Analysing Low-Loss Spectra with Drude Theory

4.3. Analysing Low-Loss Spectra with Drude Theory#

part of

MSE672: Introduction to Transmission Electron Microscopy

Spring 2024

Gerd Duscher	Khalid Hattar
Microscopy Facilities	Tennessee Ion Beam Materials Laboratory
Materials Science & Engineering	Nuclear Engineering
Institute of Advanced Materials & Manufacturing
The University of Tennessee, Knoxville

Background and methods to analysis and quantification of data acquired with transmission electron microscopes.

4.3.1. Content#

The main feature in a low-loss EELS spectrum is the volume plasmon peak.

This volume plasmon and all other features in the low-loss region of an EELS spectrum are described by Dielectric Theory of Electrodynamics.

The simplest theory to interprete this energy range is the Drude theory.

Another easy to observe component is the multiple scattering of this plasmon peak, which we can correct for or use for thickness determination.

4.3.2. Load important packages#

4.3.2.1. Check Installed Packages#

import sys
import importlib.metadata
def test_package(package_name):
    """Test if package exists and returns version or -1"""
    try:
        version = importlib.metadata.version(package_name)
    except importlib.metadata.PackageNotFoundError:
        version = '-1'
    return version

if test_package('pyTEMlib') < '0.2024.2.3':
    print('installing pyTEMlib')
    !{sys.executable} -m pip install  --upgrade pyTEMlib -q

print('done')

done

4.3.2.2. Import all relevant libraries#

Please note that the EELS_tools package from pyTEMlib is essential.

import sys
%matplotlib ipympl
if 'google.colab' in sys.modules:    
    from google.colab import output
    from google.colab import drive
    output.enable_custom_widget_manager()
    
import matplotlib.pyplot as plt
import numpy as np

# additional package 
import ipywidgets as ipyw
from scipy.optimize import leastsq  ## fitting routine of scipy

# Import libraries from the book
import pyTEMlib
import pyTEMlib.file_tools as ft          # File input/ output library
from pyTEMlib import file_tools           # File input/ output library
import pyTEMlib.kinematic_scattering as ks         # Kinematic sCattering Library
                             # Atomic form factors from Kirklands book

# For archiving reasons it is a good idea to print the version numbers out at this point
print('pyTEM version: ',pyTEMlib.__version__)

pyTEM version:  0.2024.02.2

4.3.3. Dielectric Theory#

4.3.3.1. Comparison to Optical Spectroscopy#

The interaction of a transmitted electron with a solid is here described in terms of a dielectric response function $\varepsilon(q, \omega))$.

The same response function $\varepsilon(q, \omega))$ describes the interaction of any electro-magnetic wave (such as photons) with a solid, so this formalism allows energy-loss data to be compared with the results of optical measurements.

The difference is that there is no momentum transfer in an optical transition and, therefore, the relevant dielectric function is $\varepsilon(q=0, \omega))$.

The optical dielectric function (permittivity) is a transverse property of the medium, in the sense that the electric field of an electromagnetic wave displaces electrons in a direction perpendicular to the direction of propagation, the electron density remaining unchanged.

The relation between the ac conductivity $\sigma$ (for dc conductivity the in metals $\varepsilon$ can go to $\inf$ and is therefore not well described) and $\varepsilon$ is : $$ \varepsilon = 1 + \frac{4\pi \ i \sigma}{\omega} $$

The relation between the complex refractive index $n+i\kappa$ and dielectric function is: $$ n^2-\kappa^2 ={\bf Re}(\varepsilon), \ \ 2n\kappa = {\bf Im}(\varepsilon) $$

Optical absorption spectrum is obtained through ABS= ${\bf Im}( \varepsilon(q→0,\omega) )$

So, the dielectric function describes the electrical and optical response of a material almost completely.

