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Dynamics of the Auger process in hydrocarbons

In Auger electron spectroscopy [56], core electrons in a sample are ionized using x-rays or electron impact. Once the core hole is generated, it is found that within $ \sim$10 fs, a valence electron ``falls into'' the hole [57], and the energy released causes a secondary electron to be ejected (Figure 4.21). If the two-hole state contains bonded atoms, it may relax by breaking bonds or ejecting additional electrons. Core ionized methanol, for example, breaks its OH bond over 100 fs, while core ionized formic acid breaks its OH bond over 50 fs [58].

Figure 4.21: Core holes relax via a two stage Auger decay process.

The release of low-energy secondary electrons upon core electron ionization was first observed by Auger in 1923 [59], who bombarded noble gases with x-rays in a cloud chamber, and found that in addition to a long photoelectron track, a short secondary electron track appeared. He found that the energy of the secondary electrons was dependent on the species being ionized, but not on the incident x-ray energy.

Today, Auger spectroscopy is widely used to characterize the elemental composition and chemical bonding of surfaces [60], since secondary Auger electrons can only travel a few nanometers in solids, depending on their energy, without being absorbed. Thus a signal appears only from the top layers of atoms. Furthermore, since the secondary electron energy is independent of the means used to excite the core electron, and since core electron energies are mostly the same in atoms, regardless of chemical environment, it is possible to use broad spectrum sources for excitation while still getting a clean secondary energy spectra.

Theory has mostly focused on reproducing Auger spectra by looking at the transition probabilities of moving from an initial ionized state to a final two-hole state [61,62]. Such theory has been broadly successful at reproducing the Auger spectra of atoms, atom hydrides, and substituted hydrocarbons. In those cases, an implicit assumption is that the nuclei are held stationary. Any movement of the nuclei prior to the release of secondary electrons is assumed only to broaden the spectral lines.

In the last decades, it has become apparent Auger chemistry can be used to create and modify surfaces as well as characterize them. In 1978, Knotek and Fiebelman [63] provided evidence that electron-stimulated desorption in ionic solids operating proceeded via core-hole Auger decay; a year later, Knotek, Jones, and Rehn [64] reported photon-stimulated desorption of ions from a surface via a similar mechanism. Since then, it has been proposed that covalent solids [65] may be etched via Auger chemistry as well. To study these processes, which may be key to manufacturing the next generation of semiconductors with smaller and sharper feature sizes [66] ($ \sim$20 nm, aspect ratios $ \sim$10:1), we would like to simulate how molecules fragment during the Auger process, taking into account excited electron dynamics.

Theory has only recently risen to the challenge of computing extended nuclear dynamics after the initial Auger excitation. Ab initio molecular dynamics has been used to study the dissociation of a single water molecule following core-hole excitation [67], as well as the dissociation of a water molecule in an $ \mathrm{(H_{2}O)_{5}}$ cluster [68]. Using the electron force field, we can easily model the Auger dynamics of systems containing hundreds of atoms, with all electrons included; we show below a simulation of $ C_{197} H_{112}$ over 100 fs, accomplished in two days real time.

In the previous sections, we have shown that eFF gives a reasonable model of bonding, but to ensure that eFF produces a correct distribution of secondary electrons and molecule fragments, we would like to compare the vertical ionization energies of electrons to experiment. This comparison is not entirely straightforward, since vertical ionization potentials (IPs) of the sort measured using photoelectron spectroscopy are from delocalized molecular orbitals rather than from localized orbitals of the sort eFF uses. We settle on an indirect procedure, calibrating a Hartree-Fock method against experimental IPs [69], then comparing eFF orbital energies to theoretical Boys localized Hartree-Fock orbital energies, with corrections from the calibration applied (Figure 4.22, Appendix C).

For hydrocarbons methane, ethane, neopentane, and adamantane, we find that carbon-carbon electron are bound by almost exactly the correct amount (on average, eFF 16.8 eV vs 16.7 eV corrected localized HF); but carbon-hydrogen electrons are underbound by $ \sim$2 eV (on average, eFF 13.9 eV vs 16.0 eV corrected localized HF). These differences are small in comparison to the energy difference between valence and core electrons ($ \sim$270-280 eV), and so we expect energy to be properly distributed among electrons and molecular fragments. However, we also find that eFF underbinds 1s core electrons by $ \sim$18% (236.0 eV average versus 290.6 eV experimental), due to its lack of a proper nuclear-electron cusp; this reduces the energy available in the Auger decay process.

