an exceptionally strong spin-phonon coupling in this material. This is consistent with our prediction from first principles calculations. Moreover, polar phonons become Raman active below T$_{N}$, although their activation is forbidden by symmetry in $Pm\bar{3}m$ space group. This gives evidence that the cub…
Layered inorganic materials, such as the transition metal dichalcogenides (TMDs), have attracted much attention due to their exceptional electronic and optical properties. Reliable synthesis and characterization of these materials must be developed if these properties are to be exploited. Herein, we present low-frequency Raman analysis of MoS2, MoSe2, WSe2 and WS2 grown by chemical vapour deposition (CVD). Raman spectra are acquired over large areas allowing changes in the position and intensity of the shear and layer-breathing modes to be visualized in maps. This allows detailed characterizat…
We report a detailed study of the strong near-field Raman scattering enhancement which takes place in tip-enhanced Raman scattering (TERS) in indium nitride. In addition to the well-known first-order optical phonons of indium nitride, near-field Raman modes, not detectable in the far-field, appear when approaching the plasmonic probe. The frequencies of these modes coincide with calculated energies of second order combinational modes consisting of optical zone center phonons and acoustic phonons at the edge of the Brillouin zone. The appearance of strong combinational modes suggests that TERS …
Resonance Raman (RR) peaks of and stretching modes and their higher harmonics have been observed superimposed on photoluminescence (PL) spectrum of thin films. Commercial fluorine doped thin films deposited by sputtering on glass and thin films deposited on Si by laser ablation have been studied. The dispersions of CO and OH stretching RR modes are ~ 600 cm-1/eV and 800 cm-1 respectively. The dispersion of the third harmonic of CO stretching mode is ~ 2000 cm-1/eV. Similar dispersion of RR peak of stretching modes and higher harmonics superimposed on PL spectra has been observed in Mo1-xFexO2 …
Raman spectra obtained by the inelastic scattering of light by crystalline solids contain contributions from first-order vibrational processes (e.g. the emission or absorption of one phonon, a quantum of vibration) as well as higher-order processes with at least two phonons being involved. At second order, coupling with the entire phonon spectrum induces a response that may strongly depend on the excitation energy, and reflects complex processes more difficult to interpret. In particular, excitons (i.e. bound electron-hole pairs) may enhance the absorption and emission of light, and couple str…
Resonance Raman scattering, which probes electrons, phonons and their interplay in crystals, is extensively used in two-dimensional materials. Here we investigate Raman modes in MoSe$_2$ at different laser excitation energies from 2.33 eV down to the near infrared 1.16 eV. The Raman spectrum at 1.16 eV excitation energy shows that the intensity of high-order modes is strongly enhanced if compared to the first-order phonon modes intensity due to resonance effects with the MoSe$_2$ indirect band gap. By comparing the experimental results with the two-phonon density of states calculated with dens…
We systematically study the high-resolution and polarized Raman spectra of multilayer (ML) MoTe2. The layer-breathing (LB) and shear (C) modes are observed in the ultralow-frequency region, which are used to quantitatively evaluate the interlayer coupling in ML MoTe2 based on the linear chain model, in which only the nearest interlayer coupling is considered. The Raman spectra on three different substrates verify the negligible substrate effect on the phonon frequencies of ML MoTe2. Ten excitation energies are used to measure the high-frequency modes of N-layer MoTe2 (NL MoTe2; N is an integer…
Inclusions of sp-hybridised, trans-polyacetylene [trans-(CH)x] and poly(p-phenylene vinylene) (PPV) chains are revealed using resonant Raman scattering (RRS) investigation of amorphous hydrogenated carbon (a-C:H) films in the near IR - UV range. The RRS spectra of trans-(CH)x core Ag modes and the PPV CC-H phenylene mode are found to transform and disperse as the laser excitation energy \hbarωL is increased from near IR through visible to UV, whereas sp-bonded inclusions only become evident in UV. This is attributed to \hbarωL probing of trans-(CH)x chain inhomogeneity and the distribution of …
We present a detailed study of the Ti 3$d$ carriers at the interface of LaAlO$_3$/SrTiO$_3$ heterostructures by high-resolution resonant inelastic soft x-ray scattering (RIXS), with special focus on the roles of overlayer thickness and oxygen vacancies. Our measurements show the existence of interfacial Ti 3$d$ electrons already below the critical thickness for conductivity and an increase of the total interface charge up to a LaAlO$_3$ overlayer thickness of 6 unit cells before it levels out. By comparing stoichiometric and oxygen deficient samples we observe strong Ti 3$d$ charge carrier dop…
We calculate the resonant electronic Raman scattering for the Falicov-Kimball model near the Mott transition on a hypercubic lattice. The solution is exact, and employs dynamical mean field theory.
