5.2.1 Electronic thermal conductivity and specific heat
The success of the 2T-MD model largely depends on the correct parameterisation of the elec- tronic structure system. Parameters, such as electronic thermal conductivity or electronic specific heat are well-known under ‘standard conditions’, however for high electronic excita- tions, parameterisation with respect to the electronic and ionic temperatures is required. We start our parameters discussion with the electronic specific heat (Ce), defined as the amount
of energy (Ue) required to raise the electronic temperature, Ce = ∂Ue/∂Te.
In general, the electron specific heat capacity would depend on the derivative of the Fermi function and the form of electron density of states (DOS), g(ε) [29, 225]:
Ce= Z ∞ −∞ g(ε)∂f (ε, µ, Te) ∂Te εdε, (5.1)
where ε and µ are the energy level and chemical potential, respectively. The Fermi function is defined as f (ε, µ, Te) = 1/[eβ(ε−µ)+ 1] with β = 1/(kBTe). At low electronic temperatures
(kBTe εF),where εF is the Fermi energy, the Sommerfeld expansion is typically used to
obtain a linear electronic heat capacity Te dependence (after [225]):
Ce = ΓTe, (5.2)
where Γ is the electronic heat capacity coefficient and is given by
Γ = π
2
3 k
2
Bg(εF). (5.3)
5.2. Model parameters for Fe
The expression can also be rewritten for free electron gas (FEG)
Γ = π
2
2 k
2
Bne/εF, (5.4)
where ne is the number density of free electrons. Values of Γ have been calculated exper-
imentally for various metals and several of these are collected in [225]. The value of Ce
at higher temperatures approaching the binding (or cohesive) energy (kBTe ∼ εcoh, where
εcoh= 4.28 eV/atom ∼ 50, 000 K for Fe) can be approximated as that of an ideal gas with the
heat capacity of 32kB per electron [226]. For Fe with two electrons in the 4s band, Ce would
thus saturate at around 3kB until core electrons are excited at even higher temperatures.
At intermediate (0 K kBTe < εB) and very high temperatures (kBTe > εB), however,
the Ce can vary in a non-monotonic fashion and therefore the DOS and the ∂f (ε,µ,T∂Te e) terms
should be computed directly from Eq. 5.1. Such computations were performed in [29] for eight transitions metals (several more, such as fcc and bcc iron can be found in [227]) using density functional theory (DFT). While the precise details of the calculations can be found in [29], we note here that the evaluation of the ∂f (ε,µ,Te)
∂Te term requires prior knowledge of the chemical
potential (µ(Te)) as a function of the electronic temperature. It can be obtained through the
conservation of the total number of electrons (Ne), i.e.
Ne=
Z ∞
−∞
g(ε)f (ε, µ(Te), Te)dε. (5.5)
In the following work we use the Ce(Te) ab initio parameterisation for non-magnetic bcc iron
published in [29, 227], as for the high electronic excitations produced in cascade simulations, the Sommerfeld approximation (derived for low Te) is no longer valid. The different models
of temperature dependence of Ce for Fe are presented in Fig. 5.1, with the ab initio results
presented in more detail in Fig. B.1.
In general, thermal conductivity is a measure of heat transfer and is defined as the propor- tionality constant between the heat flux and temperature gradient. It has two components, a phononic (κl) and an electronic one (κe). In metals, the electronic part dominates over the
phononic component, due to the availability of current carriers in the conduction band, by about an order or two in magnitude (for instance, in tungsten κl/κe has been measured to
be 0.25 [228]). It is the κe component that enters the electronic (continuum) system equation
in 2T-MD, whilst κl is handled by the molecular dynamics part. The thermal conductivity is
well known in low-temperature equilibrium states and in plasma. In the former, it is usually related to the electronic conduction (σ) via the Wiedemann-Franz law for metals (κ = LT σ), where L is the Lorentz number. In dielectrics its temperature dependence is described by the Debye model of phonons, whereas in the case of semiconductors both conductivity compon- ents are significant. In a plasma, thermal conductivity is described by the Coulomb collisions between atoms which results in a ∝ Te5/2 temperature dependence. For metals, such as Fe, in
an equilibrium situation κe temperature dependence is well characterised experimentally and
κ ∝ (Ti)−1. However, an effective non-equilibrium parameterisation of electronic thermal con-
0 5,000 10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5xJ10 6 ElectronicJtemperatureJ[K] ElectronicJspecificJheatJcapacityJ[Jm − 3 K − 1 ] CeJ∼tanhbΓ TeL,J Γ =J708Jm−3K−1 CeJ=JΓ Te,J Γ =J708Jm−3K−1 CeJab−initioJbLinJetJal.L
Figure 5.1: Electronic specific heat capacity for bcc Fe as a function of electronic temperature. Low temperature linear approximation (red dashed line) is valid up to 3,000 K, whilst a ∝ tanh(ΓTe) fit (blue dashed) assuming that the heat capacity saturates at 3kB (contribution
of two electrons per atom) follows the ab initio results (black) [29, 227] well up to 35,000 K, above which more than two electrons contribute. The most accurate ab initio results are used for the cascades simulations.