4.3.3.2. Dispersion#

From the Fourier representation of the Maxwell equations in the source free case, we get: $$\frac{\epsilon_i \mu_i}{c^2} = k^2$$

which we transform into:

\[k = \frac{\omega}{c} \sqrt{\varepsilon_1 \mu}\]

which gives for non-magnetic materials:

\[k = \frac{\omega}{c} \sqrt{\varepsilon_1 }\]

for $\varepsilon_1 > 0$ the wavenumber $k$ is a real function of real $\omega$.

while for $\varepsilon_1 < 0$ the wavenumber $k$ is purely imaginary.

for light in vacuum the permittivity $\varepsilon_1$ is a constant with value 1 and we get the equation for light line:

\[k = \frac{\omega}{c}\]

4.3.3.3. Cross Section#

\[ \frac{d^2\sigma}{dE d\Omega} = \frac{1}{\pi a_0 m_0 v^2 n_a} {\rm Im} \left[ \frac{-1}{\varepsilon(q,E)} \right] \left( \frac{1}{\theta^2+\theta_E^2}\right)\]

The partial cross section $\frac{d^2\sigma}{dE d\Omega}$ gives us the probability that an incident electron will be scattered into angle $q$ with energy-loss $E$.

There are three compenents, first a term that depends on the atom density $n_a$ (atoms per volume).

And the third term is the (Lorentzian) angle dependence with the characteristic angle $$\theta_E = E_E/(\gamma m_0v^2)$$

The second term is the loss-function.

The loss function of a dielectric function $\varepsilon = \varepsilon_1+ i*\varepsilon_2$ is $$ {\rm Im} \left[ \frac{-1}{\varepsilon(q,E)} \right] = \frac{\varepsilon_1(q,E)}{\varepsilon_1^2(q,E) + \varepsilon_2^2(q,E)}$$ At large energy loss and $q\approx 0$, $\varepsilon_2$ is small and $\varepsilon_1$ close to 1, the loss function becomes proportional to $\varepsilon_2$ and (apart from a factor of E$^{−3}$) the energy-loss spectrum is proportional to the X-ray absorption spectrum.

4.3.4. Load and plot a spectrum#

# ---- Input ------
load_example = True
# -----------------
if not load_example:
    if 'google.colab' in sys.modules:
        drive.mount("/content/drive")

    fileWidget = file_tools.FileWidget()

# ---- Input ------
load_example = True
file_name = 'AL-DFoffset0.00.dm3'
# -----------------
if load_example:
    if 'google.colab' in sys.modules:
      if not os.path.exists('./'+file_name):
        !wget  https://github.com/gduscher/MSE672-Introduction-to-TEM/raw/main/example_data/AL-DFoffset0.00.dm3
    else:
        datasets = file_tools.open_file('../example_data/'+file_name)
        eels_dataset = datasets['Channel_000']
      
else:
    datasets = fileWidget.datasets
    eels_dataset = fileWidget.selected_dataset
    
sum_spectrum = eels_dataset.sum()

eels_dataset = eels_dataset/sum_spectrum*100.
eels_dataset.units = '%'
eels_dataset.quantity = 'scattering probability'

view = eels_dataset.plot()

4.3.4.1. Fix energy scale and resolution function#

please see Fitting the Zero-Loss Peak for details

4.3.4.2. Relative Thickness Determination#

The probabaility of an low-loss function in a solid angle is then: $$\frac{\partial^2 P}{(\partial E \partial \Omega) }= t* \frac{e}{\pi^2 a_0 m_0 v^2} {\rm Im} \left[ \frac{-1}{\varepsilon(q,E)} \right] \left( \frac{1}{\theta^2+\theta_E^2}\right)$$

Please see Kroeger Formalism for inclusion of retardation effects, and surface plasmons.

The integration over the (effective) collection angle gives: $$\frac{\partial P}{\partial E} = \int_0^{\theta_c}\frac{\partial^2 P}{(\partial E \partial \Omega) }(E,\theta)\sin \theta\ d \theta$$

So we need to get the loss-function, calculate $\frac{\partial^2 P}{(\partial E \partial \Omega) }$ and then integrate over the angles, then we fit this to the spectrum with the sole variable of the thickness $t$.

