Plasmon energy losses in shear bands of metallic glass
Maximilian Grove, Martin Peterlechner, Harald Rösner, Robert Imlau, Alessio Zaccone, Gerhard Wilde
PPlasmon energy losses in shear bands of metallic glass
Maximilian Grove a , Martin Peterlechner a , Harald R¨osner a, ∗ , Robert Imlau b , Alessio Zaccone c,d,e , Gerhard Wilde a a Institut f¨ur Materialphysik, Westf¨alische Wilhelms-Universit¨at M¨unster, Wilhelm-Klemm-Str. 10, 48149 M¨unster, Germany b Thermo Fisher Scientific, Achtseweg Noord 5, 5651 GG Eindhoven, The Netherlands c Department of Physics ”A. Pontremoli“, University of Milan, via Celoria 16, 20133 Milano, Italy d Department of Chemical Engineering and Biotechnology, University of Cambridge, Philippa Fawcett Drive, CB3 0AS Cambridge, U.K. e Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB3 9HE Cambridge, U.K.
Abstract
Shear bands resulting from plastic deformation in cold-rolled Al Y Fe metallic glass were observed to display alternatingdensity changes along their propagation direction. Electron-energy loss spectroscopy (EELS) was used to investigatethe volume plasmon energy losses in and around shear bands. Energy shifts of the peak centre and changes in thepeak width (FWHM) reflecting the damping were precisely determined within an accuracy of a few meV using an opensource python module (Hyperspy) to fit the shapes of the plasmon and zero-loss peaks with Lorentzian functions. Themaximum bulk plasmon energy shifts were calculated for the bright and dark shear band segments relative to the matrixto be about 38 and 14 meV, respectively. The damping was observed to be larger for the denser regions. The analysispresented here suggests that the changes in the plasmons are caused by two contributions: (i) Variable damping in theshear band segments due to changes in the medium-range order (MRO). This affects the static structure factor S ( k ),which, in turn, leads to either reduced or increased damping according to the Ziman-Baym formula. (ii) The ionic densityand the effective electron mass appearing in the zero-momentum plasmon frequency formula E p ( q = 0) are coupled andgive rise to small variations in the plasmon energy. The model predicts plasmon energy shifts in the order of meV. Keywords: metallic glass; electron energy loss spectroscopy; deformation; shear band; volume plasmon
1. Introduction
Crystalline materials possess the ability to deform atconstant volume along slip planes since the periodicityof the lattice provides identical atomic positions for thesheared material to lock in to. However, in the absence ofa lattice, as for example in metallic glasses, this possibilitydoes not exist. As a consequence, the mismatch betweensheared zones (shear bands) and surrounding matrix needsto be accommodated by extra volume [1–8]. Different ex-perimental techniques have provided evidence that the ex-tra volume is present in shear bands [9–13]. The shearedzones are thus softer than the surrounding matrix enablingthe material to flow along them. Therefore, shear bandsare associated with structural changes like local dilatation,implying a volume change and thus a change in density. Animportant issue is hence the local quantification of free vol-ume inside shear bands. Recently, the local density withinshear bands of different metallic glasses has been deter-mined relative to the adjacent matrix using high angle darkfield scanning transmission electron microscopy (HAADF-STEM) [14–18]. These experiments showed an alterna-tion of higher and lower density regions along the prop-agation direction of the shear bands although on average ∗ Corresponding author
Email address: [email protected] (Harald R¨osner) shear bands were less dense than the surrounding matrix.The density changes along shear bands in Al Y Fe cor-related with small deflections along the propagation direc-tion, compositional changes and structural changes in themedium-range order (MRO) [14–20]. The observation ofshear band regions which were denser than the surround-ing matrix was initially somewhat unexpected since macro-scopic measurements reported dilation only [5, 10, 21].In this paper we focus on the changes in plasmon en-ergy losses in a shear band of cold-rolled Al Y Fe metal-lic glass. The plasmon energy shifts ∆ E p for both higherand lower density shear band segments were calculatedrelative to the surrounding matrix and found to be about38 and 14 meV, respectively. The bulk plasmon energyshifts in the shear band are discussed on the basis of inelas-tic electron-phonon scattering. According to the Ziman-Baym theory, the different local MRO in the shear bandsegments will affect the first peak of the static structurefactor S ( k ) differently and hence affect the damping of theplasmon excitation differently. Moreover, the ionic densityand effective electron mass appearing in the plasmon fre-quency formula at zero-momentum E p ( q = 0) are coupled.This gives rise to small variations in the plasmon energy E p ( q = 0) between the shear band segments and the ma-trix. Preprint submitted to Ultramicroscopy January 28, 2021 a r X i v : . [ c ond - m a t . m t r l - s c i ] J a n . Experimental Al Y Fe metallic glass is a marginal glass former.Melt-spun ribbons were produced by rapid quenching fromthe melt. Details can be found in reference [22]. Theamorphous state of the material was confirmed by x-raydiffraction (XRD) and selected area electron diffraction(SAED) prior to deformation. The ribbon material wasdeformed by cold-rolling yielding a thickness reductionof about 23 %. TEM specimens were prepared by twin-jet electro-polishing using HNO :CH OH in a ratio 1:2 at − ◦ C applying voltages of about − . V . Microstruc-tural characterization was performed in an FEI S/TEM(Themis . ×
200 nmusing a pixel size of 1 nm × . α - and β - semi-angles were 9 . . Y Fe metallic glass were ana-lyzed using automated routines based on an open sourcepython module (Hyperspy) [24] to fit the peak shapesof the zero-loss peak (ZLP) and the plasmon peak withLorentzian functions [25] in order to determine their cen-tre and width (FWHM) for each data point (pixel) of thespectrum image. The complete code and its descriptioncan be found in reference [26].