electrons or with hot ions and cooler electrons, is a more complicated task. This is because the different scattering contributions to the thermal conductivity (described below) can no longer be described by one temperature. Therefore for an effective non-equilibrium description of κe,
the ionic and electronic temperature contributions should ideally be decoupled and quantified separately.
The thermal conductivity can be related to the electron velocity v, electronic specific heat capacity Ce, and the electron characteristic scattering time τe according to the Drude model:
κe =
1 3v
2
FCeτe. (5.6)
The velocity of the electrons is typically approximated by the Fermi velocity (vF) below the
plasma temperature limit. The electron scattering time (i.e. transport time) has two contri- butions: electron-electron (τe−e) and electron-phonon scattering (τe−p), of which characteristic
times are related by 1/τe= 1/τe−e+ 1/τe−ph. We can assume that the electron-phonon scat-
tering time depends on the number of phonons only and therefore on their temperature [229], thus 1/τep= 1/(BTi), where B is a constant. This relationship is valid in the Ti > TD regime,
where TD is the Debye temperature (Fe, TD = 470 K). Furthermore, 1/τe−e = 1/(ATe2) for
Te F/kB, where A is a constant. In iron and other conductors, the electron transport time
would be dominated by the electron-phonon scattering and therefore the ATe2 term can be neg- lected for Te F/kB. Ivanov et al. [60] made a further assumption of a linear electronic heat
capacity and arrived at the following simple expression for electronic thermal conductivity:
κe= 1 3v 2 FCeτe= 1 3v 2 F ΓTe AT2 e + BTi ≈ K0Te Ti , (5.7)
5.2. Model parameters for Fe
which assumes negligible contribution from the electron-electron scattering. In the following 2T-MD cascade simulations in Fe, we expect to deposit enough energy in the electronic sys- tem to notice significant deviations from the linear specific heat (ΓTe) with respect to the
ab initio one. In fact, the maximum electronic temperature recorded in 2T-MD cascades of 50 keV/100 keV pka was of the order of 8,000 K/10,000 K (see Fig. 5.15). Furthermore, since we neglect the electron-phonons scattering dependency on the ionic temperature in the electron-phonon estimation (see Sec. 5.2.2), for consistency we should neglect it in the ex- pression for the electronic thermal conductivity, assuming these quantities are related. These approximations can be accounted for in the equation for thermal conduction as follows:
κe= κ0
Ce(Te)
Ce(300 K)
. (5.8)
A limitation of this approach is that it would result in wrong κe under equilibrium conditions
(i.e. Ti = Te) at Te > 300 K, unlike the expression in Eq. 5.7. In the simulations presented
here we have used κ0 = 80.2 W/(K m) for Fe. The conductivity would reach its peak at
around the Fermi energy (Te ∼ F/kB), when the contribution from the electron-electron
scattering becomes dominant. The thermal conductivity would subsequently decrease as the collision frequency would decrease as 1/τe−e ∝ T
−3/2
e . However, the high Fermi energy of
iron (F = 11.1 eV) means that this limit is unlikely to be reached and therefore Eq. 5.8
remains valid in the temperature regime considered in the cascades investigated here. An alternative expression for κe in the Te< F/kB limit, based on the higher order corrections in
the Sommerfeld theory, is presented in [229].