The specimen thickness $t$, it is actually the total scattering and mass thickness that is measured by EELS. If the physical density of a material were reduced by a factor $f$, the scattering per atom would remain the same (according to an atomic model) and the mean free path should increase by a factor $f$.

The big problem hoewver, is that one has to know the dielectric function. In the case of the Drude theory, that means we need to know the electron density.

Any approximation, therefore, needs to approximate this dielectric function, which cannot be generally applicable.

The relative thickness $ t_{rel} = t/\lambda$, in contrast, is relatively easy to determine. All the above problems are hidden in value of the inelastic mean free path (IMFP) $\lambda$, which is the inverse of the cross section above.

When you use a tabulated value for the IMFP, be aware that this value depends on:

acceleration voltage
effective collection angle
material density

and may not be applicable for your experimental setup.

The measurement of the relative thickness $t_{rel}$ is relative easy:

We already did this in the Fit Zero-Loss part.

The inelastic scattering can be viewed in terms of independent collisions, whose occurrences obey Poisson statistics. The probability that a transmitted electron suffers $n$ collisions is \begin{equation} \Large P_n = (1/n!)m^n \exp(-m) \end{equation}

where $m$ is the number of average collisions for electrons travel through this sample area. The number $m$ can be set to the scattering parameter $t/\lambda$. $P_n$ is represented in the EELS spectrum by the ration of the energy-integrated intensity $I_n$ of $n$-fold scattering divided by the {\bf total} integrated intensity $I_t$: \begin{equation} \Large P_n = I_n/I_t (1/n!) (t/\lambda)^n \exp(-t/\lambda) \end{equation}

For a given order $n$ of scattering, the intensity is highest when $t/\lambda =n$ In the case of the unscattered (n=0) component (zero–loss peak), the intensity is highest at t=0 and decreases exponentially with specimen thickness. For $n=0$, equation \ref{equ:poisson} gives equation \begin{equation} \Large \frac{t}{\lambda}= - \ln\left[ \frac{I_{\mbox{total}}}{I_{zl}}\right] \end{equation} I just replaced $I_{zl}$ with $I_0$.

FWHM, energy_shift = eels_tools.get_channel_zero(np.array(eels_dataset), eels_dataset.energy_loss)

print(f'Zero Loss with energy resolution of {FWHM:.2f} eV at position {energy_shift:.3f} eV')
eels_dataset.energy_loss -= energy_shift

zero_loss  = eels_tools.get_resolution_functions(eels_dataset)

plt.figure()
plt.plot(eels_dataset.energy_loss, eels_dataset, label='spectrum')
plt.plot(eels_dataset.energy_loss, zero_loss, label = 'zero-loss')
plt.plot(eels_dataset.energy_loss, np.array(eels_dataset)-zero_loss , label = 'difference')

plt.title ('Lorentzian Product Fit of Zero-Loss Peak')
#plt.xlim(-5,30)
plt.legend();
Izl = zero_loss.sum()
Itotal = np.array(eels_dataset).sum()
tmfp = np.log(Itotal/Izl)
print(f'Sum of Zero-Loss: {Izl:.0f} counts')
print(f'Sum of Spectrum: {Itotal:.0f} counts')
print (f'thickness [IMFP]: {tmfp:.5f}')

Zero Loss with energy resolution of -0.15 eV at position -0.000 eV
Sum of Zero-Loss: 84 counts
Sum of Spectrum: 100 counts
thickness [IMFP]: 0.17734

4.3.5. Single Scattering Deconvolution#

For thicker samples (relative thickness larger than half the inelastic MFP), the scattered electrons are likely to be scattered a second time by plasmon excitations. This results for thicknesses around 0.5 MFP in more intensity in the high energy tail of the plasmon peak. For thicker areas a second plasmon peak will appear at double the energy loss and for very thick areas we have a whole series of plasmon peaks. In spectra of very thick areas the zero-loss peak may completely vanish. Multiple scattering is easily recognized by the positions of the peaks (multiple of $E_{max}$).