3. Results
In this study a representative part of a shear band (SB)in Al Y Fe metallic glass was investigated by analyticalTEM. The composition was analyzed by EDX using thenewest generation of quadrupole detectors providing im-proved statistics for quantification [27]. Fig. 1a depicts aSB that exhibits a contrast change from bright to darkto bright. Slight deflections between the SB segments arecorrelated with the contrast variations [14, 15].The contrast changes were quantified as density changesusing the intensities of the HAADF-STEM signals as de-scribed in detail in references [14–16, 18]. The averagedensity change for the bright SB segments is about +3 . − . ≈ Y Fe whilethe bright SB segments showed even less structural order-ing than the surrounding matrix. [14, 15].The mean free path (MFP) was calculated for the nom-inal composition to be (130 . ± .
4) nm according to Maliset al. [28] and (162 . ± .
1) nm using the approach fromIakoubovskii et al. [29]. The values for the different localenvironments are summarized in Tab. 1.The volume plasmons in and around the SB were inves-tigated by electron-energy loss (EEL) spectrum imaging.Fig. 2a displays the HAADF signal from the spectrum im-age showing the location of the SB. From the Lorentzianfunctions fitted to the plasmon and zero-loss peaks of theindividual EEL spectra, the peak centres E max , and widths(FWHM), ¯ h Γ p , were determined. The map of the plasmonpeak energy E max (Fig. 2b) shows a noticeable energy shiftrelative to the matrix for both SB segments. The map ofthe plasmon peak width is shown in Fig. 2c. Relativeto the matrix, the plasmon peak is broader for the bright(denser) SB segment and narrower for the dark (less dense)SB segment.The results of the quantification extracted from theboxed regions in Fig. 2b are displayed in Fig. 3a andFig. 3b together with the profile of the HAADF signalshowing the SB position. Although the plasmon peak en-ergy E max is somewhat variable in the matrix, it clearlyshifts in the SB segments reaching a maximum value forE max of (15 . ± .
02) eV for the bright SB segment and(15 . ± .
4. Discussion
In the following section the experimental results arediscussed.2 able 1: Results of the EDX analyses for the two SB segments and matrix positions shown in Fig. 1b. The atomic number, molar mass andmean free path (MFP) are calculated accordingly.
Profile 1 in Fig. 1b matrix (left) bright segment matrix (right)Al [at.%] 87.5 87.3 88.4Fe [at.%] 5.6 5.5 4.6Y [at.%] 6.9 7.2 7.0Average atomic number Z 15.52 15.59 15.42Molar mass [g/mol] 32.87 33.03 32.64Mean free path [nm]calculated after Malis [30] 130.8 130.7 131.1Mean free path [nm]calculated after Iakoubovskii [31] 162.1 160.8 162.4Profile 2 in Fig. 1b matrix (left) dark segment matrix (right)Al [at.%] 87.5 88.2 88.3Fe [at.%] 5.6 5.0 4.7Y [at.%] 6.9 6.8 7.0average atomic number Z 15.52 15.42 15.42Molar mass [g/mol] 32.87 32.64 32.64Mean free path [nm]calculated after Malis [30] 130.8 131.1 131Mean free path [nm]calculated after Iakoubovskii [31] 162.1 163.8 162.4
Table 2: Calculated maximum and standard deviation based on the Lorentzian peak fitting for the plasmon energy loss (peak centre) E max and peak width ¯ h Γ p at FWHM. The expected bulk plasmon energy loss E p ( q = 0) for undamped plasmons calculated using Eq. 6 and thedifference between SB segments and the adjacent matrix are also shown. E max [eV] ¯ h Γ p [eV] E p ( q = 0)[eV] ∆ E p [meV]bright SB segment (15 . ± . . ± . . ± .
02) (37 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . Table 3: Recalculated values using processed data by applying the Fourier-log deconvolution method to the low-loss spectra in comparisonto the unprocessed data shown in Tab. 2. E max [eV] ¯ h Γ p [eV] E p ( q = 0)[eV] ∆ E p [meV]bright SB segment (15 . ± . . ± . . ± .