We report two new first-order Raman modes in the spectra of few-layer MoS$_2$ at 286~cm$^{-1}$ and 471~cm$^{-1}$ for excitation energies above 2.4~eV. These modes appear only in few-layer MoS$_2$; therefore their absence provides an easy and accurate method to identify single-layer MoS$_2$. We show that these modes are related to phonons that are not observed in the single layer due to their symmetry. Each of these phonons leads to several nearly degenerate phonons in few-layer samples. The nearly degenerate phonons in few-layer materials belong to two different symmetry representations, showi…
We perform tunable resonant Raman scattering on 17 semiconducting and 7 metallic singly resonant single wall carbon nanotubes. The measured scattering cross-section as a function laser energy provides information about a tube's electronic structure, the lifetime of intermediate states involved in the scattering process and also energies of zone center optical phonons. Recording the scattered Raman signal as a function of tube location in the microscope focal plane allows us to construct two-dimensional spatial maps of singly resonant tubes. We also describe a spectral nanoscale artifact we hav…
We present an exact solution for electronic Raman scattering in a single-band, strongly correlated material, including nonresonant, resonant and mixed contributions. Results are derived for the spinless Falicov-Kimball model, employing dynamical mean field theory; this system can be tuned through a Mott metal-insulator transition.
Resonant Raman signatures of bright and momentum-dark exciton coupled by intervalley phonon scattering in monolayer WSe2
University of Münster（明斯特大学）；Center for Soft Nanoscience (SoN), University of Münster（明斯特大学软纳米科学中心）；Technical University of Munich（慕尼黑工业大学）；Walter Schottky Institut and Physics Department, Technical University of Munich（慕尼黑工业大学沃尔特·肖特基研究所与物理系）；MCQST, München（慕尼黑量子科学与技术中心）
The appearance of a broad peak at $\sim$1550 cm$^{-1}$ (G$^{-}$ peak) in carbon nanotube resonant Raman scattering spectra has been conventionally attributed to the presence of metallic nanotubes. Here, we present resonant Raman measurements on macroscopic nanotube ensembles enriched in armchair species prepared through density gradient ultracentrifugation. Our data clearly demonstrate that the broad G$^{-}$ mode is absent for armchair structures and appears only when the excitation laser is resonant with non-armchair "metals." Due to the large number ($\sim$10$^{10}$) of nanotubes across seve…
Raman spectroscopy is the prime non-destructive characterization tool for graphene and related layered materials. The shear (C) and layer breathing modes (LBMs) are due to relative motions of the planes, either perpendicular or parallel to their normal. This allows one to directly probe the interlayer interactions in multilayer samples. Graphene and other two-dimensional (2d) crystals can be combined to form various hybrids and heterostructures, creating materials on demand with properties determined by the interlayer interaction. This is the case even for a single material, where multilayer s…
Using the methods of scanning electron (SEM) and atom force microscopy (AFM) as well as photoluminescence (PL) and Raman micro-spectroscopy, we investigated $Zn_{1-x}Co_xO$ films ($x=5$ and $15\%$) grown by molecular beam epitaxy on sapphire substrates. It is found that the films have a nanocrystalline structure with the grain size decreased from $\sim150$ down to $28$ nm at changing the $Co$ concentration from $5$ to $15\%$. High-resolution SEM images of $Zn_{0.85}Co_{0.15}O$ film have been interpreted as inhomogeneous lateral distribution of $Co$ atoms. Two broad emission bands observed in s…
The weakly screened electron-hole interactions in an atomically thin semiconductor not only downshift its excitation spectrum from a quasiparticle one, but also redistribute excitation energies and wavefunction characters with profound effects on diverse modes of material response, including the exciton-phonon scattering processes accessible to resonant Raman measurements. Here we develop a first-principles framework to calculate frequency-dependent resonant Raman intensities that includes excitonic effects and goes beyond the Placzek approximation. We show how excitonic effects in MoS2 strong…