Figure 4.22: Red points compare Boys localized Hartree-Fock orbital energies to eFF orbital energies.

Since the vertical ionization potentials of valence electrons are correct, we start by creating single valence-hole states in ethane and observing how the molecule fragments. In the simulations, we assume instantaneous removal of the initial electron. Dynamics were integrated with a time step of 0.001 fs over 100 fs, and $ m_{elec} = m_{H}$. Some of the core hole relaxation steps involved an abrupt motion of electrons, and an adaptive step size algorithm was used to shorten the time step further during those periods to ensure that energy was conserved to better than 0.0001 hartrees. Following the creation of single hole states, we find selective bond breaking: removal of a carbon-hydrogen bonding electron causes the carbon-hydrogen bond to break, while removal of a carbon-carbon bonding electron causes the carbon-carbon bond to break. In the case of CC bond dissociation, there is an additional complication in that there is no symmetry breaking, so that the remaining CC electron remains at the center of symmetry, effectively creating a two-hole state. We find that this effect disappears in larger, less symmetric molecules. The proper instability of single hole states gives us confidence that we will be able to properly describe the fragmentation of double hole states.

Figure 4.23: Removal of valence electrons from ethane results in selective bond breaking.

We next remove 1s core electrons from the hydrocarbons methane, ethane, adamantane, neopentane, and the diamondoid $ \mathrm{C_{197} H_{112}}$. We track the Auger process by plotting the potential energies of the eFF electrons over time (Figure 4.24); the advantage of having localized electrons becomes apparent here, as it is straightforward to distinguish loosely bound, valence, and core electrons. We find that the key stages of the Auger process are well reproduced: for 2-20 fs, the core-hole is stable, then there is an abrupt transition where a valence electron jumps into the hole and a valence hole is created; then over the next 20-100 fs, secondary electrons are ejected and/or fragmentation occurs. We find however that in many cases the secondary electrons are not usually released simultaneously with the filling of the core hole, but several femtoseconds afterward, as the highly excited valence hole state relaxes.

Core hole lifetimes are measured experimentally as the lifetime broadening of the x-ray photoelectron peak ( $ \Delta t = \hbar / \Delta E$). With eFF, we estimate the lifetime of the core hole as the moment when what was formerly a valence electron becomes bound by greater than 160 eV, an arbitrary threshold set to distinguish core-like and valence-like electrons. We find a core hole lifetime for methane that is comparable to experiment (9.2 fs versus 7.9 fs expt), and a lifetime for ethane that is lower then experiment (2.0 fs versus 6.7 fs). Neopentane, adamantane, and the large diamondoid particle $ \mathrm{C_{197} H_{112}}$ all have core hole lifetimes between 2 and 20 fs, in line with the ranges observed experimentally [57]. Aside from primary carbons having a particularly short core hole lifetime (2 fs), we did not observe any particular correlation between the degree of substitution of the carbon and the core hole lifetime (Table 4.7).

Figure 4.24: Electron energies show Auger process in adamantane in detail.

Table 4.7: Core-hole lifetimes are on the correct time scale.
  Core-hole lifetime (fs)
  eFF expt
methane 9.2 7.9
ethane 2.0 6.7
neopentane (C) 15.0  
neopentane ( $ \mathrm{CH_{3}}$) 2.0  
adamantane (CH) 11.7  
admantane ( $ \mathrm{CH_{2}}$) 6.1  
$ \mathrm{C_{197} H_{112}}$ (C) 4.0  


With x-ray photoelectron spectroscopy, it is also possible to measure the energy and geometry changes that occur during the initial creation of the core hole, e.g., the difference between vertical and adiabatic ionization potentials [70]. With eFF, we assume a vertical ionization, but we can estimate the magnitude of core-hole induced relaxation by averaging over all geometries from zero time to the core-hole lifetime. This is not a perfect comparison, as it assumes that the period over which the ionization occurs is comparable to the core-hole lifetime; however, such an assumption is made when the core-hole lifetime is estimated from the width of the 1s photoelectron peak, and so we regard it as justified.