Multiple scattering influences the high energy side of the first plasmon peak, and makes it difficult to quantify the spectra, whether it is in the low-loss or core–loss region. Figure \ref{fig:ssd} shows how the correction improves the determination of the width of the plasmon width. Fortunately, we can correct this multiple scattering, because mathematically, it is a kind of self-convolution. The procedure is called single scattering deconvolution (SSD).

Electrons that lost energy can interact with the sample again. The average path between two interaction is the above introduced IMFP $\lambda$. This multiple inelaxtic scattering is most obvious in the strongest energ-loss spectrum: the volume plasmon. This multiple scatering follows the Poisson statistic and the $n^{\rm th}$ scattering has the intensity: $$I_n = I_0 P_n = (I_0/n!)(t/λ)^n \exp(−t/λ)$$

It can be shown the the single scattering spectrum in fourier space $s(v)$ can be derived by:

\[s(v) = I_0 \ln[j(v)/z(v)]\]

$z(v)$ beeing the Fourier transformed zero-loss peak (resolution function) and $j(v$ the spectrum in Fourier space.

The above formula would also correct for any instument broadening and a very noisy spectrum would result, so we need to convolute (multiplication in Fourier space) a new broadening function, which could be a well behaved Gaussian with slightly smaller energy width than the original zero-loss peak.

# Single scattering deconvolution

# Use resolution Function as ZL 

j = np.fft.fft(np.array(eels_dataset))
z = np.fft.fft(zero_loss)
z2 = z ## Could be a different zl i.e. extracted from Spectrum
j1 = z2*np.log(j/z)
ssd_low_loss =np.fft.ifft(j1).real#,'fourier-log deconvolution')

plt.figure()
#plt.plot(s.tags['ene'][start:end],  zLoss[start:end]/sumSpec*1e2)
plt.plot(eels_dataset.energy_loss,  ssd_low_loss)
plt.plot(eels_dataset.energy_loss, eels_dataset)
#plt.xlim(-4,40)
plt.ylim(-.1,.1);
eels_dataset.metadata['resolution_function'] = zero_loss

4.3.6. Volume Plasmon Peak#

Most of the inelastically scattered electron arise from interaction with outer shell electrons. These interactions, therefore, have a high intensity and are easy to obtain. The quantification of this region ($<$ 100eV) in an EELS spectrum is rather hard.

The valence electrons in a solid can be considered as coupled oscillators which interact with each others and with the transmitted electrons. In a first approximation the valence electrons behave like a free electron gas (Fermi sea, jellium). The behavior of the electron gas is described in terms of the dielectric function $\varepsilon$.

The energy-loss function $F_{el}$ on the other hand is determined by the dielectric function $\varepsilon$ through:

\[ F_{el} = \Im \left[\frac{-1}{\varepsilon(\omega)} \right] \]

The maximum of the energy-loss function is the plasmon–peak, which is given by: $$ E_{max} = \left[(E_p)^2 - (\Delta E_p/2)^2\right]^{1/2} $$

4.3.6.1. Dielectric Theory#

We investigate the plasmon excitation in the so called jellium model. This model is enough to explain the essence of this excitation, but is not as complicated as the real solid state.

The displacement of a {\it quasifree} electron (with an effective mass $m$ and not the rest mass $m_0$) due to an electric field $\vec{E}$ must fulfill the equation : \begin{equation} \Large m\vec{x}’’ + m\Gamma \vec{x}’ = -e\vec{E} \end{equation}

The usage of an effective mass approximates the interaction between the valence electrons and the ion–cores. For the same reason, we introduce $\Gamma$ a damping constant. Instead of $\Gamma$, we could also use its reciprocal value $\tau = 1/\Gamma$ as in the Drude theory. This is similar to the electronic theory of conduction.