The stacking faults (deviates from Bernal) will break the translational symmetry of multilayer graphenes and modify their electronic and optical behaviors to the extent depending on the interlayer coupling strength. This paper addresses the stacking-induced band splitting and folding effect on the electronic band structure of twisted bilayer graphene. Based on the first-principles density functional theory study, we predict that the band folding effect of graphene layers may enable the G band Raman double resonance in the visible excitation range. Such prediction is confirmed experimentally wi…
The temperature evolution of the resonant Raman scattering from high-quality bilayer 2H-MoS$_{2}$ encapsulated in hexagonal BN flakes is presented. The observed resonant Raman scattering spectrum as initiated by the laser energy of 1.96 eV, close to the A excitonic resonance, shows rich and distinct vibrational features that are otherwise not observed in non-resonant scattering. The appearance of 1$^{st}$ and 2$^{nd}$ order phonon modes is unambiguously observed in a broad range of temperatures from 5 K to 320 K. The spectrum includes the Raman-active modes, $i.e.$ E$_\textrm{1g}^{2}$($Γ$) and…
Detecting Magnetic Phase Transitions in Ion-Irradiated CrSBr Through Resonant Raman Scattering
Institute for Condensed Matter Physics, TU Darmstadt（达姆施塔特工业大学凝聚态物理研究所）；Department of Inorganic Chemistry, University of Chemistry and Technology Prague（布拉格化学技术大学无机化学系）；Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research（德累斯顿罗森多夫亥姆霍兹中心离子束物理与材料研究所）
Resonant soft X-ray Raman scattering measurements on NiO have been made at photon energies across the Ni 2p absorption edges. The details of the spectral features are identified as Raman scattering due to d-d and charge-transfer excitations. The spectra are interpreted within the single impurity Anderson model, including multiplets, crystal-field and charge-transfer effects. At threshold excitation, the spectral features consists of triplet-triplet and triplet-singlet transitions of the 3d8 configuration. For excitation energies corresponding to the charge-transfer region in the Ni 2p X-ray ab…
The electronic states at the direct band gap of monolayer WSe2 at the $K^+$ and $K^-$ valleys are related by time reversal and may be viewed as pseudo-spins. The corresponding optical interband transitions are governed by robust excitons. In double resonant Raman spectroscopy, we uncover that the 2s exciton state energy differs from the 1s state energy by exactly the energy of the combination of several prominent phonons. Superimposed on the exciton photoluminescence (PL) we observe the double resonant Raman signal. This spectrally narrow peak shifts with the excitation laser energy as incomin…
We present a theoretical model to describe the double-resonant scattering process in arbitrary carbon nanotubes. We use this approach to investigate the defect-induced $D$ mode in CNTs and unravel the dependence of the $D$-mode frequency on the CNT diameter and on the energy of the resonant optical transition. Our approach is based on the symmetry of the hexagonal lattice and geometric considerations, hence the method is independent of the exact model that is chosen to describe the electronic band structure or the phonon dispersion. We finally clarify the diameter dependence of this Raman mode…
We study the effects of chemical bonding on Raman scattering from benzenethiol chemisorbed on silver clusters using time-dependent density functional theory (TDDFT). Raman scattering cross sections are computed using a formalism that employs analytical derivatives of frequency-dependent electronic polarizabilities, which treats both off-resonant and resonant enhancement within the same scheme. In the off-resonant regime, Raman scattering into molecular vibrational modes is enhanced by one order of magnitude and shows pronounced dependence on the orientation and the local symmetry of the molecu…
