Importance of Dynamical Electron
Correlation in the Stopping Power of Solids
for Slow Ions
Y. Takada
The concept of electron correlation permeatesmany subfields of condensed matter physics, but it israther difficult to find good examples of its clear
manifestation. It is even more difficult to illustrate its
importance in dynamical (i.e., time-dependent)
processes. Here we report a success in obtaining aclean theoretical formula directly relating dynamicalcorrelation,ascharacterized
byfxc
thefrequency-dependent exchange-correlation kernel ofthetime-dependent
density-functionaltheory(TDDFT), with the experimentally observed stopping
power, –dE/dx, of solids for slow ions.
The nonperturbative theory of scattering ofnoninteracting electrons by a central potential
combined with the density-functional theory (DFT)
has proved successful in calculating –dE/dx of anelectron gas for slow ions. This scheme accurately
reproduces the measured energy loss of slow protons
and antiprotons in a variety of solids. To some extent
it also successfully accounts for the observed behavior
in the energy loss of slow ions with increasing charge
Z1e. This kind of calculations were extended toprojectile velocities v approaching the Fermi velocityof the target, but so far only few attempts have been
made to include the effect of many-body dynamicalcorrelation that is absent in the existing DFT-basedschemes.
We approach the problem of including dynamicalcorrelation in the framework of the TDDFT and havesucceeded in deriving a rigorous formula for –dE/dx
of an interacting electron gas for ions in the limit oflow projectile velocities (v 0) [1]. Depending onwhether fxc is involved or not, this formula can berewritten as a sum of two terms: The first one without
fxc is shown to yield the conventional result for–dE/dx of noninteracting Kohn-Sham electrons firstreported by Echenique et al. [2]. The second one
represents a new and important result fully
incorporating the effect of dynamical correlation
through the imaginary part of fxc.
In the spirit of the widely used local-density
approximation (LDA), we can think of an appropriate
approximation to fxc in such a form as the one
proposed by Qian and Vignale [3]. Then our formulaenables us to make a reasonably accurate evaluation of
the friction coefficient (–dE/dx divided by v in thelimit of v0) of an interacting electron gas as afunction of the atomic number of the projectile ion Z1.
An example of the results obtained by suchcalculations for the homogeneous electron gas isshown in Fig. 1 in which the solid and dashed curves
correspond, respectively, to the cases with and without
dynamical correlation (or the second term in ourformula). Here the electron density characterized by

the conventional density parameter rs is taken to be
1.59 which is just the same density as the one forvalence electrons in carbon films. In view of the factthat at low velocities the energy loss of ions in solids
is due mainly to the stopping power of valence
electrons, our present theoretical results can bedirectly compared with the experimental ones incarbon.

Fig. 1. Comparison between experiment on the stopping power,–dE/dx, of carbon films for slow ions (characterized by theatomic number Z1) with three different values of the velocities vand theory on the same quantity of the homogeneous electron
gas with the electron density equivalent to the valence-electron
density in carbon in the limit of zero ion velocity (v 0).

As can be seen in Fig. 1, experiment exhibits an
oscillatory behavior in the friction coefficient withincreasing Z1. This behavior, which is attributed to theshell structure of valence electrons around the ion, isreproduced rather well in the DFT-based calculation(as labeled by LDA), but a better quantitative
agreement with experiment is obtained by our theory
based on the TDDFT, thanks to the enhancement of–dE/dx by dynamical correlation, or more specifically,
the excitonic effect which has recently been revealed
to be very important through the accurate calculationof the dynamical structure factor of the homogeneouselectron gas [4].
Agreement between theory and experimentbecomes unsatisfactory for Z1 larger than about 20,
indicating a need to go beyond the LDA-type
approximation in the description of fxc.
References[1] V. U. Nazarov, J. M. Pitarke, C. S. Kim, and Y.
Takada, Phys. Rev. B 71, 121106(R):1-4 (2005).
[2] P. M. Echenique, R. M. Nieminen and R. H.
Ritchie, Solid State Commun. 37, 779 (1981).
[3] Z. Qian and G. Vignale, Phys. Rev. B 65,235121:1-12 (2002).
[4] Y. Takada and H. Yasuhara, Phys. Rev. Lett. 89,216402:1-4 (2002).

Authors
V. U. Nazarov,a J. M. Pitarke,b C. S. Kim,a and Y.
TakadaaChonnam National University,bEuskal Herriko Unibertsitatea.