For an oscillatory field $\vec{E}= \vec{E} \exp(-i\omega t)$ above equation \ref{equ:motion} has the following solution: \begin{equation} \Large \vec{x} = (e\vec{E}/m)(\omega ^2+i\Gamma \omega)^{-1} \end{equation}

The displacement gives raise to a polarization $\vec{P} = -en\vec{x} =\varepsilon_0\chi\vec{E}$, with $\chi$ electric susceptibility and $n$ the the number of electron per unit volume. The dielectric function $\varepsilon(\omega)=1+\chi$ can be expressed as: \begin{equation} \Large \varepsilon(\omega)=\varepsilon_1+i\varepsilon_2 = 1-\frac{\omega_p^2}{\omega^2+\Gamma^2}+\frac{i\Gamma\omega^2_p}{\omega(\gamma^2+\Gamma^2)} \end{equation} Where $\omega$ is the frequency (in rad/s) of forced oscillation and $\omega_p$ is the Eigen or resonance frequency of the free electron gas, which is given by: \begin{equation} \Large \omega_p =\left[ \frac{ne^2}{\varepsilon_0 m} \right]^{1/2} \end{equation}

A sudden impulse such as a transmitted electron will contain all angular frequencies (Fourier components). It will also excite the resonance frequency of the jellium. Such an oscillation with frequency $\omega_p$ can be viewed as a pseudoparticle with (plasmon) energy $E_p =\hbar \omega_p.$ We call this pseudoparticle a plasmon.

4.3.6.2. Energy-Loss Function#

The energy-loss function (elf) is given as shown in the beginning by: \begin{equation} \Large E_{el}= \left[\frac{1}{\varepsilon(\omega)}\right] = \frac{\varepsilon_2}{\varepsilon_1^2+\varepsilon_2^2}=\frac{\omega\Gamma\omega_p^2}{(\omega^2-\omega_p^2)^2 +(\omega\Gamma)^2} \end{equation} The energy-loss is represented by $E=\hbar \omega$ and we can rewrite equation \ref{equ:eels_function2} as: \begin{equation} \left[\frac{1}{\varepsilon(E)}\right] =\frac{(E\hbar/\tau)E_p^2}{(E^2-E^2)^2 +(E\hbar/\tau)^2} % =\frac{(E \quad \Delta E_p)E_p^2}{(E^2-E^2)^2 +(E \quad \Delta E_p)^2} \end{equation}

The relaxation time $\tau = 1/\Gamma$ is directly connected to the FWHM of the energy-loss function which is given by $\Delta E_p =\hbar\Gamma = \hbar/\tau$. The effect of the effective mass on the energy-loss function is rather small.\ The maximum of the energy-loss function is given in equation \ref{equ:plasmon-maximum}.

The relaxation time $\tau$ represents the time for plasmon oscillations to decay in energy by a factor $\exp(-1)=0.37$. The number of oscillations which occur within this time is $\omega_p\tau/(2\pi)=E_p/(2\pi/\Delta E_p)$. Using experimental values for $E_p$ and $\Delta E_p$ result in 5 to 0.4 oscillations. The plasmon oscillations are, therefore, heavily damped, which depends on the band structure of the material.

4.3.6.3. Plasmon Energies#

The calculated plasmon energies of the Jellium model with equation \ref{equ:plasmon_frequency} are already quite good as can be seen in table \ref{tbl:plasmon_frequencies}. Only the number of valence electrons per unit area $n$ and the dielectric constant $\varepsilon_0$ are necessary for these calculations.

Material	$E_p$ (experimental)	$E_P$ (theoretical)
Li	7.1 eV	8.0 eV
Diamond	34 eV	31 eV
Si	16.5 eV	16.6 eV
Ge	16.0 eV	15.6 eV
InSb	12.9 eV	12.7 eV
GaAs	15.8 eV	15.7 eV
NaCl	15.5 eV	15.7 eV

A more sophisticated way of comparing the valence loss spectra to theory is to calculate the dielectric function. {\it Ab initio} density functional theory can be used to do these calculations. To obtain the dielectric function the valence band has to be convoluted with the conduction band. A comparison between experiment and theory is shown in the figure below.