University of Warsaw（华沙大学）；Wrocław University of Science and Technology（弗罗茨瓦夫科技大学）；Peking University Shenzhen Graduate School（北京大学深圳研究生院）；CNRS（法国国家科学研究中心）
In order to deepen in the knowledge of the vibrational properties of 2-dimensional MoS2 atomic layers, a complete and systematic Raman scattering analysis has been performed using both bulk single crystal MoS2 samples and atomically thin MoS2 layers. Raman spectra have been measured under non-resonant and resonant conditions using seven different excitation wavelengths from near-infrared (NIR) to ultraviolet (UV). These measurements have allowed to observe and identify 41 peaks, among which 22 have not been previously experimentally observed for this compound, characterizing the existence of d…
Understanding the stability limitations and defect formation mechanisms in 2D magnets is essential for their utilization in spintronic and memory technologies. Here, we correlate defects in mono- to multilayer CrSBr with their structural, vibrational and magnetic properties. We use resonant Raman scattering to reveal distinct vibrational defect signatures. In pristine CrSBr, we show that bromine atoms mediate vibrational interlayer coupling, allowing for distinguishing between surface and bulk defect modes. We show that environmental exposure causes drastic degradation in monolayers, with the …
The Raman effect -- inelastic scattering of light by lattice vibrations (phonons) -- produces an optical response closely tied to a material's crystal structure. Here we show that resonant optical excitation of IR and Raman phonons gives rise to a Raman scattering effect that can induce giant shifts to the refractive index and induce new optical constants that are forbidden in the equilibrium crystal structure. We complete the description of light-matter interactions mediated by coupled IR and Raman phonons in crystalline insulators -- currently the focus of numerous experiments aiming to dyna…
The nature of the interface in lateral heterostructures of 2D monolayer semiconductors including its composition, size, and heterogeneity critically impacts the functionalities it engenders on the 2D system for next-generation optoelectronics. Here, we use tip-enhanced Raman scattering (TERS) to characterize the interface in a single-layer MoS2/WS2 lateral heterostructure with a spatial resolution of 50 nm. Resonant and non-resonant TERS spectroscopies reveal that the interface is alloyed with a size that varies over an order of magnitude-from 50-600 nm-within a single crystallite. Nanoscale i…
We develop a general, fully quantum mechanical theory of Raman scattering from first principles in terms of many-body correlation functions. In order to arrive at expressions that are practically useful in the context of condensed matter physics, we adopt the Lehmann-Symanzik-Zimmermann reduction formula from high-energy physics and formulate in the modern language of many-body perturbation theory. This enables us to derive a general and practically useful expression for the Raman scattering rate in terms of quantities that can be computed \textit{ab initio}. Our work paves the way toward a co…
Resonant Raman spectra (RRS) of O-H and O-D vibration and libration modes, their combinations and higher harmonics have been observed in LiTaO3 polycrystalline thin films. RRS peaks are superimposed on photoluminescence (PL) spectrum. Monochromatic light from a xenon lamp is used as excitation source. PL spectrum shows two broad peaks, first near the band gap in UV (4.4-4.8eV) and another in the sub band gap region (< 4.0 eV). Band gap PL along with RRS peaks are reported for the first time. Photoluminescence excitation spectrum (PLE) shows a peak at 4.8 eV. Peak positions and full width at ha…
Transition metal dichalcogenides (TMDs) are layered materials which show excellent potential for nanoelectronic and optoelectronic applications. However, as many of the exciting features of these materials are controlled by the anharmonic effects, a proper understanding of the phonon properties and anharmonicity associated with these materials is essential for the proposed applications to be realized. We present here a comprehensive study on the phonon properties of two different TMDs; viz., MoS2 and MoTe2, as a function of temperature, laser excitations, and polarization, as well as flake-thi…