Comparing eFF averaged relaxations to experiment [71,72], we found that (1) geometry changes for methane were comparable to those observed experimentally, but for ethane the C-H bonds stretched and bent in an opposite manner to what was observed (Table 4.8), and (2) the core hole showed a relaxation energy nearly eighty times greater than what was observed experimentally.

The larger-than-observed energy relaxation of the core-hole state suggests that eFF has made an uneasy truce between adiabatic curve hopping abrupt transitions and a mean-field description of electron dynamics. Experiments [70] suggest that the core-hole is filled in an abrupt way: while the methane C 1s electrons is being ionized, for instance, the $ \mathrm{CH_{4}}$ molecule only lowers its energy by -0.15 eV, but we know that within 8 fs the energy of the molecule drops by 290 eV. In eFF, core-hole filling is abrupt, but the ``plateau before the cliff'' is slightly steeper, which suggests that a portion of the valence-hole state is mixing in with the core-hole state prior to the electron jump. This larger core-hole prerelaxation may explain why we are able to obtain core-hole lifetimes comparable to experiment, even with massive electrons.


Table 4.8: In eFF, core hole shows strong relaxation even before it is filled.
    Core-hole relaxation
    eFF expt
methane $ \mathrm{\Delta E (eV)}$ -12.43 -0.15
  $ \mathrm{\Delta d_{CH}}$ (A) -0.07 -0.05
       
ethane $ \mathrm{\Delta E (eV)}$ -12.03 -0.17
  $ \mathrm{\Delta d_{CH}}$ (A) 0.03 -0.05
  $ \mathrm{\Delta d_{CC}}$ (A) 0.00 0
  $ \mathrm{\Delta d_{HCC}}$ (degrees) 0.36 -3.00


We consider now the nuclear dynamics after the core hole has been filled by a valence electron. In the case of methane, we find the following sequence of events (Figure 4.25):
$\displaystyle \mathrm{CH_{4}^{+}}$ $\displaystyle \rightarrow$ $\displaystyle \mathrm{core-hole\ collapse\ (9\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{CH_{3}^{+} + H} \ \mathrm{(17\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{CH_{2}^{+} + 2 H} \ \mathrm{(23\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{CH_{2}^{2+} + e^{-} + 2 H} \ \mathrm{(25\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{CH^{2+} + e^{-} + 3 H} \ \mathrm{(49\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C^{+} + H^{+} + e^{-} + 3 H} \ \mathrm{(79\ fs)}.$  

In our simulations, the secondary electron is not ejected from the highly excited $ \mathrm{CH_{4}^{+}}$ until two hydrogen atoms have already dissociated from it. Experimentally, it is possible to find out which fragments are present when the secondary electron is released through the use of energy-resolved electron-ion coincidence (EREICO). Kukk et al. [73] found that core-ionized deuteromethane produces along with the secondary electron the major fragment $ \mathrm{CD_{2}^{+}}$, with $ \mathrm{CD^{+}}$ and $ \mathrm{C^{+}}$ also present, and almost no $ \mathrm{CD_{3}^{+}}$. It may seem curious that $ \mathrm{CD_{3}^{+}}$ and $ \mathrm{CD_{4}^{+}}$ do not appear in the spectra, especially given that ionization of the valence electrons have been shown by the same method to produce only $ \mathrm{CD_{3}^{+}}$ and $ \mathrm{CD_{4}^{+}}$.

Most likely $ \mathrm{CH_{4}^{+}}$ (or by their finding $ \mathrm{CD_{4}^{2+}}$) is created as a very hot molecule, and it is only by detaching bound hydrogen atoms that it becomes stable enough to detect. This is consistent with our model. It may also be possible that our unusual observation that the secondary electron is only released after two hydrogen atoms have dissociated is correct, and the two-hole state is created in a nonconcerted fashion. This would also explain the lack of $ \mathrm{CD_{3}^{+}}$ and $ \mathrm{CD_{4}^{+}}$ in the experimental spectra.

Figure 4.25: Auger dissociation of methane and ethane following creation of a core hole.

