DEVELOPMENT OF EWALD’S METHOD FOR DETERMINATION OF PHONON PROPERTIES IN
PLANAR NANOSTRUCTURES WITH IONIC CRYSTAL LATTICE
CRÎŞMARI Dmitrii, POKATILOV Evghenii
Moldova State University
Reviewer: NIKA D., dr.
Keywords: zincblende, Valence Force Field model, Ewald’s method
Currently the question naturally arises as to the effect of dimensional confinement on the properties on the
phonons in ionic nanostructures as well as the properties of the phonon interactions in such nanostructures [1]. In the
given work by Ewald’s method was calculated the Coulomb field in dipole lattice with zincblende structure. Thus,
the previously developed covalent molecular-dynamic Valence Force Field model [2,3] was supplemented by
Coulomb interactions between ions. This allowed to describe theoretically phonon dispersion dependences of
nanodimensional slabs from GaP.
In the work we consider a crystal slab consisting of N one-cell layers, parallel to the (hkl ) plane of the
crystal structure [4]. By appropriate choice of the unit cell, the primitive translation vectors a1 and a2 can be taken
to lie on the (hkl ) plane, while a3 lies out of it. The slab is considered as a two-dimensional periodic structure and
a two-dimensional reciprocal lattice is associated with it.
The field at a point x due to all point dipoles p(l ', k ') at lattice sites x (l ', k ') , l '  (l1 ', l2 ', l3 ') , is given by
E ( x ) 
p (l


3
', k ')
l3 ', k '
2
x x
exp[iq  x (l ', k ')]
,
l1 ',l2 ' | x (l ', k ')  x |

(1)
where  ,   (1, 2,3) number the axis of some Cartesian coordinate system Ox1 x2 x3 , k ' runs over all ions in the
unit cell, l ' runs over all unit cells from (, ,0) to (, , N ) , and q is a two-dimensional wave vector. We
note that the index l3 labels the one-unit-cell layers of the slab. Hence the index (l3 , k ) labels the atoms in the long
unit cell of the two-dimensional periodic structure defined by the vectors (a1 , a2 , Na3 ) .
The sum over l1 ', l2 ' on the right-hand side of Eq. (1) is a periodic function in two dimensions and can be
expanded in Fourier series
2
 exp{ | x (l1 ', l2 ')  x|| (l3 ', k ')  x|| |2  2  iq  [ x (l1 ', l2 ')  x|| (l3 ', k ')  x|| ]}

l1 ', l2 '
2 

sa  2
,
 exp{ | K (h , h )  q |
2
1
2
(2)
/4   iK (h1 , h2 )  [ x|| (l3 ', k ')  x|| ]}
2
h1 , h2
where sa | a1  a2 | and K (h1 , h2 ) is a vector of the two-dimensional reciprocal lattice. Further necessary to
introduce a dividing point R on the  axis and evaluate the integral in Eq. (1) separately over the interval (0, R)
using the right-hand side of relation (2) and over the interval ( R, ) keeping the integrand as it is. So twodimensional variant of Ewald’s method provides convergence of lattice sums in the case of ionic planar
nanostructures.
The described theoretical method allowed to carry out the lattice dynamics research of nanostructured ionic
quasi-two-dimensional compounds and calculate energy spectra of such compounds from gallium phosphide in
order to determine further their kinetic coefficients.
References:
1. Stroscio М.A., Dutta М. Phonons in Nanostructures. - Cambridge: Cambridge University Press, 2004. 288 p.
2. Крышмарь Д., Покатилов Е. Вычисление кулоновского поля в дипольной решётке методом Эвальда
в дополнение к модели VFF для расчёта колебаний ионного кристалла GaP // Stud. Univ. - 2010. - Vol.37. P.65-69.
3. Biswas K., Franceschetti A., and Lany S. Generalized valence-force-field model of (Ga,In)(N,P) ternary
alloys // Phys. Rev. B. - 2008. - Vol.78. - P.085212-1-085212-10.
4. Холопов Е.В. Проблемы сходимости кулоновских и мультипольных сумм в кристаллах // Успехи
физ. наук. - 2004. - Vol.174. - P.1033-1060.
The work was supported in part by the Moldova State under Project No. 10.819.05.02F (17/ind) and
11.817.05.10F.