Plasmon–Loss peak of Li (black circles), Gibbson et al Phys.~Rev.~B {\bf 13}:2451 (1979) and calculated energy-loss function by S.~Ramachandran et al. (unpublished)}

The plasmon–loss enables us to determine the plasmon energy, a materials parameter. The width plasmon peak is also a characteristic of the material and is needed to determine the plasmon energy.

Plasmon–Loss peak of SrTiO$_3$ with highlighted FWHM

The spectrum here is quite complicated due to the surface plasmon–peak which shows up to the left of the volume plasmon–peak. We measure the energy of the maximum of the plasmon–peak (here $E_{\mbox{max}} = 31$ eV) and the width of the plasmon–peak (here $\Delta E_P = 5.3$ eV). While it is easy to determine the plasmon–peak energy , the plasmon–peak width can only be determined approximately, because we don’t know how to subtract the background.

Using equation above, we find that the plasmon energy is 30.5 eV for SrTiO$_3$.

This spectrum shows the high dynamic range of an EELS spectrum. The valence loss part of the spectrum is noisy, but the zero-loss peak is not saturated. The surface plasmon peak is strong, which indicates a rather thin area. May be a little too thin. Without loss of information we can get spectra from thicker areas.

Not only the plasmon energy (or the plasmon maximum) can be used for an identification of materials, but also the shape, especially the width of the plasmon peak.

As seen above, it is rather difficult to determine the FWHM of a plasmon peak. However, it is much easier after subtraction of the zero-loss peak, as can be seen in figure below.

Plasmon–Loss peak of Ni (black) after subtraction of the zero-loss peak and with highlighted FWHM. The original spectrum and the zero-loss peak are displayed in the background.

A comparison of the plasmon peak width of Ni (above) and Si (figure below) makes it apparent how different the shape of plasmon peaks can be. The Si plasmon peak is sharp (FWHM: 7.3 eV) and has a well defined maximum, while the Ni plasmon peak is very broad (FWHM: 40 eV).

*Plasmon--Loss peak of Si (black) after subtraction of the zero-loss peak and with highlighted FWHM. The original spectrum and the zero-loss peak are displayed in the background.*

The Si plasmon peak is obtained from a rather thin area, which also makes it easier to determine the width, because the right hand tail is not broadened by the thicker sample (please see section above about single scattering deconvolution).

4.3.7. Drude Function#

The dielectric function in the Drude theory is given by two input parameters the position of the plasmon energy $E_p$ and the width of the plasmon $\Gamma$

\[ ε(ω) = ε1 + iε2 = 1 + χ = 1 − \frac{\omega_p^2}{\omega^2+\Gamma^2} + \frac{i\Gamma \omega_p^2}{\omega(\omega^2+\Gamma^2)}\]

Here $\omega$ is the angular frequency (rad/s) of forced oscillation and $\omega_p$ is the natural or resonance frequency for plasma oscillation, given by $$ ω_p = \sqrt{\frac{ne^2}{(ε_0m_0)}} $$ A transmitted electron represents a sudden impulse of applied electric field, containing all angular frequencies (Fourier components). Setting up a plasma oscillation of the loosely bound outer-shell electrons in a solid is equivalent to creating a pseudoparticle of energy $E_p = \hbar \omega_p$, known as a plasmon (David Pines, Elementary Excitations in Solids, 1963).