02) (33 . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . . ± . igure 1: (a) Top: HAADF-STEM image of cold-rolled Al Y Fe metallic glass showing a shear band with contrast reversal (bright-dark-bright). Bottom: Quantified density along the shear band. (b)Top: HAADF-STEM image showing the regions used for the EDXquantification. Below: Elemental profiles based on the EDX mea-surements extracted from the indicated boxes over the bright (1)and dark (2) shear band segment. The grey shaded box indicatesthe position of the shear band. Figure 2: (a): HADDF-STEM signal from the spectrum image. (b):Map of the plasmon peak centre E max . (c): Map of the plasmonpeak width (FWHM) reflecting the damping. To discuss the influence of the background and the de-gree of plural scattering on the energy shifts and the peakbroadening, three representative spectra (raw data) aredepicted in Fig. 4 which are taken from different positionsof the linescan shown in Fig. 3a. A small second plas-mon peak can be noticed in all spectra indicating the pres-ence of plural scattering due to the foil thickness, whichranges from 92 - 102 nm (see Fig. 3c) or 0.7 - 0.8 mean freepath. While a visual inspection of Fig. 4 shows no apparentchange in the background, an indiscernible effect may arisefrom subtle changes in the second plasmon peak. To evalu-ate this contribution, plural scattering was removed usingthe Fourier-log deconvolution technique [30–32]. Subse-quently, all values were recalculated using deconvolutedspectra and the results are given in Tab. 3. A compari-son of the data in Tabs. 2 and 3 shows that, while thereare small changes in the values as a result of deconvolu-tion, the trends for the peak shifts and widths remain thesame. From this we conclude that in our case the influenceof background and plural scattering are minor relative tothe total error and thus justifies the use of unprocessedlow-loss data, which are preferred for practical reasons.
While there are small changes in the composition of thematrix and the SB regions (Fig. 1b), there is no appar-ent correlation with the shifts in the plasmon peak energy(Figs. 3a and b). According to the experimental work ofHibbert et al. [33], the variation of the plasmon energy losswith composition in Al-rich Al-Mg solid solutions showedthat an addition of 1 at.% Mg as impurity caused a de-crease in the plasmon energy loss of about -50 meV. Con-sidering the additions of Fe and Y in the first instance to4 igure 3: (a, b) Profiles (HAADF-STEM intensity and plasmonenergy loss E max ) corresponding to the boxes shown in Fig. 2b.The red and blue points correspond to the profiles across the bright(denser)and dark (less dense) SB segments. (c) Foil thickness profiledrawn (top to bottom) across the SB shown in Fig. 2a, respectively. have similar effects to the plasmon energy loss of Al as Mg,a compositional difference of 1 at.% Al between the brightand dark SB segments (see Tab. 1) should then result ina similar energy shift of about - 50 meV. However, in ourcase the bright SB segment in Fig. 1b, which has less Al,shows an increased energy shift of about 24 meV. Thus,the compositional changes are insufficient to explain theobserved plasmon energy shift in the SB segments.
The plasmon resonance peak may be modeled as aLorentzian function centered at the plasmon energy loss E max ≈ . h Γ p ≈ . p denotes the damping coeffi-cient of the plasmon excitation. The observed values areclose to typical values for crystalline Al, i.e. E p = 15 eV[34]. However, it is worth noting that crystalline Al showsa peak shift of about 0 . E p = ¯ h (cid:113) ne m(cid:15) , where ¯ h is the reducedPlanck constant, n is the electron density, e and m are theelectron charge and mass, and (cid:15) is the vacuum permittiv-ity, the shifts of the plasmon energy losses in the SB cannotbe explained simply in terms of a uniform nearly-free elec-tron concentration over the entire sample. Thus, although
01 0 0 0 02 0 0 0 03 0 0 0 04 0 0 0 05 0 0 0 06 0 0 0 07 0 0 0 0
02 5 0 05 0 0 07 5 0 01 0 0 0 01 2 5 0 0
Intensity [Counts]
E n e r g y l o s s [ e V ] m a t r i x s t a r t m a t r i x e n d S B a p e x
Figure 4: Three representative EEL spectra (raw data) taken fromthe linescan shown in Fig. 3a representing the start (matrix black),the end (matrix red) and the apex of bright SB segment (blue). the electrons are treated as delocalized in the nearly-freeelectron model, there are differences in concentration ofthe nearly-free electrons in different parts of the sample.These differences reflect the underlying differences of ionicdensity and the individual damping characteristics of thedifferent regions. Thus, the measured plasmon energy lossin our material is not given by E p = ¯ h (cid:113) ne m(cid:15) , becausethis simple estimate holds only in the absence of damp-ing (hence, it works only for perfect defect-free crystallinemetals at low temperature). In a metallic glass at roomtemperature, damping is of course a very important con-tribution to the maximum, E max , occurring. According totheory [34, 36], the maximum occurs at an energy approx-imately proportional to the inverse of the imaginary partof the dielectric function, given by [34]max {(cid:61) (cid:18) − (cid:15) ( ω ) (cid:19) } ≈ ¯ hω p Γ p = E p Γ p and reaches a maximum value of ω p Γ p at an energy