For completeness, we briefly describe an expression which would describe the electronic thermal conductivity above the Fermi energy in a situation when electron gas becomes non- degenerate, i.e. in a low density plasma limit. Such an expression was described further in [60, 230] and given as κe= C (ϑ2e+ 0.16)5/4(ϑ2e+ 0.44)ϑe (ϑ2 e+ 0.092)1/2(ϑ2e+ bϑi) , (5.9)
where C and b are experimentally determined constants, ϑe = Te/TF, and ϑi = Ti/TF. This
expression correctly reproduces the low (Eq. 5.8) and the high temperature limits (κe ∝ Te5/4).
The most commonly used formula for κe in the two-temperature laser-matter interaction
studies is Eq. 5.7. It is also argued that this form leads to better theoretical predictions of desorption for laser heating experiments [231]. Due to the complexity of some of the expressions and uncertainties involved (none of these expressions can be directly verified by experiments), some authors used the simplest approach and employed a constant electronic conductivity value (as in [76]). Nonetheless, it needs to be noted that a non-equilibrium thermal conduction theory describing a two-temperature state has not been developed yet.
5.2.2 Electron-phonon coupling and electronic stopping
The electronic stopping is a measure of energy loss of a ballistically travelling ion to the electronic excitations per unit distance. This process is agnostic to the motion of other atoms
in a cascade and assumes only small trajectory deviations in electron-ion encounters. The projectile interaction cross-section is predicted to be proportional to its velocity in the Firsov [22] and Lindhard and Scharff [23] models. This feature makes electronic stopping very easy to implement in the MD equations of motion, as it can be represented by a simple damping term proportional to the ionic velocity. The value of this damping term can be estimated from the SRIM [54] code. In the case of iron projectile in iron the damping time (τes) is obtained
from the stopping strength of λ = 0.1093 eV/˚A via 1/τes = χes = λ(m2)1/2 and is set to
τes= 0.984 ps, where λ is the constant of proportionality [76] in the Firsov [22] (Se= λE1/2)
or Lindhard and Scharff [23] (F = λ(m/2)1/2v) models.
The electronic stopping cutoff energy, EC, is a practical way of preventing over-damping of
the ionic system. In particular, for a system at equilibrium (Ti = Te) the ionic temperature in
damped MD would tend to zero. It is therefore assumed that below a certain cutoff the motion of atoms is too correlated to be represented as a projectile through an electron sea and that the electron-phonon coupling (which allows for two-way energy transfer) is dominant. Since there is no theoretical justification for the value of the cutoff, rather than a practical one, it is here assumed to be 8.4 eV [76], 4.2 eV and 0.6 eV [128]. Recently, approximation to Ehrenfest time-dependent tight-binding [128], and real-time time-dependent density functional theory (RT TD-DFT) [91] have not conclusively shown that such a well-defined cutoff exists.
The evaluation of the electron-phonon coupling is more difficult. A value for e-p is typically obtained through low energy laser excitations experiments by fitting the temperature relaxation data to the two-temperature models. Here, we are using the electron-phonon coupling constant reported by Zhigilei et al. [29, 227]:
G0= 5.5 · 1018 W m−3K−1, (5.10)
and we can translate it to the electron-phonon relaxation time (analogously to Eq. 2.11)
τep = Ce(300 K)/G0 = 1.404 ps. (5.11)
The value of G0 = G(300 K) is kept constant throughout the simulations for simplicity. In
general, it will depend quite strongly on Te and less on Ti. However, due to electron-phonon
coupling inherent uncertainty even at equilibrium conditions (Ti = Te = 300 K) the addition
of further complexity in the model is unnecessary. The G0 dependence on Te for iron was
characterised in [29].
5.2.3 Interatomic potential
We chose a potential based on the many-body embedded atom model (EAM). Details of this potential form are described in Sec. 4.1.4.1. The EAM parameterisation used here was optim- ised [232] to reproduce the defect energetics, as well as other bulk properties of bcc-iron. It is therefore a suitable and a widely chosen interatomic interaction model for cascade simula- tions (see ‘M07’ in [233] for a comparison). The EAM potential was joined to a ZBL form [206] at short distances (< 1 ˚A). This joining process was calibrated against the threshold