In the case of ethane, ionizing a core electron causes a $ \mathrm{\sigma_{CC}}$ electron to fill in the core hole. At that point, the carbon-carbon bond breaks, lengthening steadily from 1.48 $ \mathrm{\AA}$; 12.5 fs after the core hole fills, the carbon-carbon bond is already 2 $ \mathrm{\AA}$ long. During this period, the single remaining $ \mathrm{\sigma_{CC}}$ electron remains at the center of the bond to maintain the system's overall symmetry, making it appear as if bond breaking and secondary electron ionization happen at the same time. In contrast to the methane dynamics, and perhaps because the secondary electron is released early, the $ \mathrm{CH_{3}^{+}}$ fragments are not left with enough energy to break carbon-hydrogen bonds, and no further fragmentation is observed.

We continue to the larger hydrocarbons neopentane and adamantane (Figure 4.26). In the case of neopentane, there are two different carbons whose 1s electrons we may ionize: the quaternary carbon at the center, or the four primary carbons at the periphery. Ionizing the quaternary carbon causes four surrounding valence electrons with the same spin as the ionized electron to simultaneously move inward to fill the vacancy; symmetry breaks, and after 15 fs only one valence electron fills in to occupy the core. As in ethane, the loss of a $ \mathrm{\sigma_{CC}}$ electron causes the neopentane to dissociate into $ \mathrm{(CH_{3})_{3} C^{+}} + \mathrm{CH_{3}^{+}}$ plus a secondary electron.

Unlike ethane, however, the $ \mathrm{(CH_{3})_{3} C^{+}}$ fragment is released highly excited -- recall that we had three valence electrons surrounding the central carbon that were drawn inward but did not fall into the core. These valence electrons now transfer their energy to the C-C bonds of $ \mathrm{(CH_{3})_{3} C^{+}}$, causing the C-C bond lengths to increase to $ \sim$2.3 $ \mathrm{\AA}$ before collapsing back down to an equilibrium size. Ultimately the $ \mathrm{(CH_{3})_{3} C^{+}}$ remains intact and does not dissociate further.

Ionizing a primary carbon in neopentane causes prompt fragmentation (2 fs) into $ \mathrm{(CH_{3})_{3} C^{+}} + \mathrm{CH_{3}^{+}} + e^{-}$, and, as in ethane, the two fragments have minimal excess vibrational energy. It is interesting to note that the bond connecting the excited carbon is selectively broken. This is relevant for understanding how photon and electron stimulated diffusion operates; if both holes are localized on the same bond, we can have selective bond breaking dominated by excited state kinetics rather than an overall heating of the molecule and statistical bond breaking. Jennison et al. has noted both experimental and theoretical evidence for localization of two-hole final states in hydrocarbons, including neopentane [74]. We observe such localization, i.e., the electron that falls into the hole stimulates nearby electrons to be ionized, in both neopentane and the next molecule to be discussed, adamantane.

Adamantane contains two different types of carbons that may be ionized: four tertiary carbons (CH) and six secondary carbons ( $ \mathrm{CH_{2}}$). Removing a 1s electron from a tertiary carbon causes the following events to occur:

$\displaystyle \mathrm{C_{10}H_{16}^{+}}$ $\displaystyle \rightarrow$ $\displaystyle C_{10}H_{15}^{2+} + H^{-} \ \mathrm{(6\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{core-hole\ collapse} \ \mathrm{(11.7\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C_{10}H_{15}^{3+} + e^{-} + H^{-}} \ \mathrm{(13\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C_{9}H_{15}^{3+} + C + e^{-} + H^{-}} \ \mathrm{(20\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C_{9}H_{15}^{3+} + C^{+} + 2 e^{-} + H^{-}} \ \mathrm{(48\ fs)}.$  

The core-hole relaxation causes a hydride to be dissociated even before the core hole is filled; only 1 fs after the core hole is filled, a secondary electron is ejected, followed by a neutral carbon atom 7 fs later. The carbon that is ejected is the carbon that was initially ionized -- more evidence of two hole localization. The system is mostly stable at this point, only stopping to release an electron from the neutral carbon atom after another 28 fs. All these steps take $ \sim$48 fs from the initial formation of the core hole, in line with the typical nuclear relaxation time of first row Auger dissociative processes.