Excitation of four coherent phonon modes of different symmetries has been realized in copper metaborate CuB$_2$O$_4$ via impulsive stimulated Raman scattering (ISRS). Phonons were detected by monitoring changes in the linear optical birefringence using the balanced-detection (BD) technique. We compare the results of BD-ISRS experiment to the polarized spontaneous Raman scattering spectra. We show that the agreement between the two sets of data obtained by these allied techniques in a wide phonon frequencies range of 4-14 THz can be achieved by rigorously taking into account the symmetry of the…
The interaction between excitons and phonons in semiconductor nanocrystals plays a crucial role in the exciton energy spectrum and dynamics, and thus in their optical properties. We investigate the exciton2 phonon coupling in giant-shell CdSe/CdS core-shell nanocrystals via resonant Raman spectroscopy. The Huang-Rhys parameter is evaluated by the intensity ratio of the longitudinal-optical (LO) phonon of CdS with its first multiscattering (2LO) replica. We used four different excitation wavelengths in the range from the onset of the CdS shell absorption to well above the CdS shell band edge to…
We present measurements of the $D''$ Raman mode in graphene and carbon nanotubes at different laser excitation energies. The Raman mode around 1050 - 1150\,cm$^{-1}$ originates from a double-resonant scattering process of longitudinal acoustic (LA) phonons with defects. We investigate its dependence on laser excitation energy, on the number of graphene layers and on the carbon nanotube diameter. We assign this Raman mode to so-called 'inner' processes with resonant phonons mainly from the $Γ-K$ high-symmetry direction. The asymmetry of the $D''$ mode is explained by additional contributions fr…
Resonantly-enhanced Raman response in graphene-capped bismuthene on SiC
Universität Würzburg（维尔茨堡大学）；Würzburg-Dresden Cluster of Excellence ctd.qmat, Universität Würzburg（维尔茨堡大学 würtzburg-dresden 卓越集群 ctd.qmat）；Sapienza University of Rome（罗马第一大学）；RWTH Aachen University（亚琛工业大学）；Forschungszentrum Jülich（于利希研究中心）；Institut für Theoretische Physik und Astrophysik, Universität Würzburg（维尔茨堡大学理论与天体物理研究所）；TUD Dresden University of Technology（德累斯顿工业大学）
The temperature effect on the Raman scattering efficiency is investigated in $\varepsilon$-GaSe and $γ$-InSe crystals. We found that varying the temperature over a broad range from 5 K to 350 K permits to achieve both the resonant conditions and the antiresonance behaviour in Raman scattering of the studied materials. The resonant conditions of Raman scattering are observed at about 270 K under the 1.96 eV excitation for GaSe due to the energy proximity of the optical band gap. In the case of InSe, the resonant Raman spectra are apparent at about 50 K and 270 K under correspondingly the 2.41 e…
Since the stacking order sensitively affects various physical properties of layered materials, accurate determination of the stacking order is important for studying the basic properties of these materials as well as for device applications. Because 2H-molybdenum disulfide (MoS2) is most common in nature, most studies so far have focused on 2H-MoS2. However, we found that the 2H, 3R, and mixed stacking sequences exist in few-layer MoS2 exfoliated from natural molybdenite crystals. The crystal structures are confirmed by HR-TEM measurements. The Raman signatures of different polytypes are inves…
The phonon modes of self-assembled Ge/Si quantum dots grown by molecular-beam epitaxy in an apparatus integrated with a chamber of the scanning tunneling microscope into a single high-vacuum system are investigated using Raman spectroscopy. It is revealed that the Ge-Ge and Si-Ge vibrational modes are considerably enhanced upon excitation of excitons between the valence band $Λ_3$ and the conduction band $Λ_1$ (the E1 and E1 + $Δ_1$ transitions). This makes it possible to observe the Raman spectrum of very small amounts of germanium, such as one layer of quantum dots with a germanium layer thi…