# ----------- Input -------------
E_p = plasmon_energy = 17. # in eV
E_w = plasmon_gamma = .7 # in eV
# -------------------------------
energy_scale = eels_dataset.energy_loss+1e-18 #= np.linspace(0,50,1024)+1e-18

def Drude(E,E_p,E_w):
    eps = 1 - E_p**2/(E**2+E_w**2) +1j* E_w* E_p**2/E/(E**2+E_w**2)
    elf = (-1/eps).imag
    return eps,elf

eps,elf = Drude(energy_scale, plasmon_energy, plasmon_gamma)

plt.figure()
plt.plot(energy_scale, eps.real, label='Re($\epsilon_{Drude}$)')
plt.plot(energy_scale, eps.imag, label='Im($\epsilon_{Drude}$)')
plt.plot(energy_scale, elf, label='loss function$_{Drude}$')
plt.plot([0,energy_scale[-1]],[0,0],c='black')

plt.legend()
plt.gca().set_ylim(-2,max(elf)*1.05);
plt.xlim(-1,40);
plt.xlabel('energy loss (eV)')

Text(0.5, 0, 'energy loss (eV)')

4.3.8. Fitting a Drude Function to Plasmon#

The position and the width are important materials parameters and we can derive them by fitting the Drude function to the volume plasmon region.

from scipy.optimize import leastsq

def Drude(E,Ep,Ew):
    eps = 1 - Ep**2/(E**2+Ew**2) +1j* Ew* Ep**2/E/(E**2+Ew**2)
    elf = (-1/eps).imag
    return eps,elf

def errfDrude(p, y, x):
    eps,elf = Drude(x,p[0],p[1])
    err = y - p[2]*elf
    #print (p,sum(np.abs(err)))
    return np.abs(err)#/np.sqrt(y)


pin2 = np.array([15,1,.7])
E = energy_scale
startFit =np.argmin(abs(energy_scale-13))
endFit = np.argmin(abs(energy_scale-18))
    
p2, lsq = leastsq(errfDrude, pin2, args=(ssd_low_loss[startFit:endFit], energy_scale[startFit:endFit]), maxfev=2000)

eps, elf =Drude(energy_scale,p2[0],p2[1])
drudePSD = p2[2]* elf
plt.figure()

plt.plot(energy_scale,eels_dataset)
plt.plot(energy_scale,drudePSD)
plt.plot(energy_scale,eels_dataset-drudePSD)
plt.axhline(0, color='black')

plt.gca().set_xlim(0,40)
plt.gca().set_ylim(-0.01,0.2)
print(f"Drude Theory with Plamson Energy: {p2[0]:2f} eV and plasmon Width {p2[1]:.2f} eV") 
print(f"Max of Plasmon at {energy_scale[drudePSD.argmax(0)]:.2f} eV")
print(f"Amplitude of  {p2[2]:.2f} was deteremined by fit ")

Drude Theory with Plamson Energy: 15.049794 eV and plasmon Width 0.72 eV
Max of Plasmon at 15.04 eV
Amplitude of  0.01 was deteremined by fit 

plt.figure()
plt.title ('Drude Fit: Dielectric Function - Permittivity')
plt.plot(energy_scale,eps.real,label = 'Re($\epsilon)$')
plt.plot(energy_scale,eps.imag,label = 'Im($\epsilon)$')
plt.plot(energy_scale,drudePSD,label = 'loss-function$_{Drude}$')
plt.plot(energy_scale,eels_dataset,label = 'loss-function$_{exp}$')
plt.axhline(0, color='black')

plt.gca().set_xlim(0,40)
plt.gca().set_ylim(-2.5,5.3)

plt.legend();

C:\Users\gduscher\AppData\Local\Temp\ipykernel_28844\467939617.py:1: RuntimeWarning: More than 20 figures have been opened. Figures created through the pyplot interface (`matplotlib.pyplot.figure`) are retained until explicitly closed and may consume too much memory. (To control this warning, see the rcParam `figure.max_open_warning`). Consider using `matplotlib.pyplot.close()`.
  plt.figure()

4.3.8.1. Plasmon Frequency and Phase Identification#

Plasmon peaks occurring in EELS is directly related to valence electron density and can thereby be used as a means for materials characterization and phase identification. Free-electron metals have very sharp plasmon peaks as compared to semiconductors and insulators with broader peaks, where the valence electrons are no longer free (covalent or ionic bonding). While many materials properties are a function of valence electron density, knowing the plasmon energy can prove to be a useful tool in identifying micro structures.