loss E max = (cid:34) ( E p ) − (cid:18) ¯ h Γ p (cid:19) (cid:35) , (1)which is close to E p , in the case of small damping. In areal system, the measured plasmon energy loss is E max ,not E p and the two values coincide only for ideal perfectcrystals at low temperature. The undamped plasmon fre-quency is related to the undamped plasmon energy lossvia ¯ hω p = E p . The damping coefficient Γ p depends on thelocal scattering events between conduction electrons andion-cores and hence the local microstructure. Therefore,Γ p may vary spatially because in some portions of the ma-terial scattering events are more frequent than in otherparts of the material (cf. Fig. 2c). In general, there are5everal contributions to damping besides ion scattering:important effects are Landau damping, inter-band tran-sitions [37] and electron-phonon scattering. It should bementioned that damping of collective excitations in amor-phous solids may be strongly non-local: as recently re-ported, long-range (power-law) stress correlations in amor-phous solids cause logarithmic decay of phonon dampingwith the wave vector [38]. However, for Al-rich materi-als the Fermi energy lies well below the band top so thatsingle-particle excitations are very unlikely, and dampingcan be assumed to be caused predominantly by electron-ion and electron-phonon scattering. Hence, the dampingcoefficient may be estimated from its definition in termsof the relaxation time (mean time between scattering col-lisions with ions), Γ p = τ − . The relaxation time τ isrelated to the resistivity ρ ∗ via ρ ∗ = (cid:16) ne τm (cid:17) − .The resistivity of metallic glasses is described by theZiman theory [35, 38], which was originally developed forliquid metals but works also for certain glasses. The fa-mous Ziman formula for the resistivity of amorphous met-als reads ρ ∗ = 3 πm n i e ¯ h k F (cid:90) k F k S ( k ) | V ( k ) | dk (2)which gives the damping coefficient as [36]:Γ p = τ − = 3 πnm n i ¯ h k F (cid:90) k F k S ( k ) | V ( k ) | dk (3)where k denotes the wave vector (here defined as the recip-rocal of the radial distance r measured from one ion takenas the centre of a spherical frame and not to be confusedwith the wave vector of the incident electron beam), k F is the Fermi wave vector, and n i is the number density ofions in the material. Furthermore, S ( k ) is the static struc-ture factor of the metallic glass (i.e. the spatial Fouriertransform of the radial distribution function g ( r )), whichagain is a local quantity that significantly differs for thebright and dark SB segments, since the local atomic struc-ture (topological order) is different for the two segments[15]. Finally, V ( k ) is the average Thomas-Fermi-screenedelectron-ion pseudopotential form factor for elastic scat-tering at the Fermi surface [38]. The Ziman formula ac-counts for elastic electron-ion scattering only, and thus istypically valid at temperatures well below the Debye tem-perature. Since the Debye temperature for Al Y Fe is(360 ±
6) K [39], the Ziman formula should be replaced bythe Baym formula [38]. Thus S ( k ) in Eq. 3 needs to be re-placed by a frequency integral over the dynamic structurefactor S ( k, ω ) times a factor ¯ hωkT (exp ¯ hωkT ) − . Using the Vine-yard approximation S ( k, ω ) S ( k ) S s ( k, ω ), where S s ( k, ω )denotes the self-part of the atomic dynamics, the dynamicstructure factor S ( k, ω ) can be extracted from the fre-quency integral, and upon performing the frequency in- tegral, the damping coefficient becomesΓ p = τ − = 3 πnm n i ¯ h k F (cid:90) k F k S ( k ) g ( k ) | V ( k ) | dk (4)where g ( k ) is a function of k only. As a first approximation,the integral is dominated by S ( k ), especially for the low- k region. Alternatively, one can manipulate the Baymformula after Meisel and Cote [40] for metallic glass andarrive at the same simplification.Regarding the integration limit above, it is importantto note that for some metallic glasses the Nagel-Tauc rule[41] for metallic glass stability and formability stipulatesthat 2 k F ≈ k max , where k max denotes the wave vector ofthe first peak in the structure factor. However, for pure Alit is known that 2 k f ≈ k min , where k min is the minimumafter the first peak in S ( k ), as is the case for all three-valent metals [38]. Since our system is very rich in Al, itis likely that the value of k , which satisfies 2 k F ≈ k , liessomewhere between k min and k max . Figure 5: Schematic illustration of the static structure factor S ( k )for metallic glass. S ( k ) is the spatial Fourier transform of the radialdistribution function g(r). Pronounced structural order in terms ofmedium-range order within the dark shear band (dashed line) leadsto broadening and lowering of the first peak of S ( k ) relative to thematrix. The contribution to the integral in the Ziman-Baym formulaEq. 