Removing a 1s electron from a secondary carbon of adamantane causes the following events to occur:

$\displaystyle \mathrm{C_{10}H_{16}^{+}}$ $\displaystyle \rightarrow$ $\displaystyle \mathrm{core-hole\ collapse} \ \mathrm{(6.1\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C_{10}H_{14}^{2+} + e^{-} + 2 H^{-}} \ \mathrm{(8\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C_{9}H_{14}^{2+} + C^{2+} + e^{-} + 2 H^{-}} \ \mathrm{(15\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C_{9}H_{14}^{2+} + CH_{2}^{+} + 2 e^{-}} \ \mathrm{(40.5\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C_{9}H_{14}^{2+} + CH_{2}^{2+} + 3 e^{-}} \ \mathrm{(51.5\ fs)}$  
  $\displaystyle \rightarrow$ $\displaystyle \mathrm{C_{9}H_{14}^{2+} + CH^{+} + H + 2 e^{-}} \ \mathrm{(80\ fs)}$  

In this case, core-hole relaxation causes the C-H bonds attached to the ionized carbon to stretch out but not break before the hole is filled; after the core hole is filled, two hydrides and one secondary electron are promptly (2 fs) released. The next steps involve release of an ionic carbon, recombination of carbon and hydrides to form a stable $ \mathrm{CH_{2}}$ ion, and the ultimate dissociation of the $ \mathrm{CH_{2}^{2+}}$ into $ \mathrm{CH^{+} + H + e^{-}}$.

In these larger hydrocarbons, we observe (1) significant core-hole relaxations in C-H bonds attached to the ionized carbon, but not attached C-C bonds, (2) a tendency to eject the ionized carbon atom and a secondary electron very soon after the core hole is filled, and (3) fragmentation and electron ion recombination events over the next tens of picoseconds.

Figure 4.26: Auger dissociation of neopentane and adamantane following creation of a core hole.

In even larger hydrocarbons, we enter a regime where secondary electrons may be produced, but trapped inside a bulk solid and recombined. This effect is the reason Auger spectroscopy can be used to analyze surfaces -- secondary electrons can only escape from the top monolayers of a surface. To test whether eFF can simulate this effect, we ionize a 1s electron from the center of a roughly spherical diamondoid $ C_{192} H_{112}$ (Figure 4.27). The diamondoid was constructed by starting with a 3x3x3 diamond lattice taken from a periodic structure, truncating primary carbons, then manually reconstructing (100) faces via dehydrogenation. This process introduces some strain into the particle, and we found smaller diamond lattices tended to relieve this strain by forming $ sp^{2}$ carbons and sheet-like structures. As we are interested in the case of saturated hydrocarbons, we chose a larger particle to ionize.

As in neopentane, removing a center core electron causes four surrounding valence electrons to move inward. One valence electron fills the core (4 fs), causing the other three valence electrons to make large amplitude motions and move a short distance through the lattice. The carbon lattice expands slightly around the excitation site, then recontracts as the excited valence electrons recombine with the core. Plotting the trajectories of all the electrons around the excited core, we find that after 5 fs the three valence electrons have moved; after 10 fs motion has been transferred to adjacent electrons; and after 50 fs the motion has been dissipated into thermal motion throughout the lattice (Figure 4.28). Plotting the energy distribution of the electrons over time shows the same effect (Figure 4.29).

Figure 4.27: We remove a core electron from a central carbon of a diamondoid particle.

Figure 4.28: Trajectories of electrons after removal of a core electron.

Figure 4.29: Excited electrons dissipate their energy into their surroundings.

In conclusion, we reproduce nearly all the qualitative aspects of the Auger process -- abrupt core-hole filling, followed by fragmentation and secondary electron generation; localization of two hole states; and trapping of secondary electrons in a bulk solid. We also remarkably reproduce some key quantitative aspects as well, such as the core-hole lifetime and time scale for fragmentation. The theory suggests that in some cases, secondary electron emission may occur only some time after the core hole has been filled, and we speculate that this nonconcerted formation of the two-hole state may explain the lack of $ \mathrm{CD_{3}^{+}}$ and $ \mathrm{CD_{4}^{+}}$ ions following core-ionization of $ \mathrm{CD_{4}}$. We hope that the eFF method will lead eventually to simulations of electron and photon stimulated desorption processes on realistic surfaces and bulk solids, and provide correct microscopic mechanisms for observed macroscopic behavior, such as the selectivity in etch rates that make it possible to create small sharp surface features.


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Julius 2008-04-29