Microstructural phases are observed by shifts in the plasmon energy. Examples of phase identification are observed easily in alloys and also precipitate structures. A common textbook example is EELS low loss comparison of diamond and graphite. The example given in Figure is a line scan showing the variation of plasmon energies with position. The plasmon energies range from 21 to 23 eV, corresponding to SiC (20.8 eV) and SiN3 (22.5 eV) respectively. Figure \ref{plasmon-ls2} is a reconstructed map of the plasmon energies. Blue regions correspond to SiC, while the red regions correspond to SiN3.

Appendix C in Egerton’s Electron Energy Loss Spectroscopy in the Electron Microscope lists plasmon energies of some elements and compounds [Egerton-1999].

4.3.9. Surface Plasmon#

Spectra from thin specimen show the excitations of the surface plasmons on each side of the specimen. For any normal specimen these surface plasmons do interact, but this is not true for extremely thick specimen ($>> 10$nm). The surface plasmon frequency $\omega_S$ for thin specimen is related to the bulk plasmon frequency $\omega_P$ by Ritchie [Ritchie-PR1957]:
$$ \omega_S=\omega_P\left[ \frac{1\pm \exp(-q_st) }{1+\varepsilon} \right]^{1/2} $$

The symmetric mode, where like charges face one another, corresponds to the higher angular frequency $q_s$. Please note, that this relationship does only apply for large $q_s$

The differential probability for surface excitation at both surfaces of a sample with thickness $t$ can be expressed (normal incident, no retardation effects) by: $$ \frac{d^2 P_s}{d\Omega d E}=\frac{2\hbar}{\pi^2 \gamma a_0 m_0^2 \mu^3}\frac{\theta}{(\theta^2+\theta^2_E)^2} \Im\left[ \frac{(\varepsilon_a - \varepsilon_b)^2 } {\varepsilon_a^2 \varepsilon_b}\right] $$ with $$ R_c = \frac{\varepsilon_a \sin^2(tE/2\hbar\mu)}{\varepsilon_b + \varepsilon_z }\tanh (q_s t/2)

\frac{\varepsilon_a \cos^2(tE/2\hbar\mu)}{\varepsilon_b + \varepsilon_a} \coth (q_s t/2) $$ and $\varepsilon_a$ and $\varepsilon_b$ are the permitivities of the two surfaces.

A secondary effect of the surface excitation is the reduced intensity of the bulk plasmon peak. The effect is usually smaller than 1%, but can be larger for spectra with small collection angle, because the preferred scattering of surfuce losses into small angles. The correction for surface plasmon will be discussed in the Kramers–Kronig Analysis.

4.3.10. Summary#

The beauty of Low--Loss spectroscopy is its derivation of the dielectric function to high energies without prior knowledge of the composition. The signal is strong and the acquisition time is mostly restricted by the dynamic range of the spectrum.

Think of low-loss spectroscopy as Electrodynamics

The advantages of EELS is the derivation of these values spatially resolved. And from a linescan across an Si/SiO$_2$ interface the dielectric function per pixel can be obtained. From that we can calculate the dielectric polarizability $\alpha_e (E)$, which may be a measure of the dielectric strength.

We obtain more or less easily:

relative thickness
absolute thickness
inelastic mean free path
plasmon frequency
plasmon width
band gap
dielectric function
reflectivity
absorption
effective number of electrons per atoms

The analysis of the optical data requires the exact knowledge of the zero-loss peak. Because of the weighting in the Fourier Analysis, the low energy part contributes heavily to the dielectric function. Therefore, energy resolution is critical for an exact determination of all the optical values from EELS. The new monochromated TEMs are now able to achieve an energy resolution of 10 meV (one is at the oak Ridge National Laboratory), which allows for a sharper zero-loss peak. Such a sharp zero-loss peak will enable us to extract this low energy data more accurately. The dielectric function and the parameters derived from it, can be more precisely determined from such EELS spectra.