4 leads to less plasmon damping in the dark SB. Since the pseudopotential V ( k ) is approximately givenby the Fourier transform of the Thomas-Fermi screenedCoulomb attraction, the function V ( k ) is relatively shal-low within the range of integration. Hence, the integral isdominated by the first peak in the static structure factor S ( k ), i.e. by the fraction of short and medium-range orderin the atomic coordination (Fig. 5). The dependence of the damping coefficient on the localfree volume is clearly the same as the volume- (or equiv-alently, density-) dependence of the resistivity. The effectof dilation on resistivity has been studied extensively byZiman and Faber on the basis of Eq. 2. In particular,6aber [35] has provided the following expression for thevolume-dependence of the resistivity: Vρ ∗ (cid:18) ∂ρ ∗ ∂V (cid:19) T = 23 ξ − (cid:82) k F k (cid:16) ∂S ( k ) ∂V (cid:17) T | V ( k ) | dk (cid:82) k F k S ( k ) | V ( k ) | dk + (cid:82) k F k (cid:16) ∂ | V ( k ) | ∂V (cid:17) T S ( k ) dk (cid:82) k F k S ( k ) | V ( k ) | dk (5)where ξ = − k F ρ (cid:16) ∂ρ∂k F (cid:17) is a dimensionless parameter. Eq. 5is attractive because it clearly singles out the three maincontributions to the change of resistivity upon dilation: i)the effect of expanding the Fermi sphere, encoded in ξ ; ii)the change in short and medium-range order encoded inthe derivative ∂S ( k ) ∂V ; iii) the effect on the pseudopotentialof any change in the screening properties of the conduc-tion electron gas. Faber [35] concluded from his analysisthat the second term in Eq. 5 is the one which usuallydominates the overall dependence of resistivity on dila-tion. Assuming that the position and the width of thefirst peak of S ( k ) scale with V − / , Faber concluded that ∂S ( k ) ∂V ∼ ∂S ( k ) ∂k .In the absence of a more quantitative theory, we resortto the following argument based on the consideration ofthe microstructure in the SB. In the Ziman-Baym formula S ( k ) is related to the pair-correlation function or radialdistribution function, RDF in isotropic systems, giving theaveraged probability of finding any atom at a distance rfrom the atom at the centre of the frame.We now apply the Ziman-Baym theory to estimate thechange in the plasmon energy loss in the SB with respectto the matrix due to damping. We know from the plas-mon peak fitting of the EELS data that the FWHM in thematrix adjacent to the bright SB segment is ¯ h Γ matrix bright =2 .
688 eV, whereas in the matrix adjacent to the dark SBsegment it is ¯ h Γ matrix dark = 2 .
679 eV. For the SB segments weobtain 2 .
796 eV and 2 .
63 eV, for bright and dark, respec-tively. The damping of the excitation is reduced in thedark SB segment due to higher structural order in termsof MRO (small crystal-like Al-rich clusters embedded inamorphous material) resulting in a broader and loweredfirst peak of S ( k ) [14, 15], which in turn results in a re-duced resistivity (hence a lower damping of the plasmon),because the lower peak in S ( k ) gives a smaller value forthe integral in Eq. 4. For the bright SB segment the damp-ing is higher than in the matrix because there is even less‘structure’ in terms of MRO and hence less dispersion withthe result that the first peak of S ( k ) is higher. Now, the energy of the plasmon loss is given by Eq. 1as E p ( q = 0) = (cid:115) E + (cid:18) ¯ h Γ p (cid:19) , (6)and for the relative shift in each segment we have then E SB p ( q = 0) − E matrix p ( q = 0)= (cid:118)(cid:117)(cid:117)(cid:116)(cid:34) ( E SBmax ) + (cid:18) ¯ h Γ p (cid:19) (cid:35) − (cid:118)(cid:117)(cid:117)(cid:116)(cid:34) ( E matrixmax ) + (cid:18) ¯ h Γ p (cid:19) (cid:35) , (7)where E p denotes the bulk plasmon energy loss in the com-plete absence of damping or at momentum transfer q = 0.Using the measured values of ¯ h Γ p in the various regionstogether with the measured values of the plasmon energyloss E max , one can infer the values of undamped plasmonenergy E SB p, in the SB segments and compare those val-ues to the value measured in the matrix E matrix p, . Thesevalues are then related to the ionic density and the effec-tive electron mass m ∗ in the various regions via E p, =¯ h (cid:113) n e · e m ∗ · (cid:15) = ¯ h (cid:113) n i · ( ze ) m ∗ · (cid:15) . Neglecting the variation of elec-tron effective mass in the first instance, we arrive at the fol-lowing enhancement for the bright (densified) SB segment: E SB,bright p, E matrix p, = 1 . ≈ (cid:114) n SB,bright i n matrix i , and for the dark (di-lated) SB segment we get: E SB,dark p, E matrix p, = 1 . ≈ (cid:114) n SB,dark i n matrix i .This result can be explained as follows. The electron ef-fective mass in metallic glasses is controlled by electron-phonon coupling via [35]: m ∗ = m (1 + λ ), where themass-enhancement factor (electron-phonon coupling pa-rameter) is given through the Eliashberg function, validalso for metallic glasses [42], λ = (cid:82) ω D dωω α F ( ω ). Here, α is the electron-phonon matrix element and F ( ω ) is the vi-brational density of states (VDOS). The above formula forthe mass-enhancement factor λ can be expressed in termsof an integral over the dynamical structure factor S ( k, ω )times the matrix element via the Eliashberg theory. Meiseland Cote [43] and independently Jaeckle and Froboese [44]using a slightly different derivation, have shown that inmetallic glasses λ ∝ Λ − , where Λ denotes the mean-freepath of the electron in the ionic environment. The pref-actors in this relation are density-independent constants,hence they do not change from region to region, whereasthe mean free path is roughly inversely proportional to thelocal ionic density n SB i in the region. The mean free pathwas calculated for the different regions according to Maliset al. [28] and found to be around (130 . ± .
4) nm. Us-ing the model (Kramers-Kronig sum rule) of Iakoubovskiiet al. [29] we find a mean free path of (162 . ± .
1) nm.7hile the absolute values of the mean free path differ to agreat deal depending on the used model [28, 29], the val-ues calculated within one model do not vary significantlyin space (Tab. 1). Thus, assuming the mean free path tobe constant, we get E SB p, E matrix p, ≈ (cid:115) n SB i n matrix i (cid:114) λ matrix λ SB . (8)Moreover, since in metallic glasses the mass-enhancementparameter is large ( λ (cid:29) λ matrix λ SB ∝ Λ SB Λ matrix ∝ n matrix i n SB i in the second radical of Eq. 8 almost can-cels the effect of the first radical. Accordingly, the plasmonenergy shift for the dark SB segment is mostly related toa reduced damping. In fact, the increased MRO in thedark SB segment gives rise to a reduced first peak of S ( k )and thus leads to reduced damping because of the smallercontribution of the lowered peak of S ( k ) to the integral inEq. 2. Somewhat different is the case of the bright SB seg-ment, where the coupling of effects is comparatively moreeffective and leaves a slightly larger enhancement for thevalue of E SB p, with respect to the matrix due to the ionicdensity.In essence, we have two contributions that are respon-sible for the plasmon energy shifts; that is, (i) dampingdue to electron-phonon scattering and (ii) the ionic den-sity. The increase in damping for the denser (bright) SBsegment and the decrease in the dilated (dark) one withrespect to the matrix is related to the level of structuralorder present in the different regions [14, 15, 18, 45, 46].This also fits to recent findings observed in granular me-dia where sound damping is determined by the interplaybetween elastic heterogeneities and inelastic interactions[47]. The second contribution is related to the efficacy ofthe coupling between the ionic density and the effectiveelectron mass appearing in the plasmon frequency formulaat zero-momentum E p ( q = 0), which is less effective in thedark SB segment than in the bright one.
5. Conclusions
Shifts and widths of plasmon energy losses were ex-perimentally determined from a sheared zone (shear bandand its immediate environment) of a metallic glass usingautomated routines based on an open source python mod-ule (Hyperspy) to fit the peak shapes of the zero-loss peakand the plasmon peaks [26]. These signals are characteris-tic fingerprints and therefore suitable to visualize deforma-tion features in amorphous materials such as shear bands.The model presented here suggests two reasons for the theplasmon energy shifts. First, variable plasmon dampingin the shear band segments caused by differences in themedium-range order present. This affects the first peak ofthe static structure factor S ( k ), which leads to either low-ered or increased damping according to the Ziman-Baymresistivity formula. The second reason is that the ionic density and the effective electron mass appearing in thezero-momentum plasmon frequency formula E p ( q = 0) arecoupled and give rise to small variations in the plasmon en-ergy between the shear band and the matrix. The modelpredicts plasmon energy shifts in the order of meV [48].
6. Acknowledgments
We gratefully acknowledge financial support from theDeutsche Forschungsgemeinschaft (WI 1899/29-1; projectnumber 325408982). We thank Dr. P. Schlossmacher(Thermo Fisher Scientific, FEI Deutschland GmbH) forenabling the measurements at the Nanoport in Eindhovenduring a demonstration of a Themis
300 microscope. A.Z.gratefully acknowledges financial support from the US ArmyResearch Office through contract no. W911NF-19-2-0055.Fruitful discussions with Drs. Vitalij Hieronymus-Schmidtand Sven Hilke are acknowledged.
References [1] F. Spaepen, A microscopic mechanism for steady state inhomo-geneous flow in metallic glasses, Acta Metallurgica 25 (4) (1977)407–415.[2] A. Argon, Plastic deformation in metallic glasses, Acta Metal-lurgica 27 (1) (1979) 47–58.[3] P. Donovan, W. Stobbs, The structure of shear bands in metallicglasses, Acta Metallurgica 29 (8) (1981) 1419–1436.[4] M. Heggen, F. Spaepen, M. Feuerbacher, Creation and anni-hilation of free volume during homogeneous flow of a metallicglass, Journal of Applied Physics 97 (3) (2005) 033506.[5] D. Klaum¨unzer, A. Lazarev, R. Maaß, F. Dalla Torre, A. Vino-gradov, J. F. L¨offler, Probing shear-band initiation in metallicglasses, Physical Review Letters 107 (18) (2011) 185502.[6] J. Pan, Q. Chen, L. Liu, Y. Li, Softening and dilatation in asingle shear band, Acta Materialia 59 (13) (2011) 5146–5158.[7] A. L. Greer, Y. Q. Cheng, E. Ma, Shear bands in metallicglasses, Materials Science and Engineering: R: Reports 74 (4)(2013) 71–132.[8] E. Ma, Tuning order in disorder, Nature materials 14 (6) (2015)547.[9] J. Li, F. Spaepen, T. C. Hufnagel, Nanometre-scale defectsin shear bands in a metallic glass, Philosophical Magazine A82 (13) (2002) 2623–2630.[10] W. Lechner, W. Puff, G. Wilde, R. W¨urschum, Vacancy-typedefects in amorphous and nanocrystalline al alloys: Variationwith preparation route and processing, Scripta Materialia 62 (7)(2010) 439–442.[11] J. Bokeloh, S. V. Divinski, G. Reglitz, G. Wilde, Tracer mea-surements of atomic diffusion inside shear bands of a bulk metal-lic glass, Physical Review Letters 107 (23) (2011) 235503.[12] J. B¨unz, T. Brink, K. Tsuchiya, F. Meng, G. Wilde, K. Albe,Low temperature heat capacity of a severely deformed metallicglass, Physical Review Letters 112 (13) (2014) 135501.[13] Y. P. Mitrofanov, M. Peterlechner, S. Divinski, G. Wilde, Im-pact of plastic deformation and shear band formation on theboson heat capacity peak of a bulk metallic glass, Physical Re-view Letters 112 (13) (2014) 135901.[14] H. R¨osner, M. Peterlechner, C. K¨ubel, V. Schmidt, G. Wilde,Density changes in shear bands of a metallic glass determinedby correlative analytical transmission electron microscopy, Ul-tramicroscopy 142 (2014) 1–9.[15] V. Schmidt, H. R¨osner, M. Peterlechner, G. Wilde, P. M. Voyles,Quantitative measurement of density in a shear band of metal-lic glass monitored along its propagation direction, Physical Re-view Letters 115 (3) (2015) 035501.
16] V. Hieronymus-Schmidt, H. R¨osner, G. Wilde, A. Zaccone,Shear banding in metallic glasses described by alignments ofeshelby quadrupoles, Physical Review B 95 (13) (2017) 134111.[17] C. Liu, V. Roddatis, P. Kenesei, R. Maaß, Shear-band thick-ness and shear-band cavities in a zr-based metallic glass, ActaMaterialia 140 (2017) 206–216.[18] S. Hilke, H. R¨osner, D. Geissler, A. Gebert, M. Peterlechner,G. Wilde, The influence of deformation on the medium-rangeorder of a zr-based bulk metallic glass characterized by variableresolution fluctuation electron microscopy, Acta Materialia 171(2019) 275–281.[19] S. Balachandran, J. Orava, M. K¨ohler, A. J. Breen, I. Kaban,D. Raabe, M. Herbig, Elemental re-distribution inside shearbands revealed by correlative atom-probe tomography and elec-tron microscopy in a deformed metallic glass, Scripta Materialia168 (2019) 14–18.[20] C. Liu, Z. Cai, X. Xia, V. Roddatis, R. Yuan, J.-M. Zuo,R. Maaß, Shear-band structure and chemistry in a zr-basedmetallic glass probed with nano-beam x-ray fluorescence andtransmission electron microscopy, Scripta Materialia 169 (2019)23–27.[21] H. Shao, Y. Xu, B. Shi, C. Yu, H. Hahn, H. Gleiter, J. Li, Highdensity of shear bands and enhanced free volume induced inzr70cu20ni10 metallic glass by high-energy ball milling, Journalof Alloys and Compounds 548 (2013) 77–81.[22] J. Bokeloh, N. Boucharat, H. R¨osner, G. Wilde, Primary crys-tallization in al-rich metallic glasses at unusually low tempera-tures, Acta Materialia 58 (11) (2010) 3919–3926.[23] J. Hunt, D. B. Williams, Electron energy-loss spectrum-imaging, Ultramicroscopy 38 (1) (1991) 47–73.[24] F. de la Pe˜na, E. Prestat, V. Tonaas Fauske, P. Burdet,P. Jokubauskas, M. Nord, T. Ostasevicius, K. E. MacArthur,M. Sarahan, D. N. Johnstone, et al., hyperspy/hyperspy: Hy-perspy v1. 5.2, zndo (2019).[25] D. McComb, A. Howie, Characterisation of zeolite catalysts us-ing electron energy loss spectroscopy, Ultramicroscopy 34 (1-2)(1990) 84–92.[26] https://github.com/mgrove-wwu/EELS-LL-image-fittingM. Grove (2019).[27] P. Schlossmacher, D. O. Klenov, B. Freitag, S. von Harrach,A. Steinbach, Nanoscale chemical compositional analysis withan innovative s/tem-edx system, Microscopy and Analysis 5(2010).[28] T. Malis, S. Cheng, R. Egerton, Eels log-ratio technique forspecimen-thickness measurement in the tem, Journal of Elec-tron Microscopy Technique 8 (2) (1988) 193–200.[29] K. Iakoubovskii, K. Mitsuishi, Y. Nakayama, K. Furuya, Thick-ness measurements with electron energy loss spectroscopy, Mi-croscopy Research and Technique 71 (8) (2008) 626–631.[30] D. Johnson, J. H. Spence, Determination of the single-scatteringprobability distribution from plural-scattering data, Journal ofPhysics D: Applied Physics 7 (6) (1974) 771.[31] R. Egerton, B. Williams, T. Sparrow, Fourier deconvolution ofelectron energy-loss spectra, Proceedings of the Royal Societyof London. A. Mathematical and Physical Sciences 398 (1815)(1985) 395–404.[32] R. Egerton, P. Crozier, The use of fourier techniques in electronenergy-loss spectroscopy, Scanning Microscopy, Supplement 2(1988) 245–254.[33] G. Hibbert, J. Edington, D. Williams, P. Doig, The variationof plasma energy loss with composition in dilute aluminium-magnesium solid solutions, Philosophical Magazine 26 (6)(1972) 1491–1494.[34] R. F. Egerton, Electron energy-loss spectroscopy in the electronmicroscope, Springer Science & Business Media, 2011.[35] T. E. Faber, Introduction to the theory of liquid metals, Cam-bridge University Press, 2010.[36] H. Nikjoo, S. Uehara, D. Emfietzoglou, Interaction of radiationwith matter, CRC press, 2012.[37] S. Gelin, H. Tanaka, A. Lemaˆıtre, Anomalous phonon scatteringand elastic correlations in amorphous solids, Nature Materials 15 (11) (2016) 1177–1181.[38] N. H. March, M. P. Tosi, et al., Coulomb liquids, AcademicPress, 1984.[39] M. Gerlitz, Excess heat capacity contributions of marginallyglass forming metallic glasses (2018).[40] L. Meisel, P. Cote, Structure factors in amorphous and disor-dered harmonic debye solids, Physical Review B 16 (6) (1977)2978.[41] S. Nagel, J. Tauc, Nearly-free-electron approach to the theoryof metallic glass alloys, Physical Review Letters 35 (6) (1975)380.[42] M. Baggioli, C. Setty, A. Zaccone, Effective theory of supercon-ductivity in strongly coupled amorphous materials, Phys. Rev.B 101 (21) (2020) 214502.[43] L. Meisel, P. Cote, Eliashberg function in amorphous metals,Physical Review B 23 (11) (1981) 5834.[44] J. Jackle, K. Frobose, The electron-phonon coupling constant ofamorphous metals, Journal of Physics F: Metal Physics 10 (3)(1980) 471.[45] S. Hilke, H. R¨osner, G. Wilde, The role of minor alloying in theplasticity of bulk metallic glasses, Scripta Materialia 188 (2020)50–53.[46] F. A. Davani, S. Hilke, H. R¨osner, D. Geissler, A. Gebert,G. Wilde, Correlation between the ductility and medium-rangeorder of bulk metallic glasses, J. of Appl. Phys. 128 (1) (2020)015103.[47] K. Saitoh, H. Mizuno, Sound damping in frictionless granularmaterials: The interplay between configurational disorder andinelasticity, arXiv preprint arXiv:2008.09760 (2020).[48] E. W. Huang, K. Limtragool, C. Setty, A. A. Husain, M. Mi-trano, P. Abbamonte, P. W. Phillips, Extracting correlationeffects from momentum-resolved electron energy loss spec-troscopy (m-eels): Synergistic origin of the dispersion kink inbi . sr . cacu o x , arXiv preprint arXiv:2010.02947 (2020)., arXiv preprint arXiv:2010.02947 (2020).