XVI National Symposium on Condensed Matter Physics, Sokobanja 2004
Phonon Spectra And Possible States In
Superlattices
D.I.Ilić 1, S.M.Vučenović 2, S.K.Jaćimovski 3, J.P. Šetrajčić 4, D.Raković 5
1
Faculty of Technical Sciences, University of Novi Sad,
Trg D.Obradovića 6, 21000 Novi Sad, Serbia and Montenegro
2
Faculty of Medicine, University of Banja Luka,
Save Mrkalja 14, 78 000 Banja Luka, Republic of Srpska, Bosnia and Hercegovina
3
High School "D.Obradović",
Kralja Petra I Karađorđevića, 23330 Novi Kneževac, Serbia and Montenegro
4
Institute of Physics, Faculty of Sciences, University of Novi Sad,
Trg D.Obradovića 4, 21000 Novi Sad, Serbia and Montenegro
5
Faculty of Electrical Engineering, University of Belgrade,
Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia and Montenegro
Abstract. The Green's functions method, adjusted to bounded crystalline structures, is ap-
plied to obtain the phonon dispersion law in superlattices. Poles of Green's functions defining
phonon spectra can be found by solving of the secular equation. For different boundary parameters, this problem is presented graphically. The presence of boundaries as well as the
change of boundary parameters lead to appearance of new properties of layered structure. The
most important feature is that beside allowed energy zones (which are continuous as in the
bulk structure), zones of forbidden states appear. Different values of boundary parameters
lead to appearance of lower and upper energy gaps, or dispersion branches spreading out of
bulk energy zone. The correlation with spectra of phonons in corresponding unbounded
structures is maintained in the work.
Superlattices are ultrathin layered crystal structures, periodical in one direction,
with the period exceeding the constant of the lattice about twenty times [1–3]. The
scope of our study in this paper is the superlattice, the basic motive of which is
composed of na layers of one, nb layers of another and nc layers of the third type of
atoms, successively arranged along the z − direction, while it is unbounded along x
and y directions. To make connected layers consisting of different atoms possible, the
lattice constants along x and y directions must respectively be equal, i.e.
a xa = a xb = a xc ≡ a x and a ya = a by = a cy ≡ a y , whereas along z − direction they may be
different a za ≡ a a ≠ a bz ≡ a b ≠ a zc ≡ a c and a za / b ≡ a a / b ≠ a bz / c ≡ a b / c ≠ a za / c ≡ a a / c .
We introduce the following notation:
r
n ≡ {n x , n y , n z } , n x / y / z ∈ (− N x / y / z 2 , + N x / y / z 2) ,
where: n x / y − is the atom site counter along x , i.e. y − direction, n z − is the position
counter of the basic motive of superlattice ( z − direction), while: nl − is the atom site
counter in the basic motive. Starting point of our study is the standard Hamiltonian of
the phonon subsystem for bulk structures [4–7], written in the harmonic as well as in
the nearest neighbours approximations, which is adapted to the model-structure of
superlattice presented on Fig.1.
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XVI National Symposium on Condensed Matter Physics, Sokobanja 2004
FIGURE 1. Arrangement of atoms in the basic motive of the superlattice
Taking into account that the superlattice represents the periodical crystal structure,
for an arbitrary function of the position the cyclic conditions of x, y and z indexes are
valid, by which the permitted validities of x, y and (particularly) z − components of
the wave vector can be obtained:
2πν i
iN
k a
f mx m y m z ml + N x / y = f mx m y mz ml
⇒ e x/ y x/ y x/y = e x/ y ,
(1)
~
f mx m y m z ml + (na + nb + nc )N z = f mx m y mz ml ⇒ e i (na + nb + nc )N z k z a = e 2πν z i .
For counting of allowed values of z − component of the wave vector k z , the
counter ν z ∈ {0, ± 1, ± 2, ...,± N z 2 } is used, by which the boundaries of the first
Brillouin zone along the z − direction are defined:


π
π
(2)
k z ∈ −
, +
~
~
(n a + nb + nc ) a 
 (na + nb + nc ) a
where the notation a~ for the average validity of the lattice-constant along the z −
direction has been introduced:
(n − 1) a a + (nb − 1) a b + (nc − 1) a c + a a / b + a b / c + a a / c
(3)
a~ = a
n a + nb + nc
We are looking for the phonon dispersion law with the help of the phonon two-time
commutator Green's function [8–10]:
(4)
G nr ,nl ;mr ,ml (t − t ′ ) ≡ u nr ,nl (t ) u mr , ml (t ′) = Θ(t − t ′) [u nr , nl (t ) , u mr ,ml (t ′)]
which satisfies the equation of motion
d2
Θ (t − t ′ ) r
[[ p nr ,nl (t ), H (t )], u mr , ml (t ′)]
M i 2 Gnr , nl ;mr , ml (t − t ′) = −ihδ nr , mr δ nl , ml δ (t − t ′) +
ih
dt
where M i ∈ (M a , M b , M c ) . Assuming that t ′ = 0 and after performing time Fourier
transform, the last equation passes into
r
ih
1
− M i ω 2 Gnr , nl ;mr , ml (ω ) = −
δ nr ,mr δ nl ,ml +
[ p nr ,nl , H ], u mr , ml
(5)
ω
ih
2π
Next, we are going to calculate the commutators in the Green's function which
appears in the equation (5). Since the translational invariance of the system we are
studying is broken, we introduce the partial spatial Fourier-transformation by indexes
x, y and z (because by index l, the translational symmetry has been disturbed)
1
i [ax k x ( nx − m x )+ a y k y (n y − m y )+ a~ ( na + nb + nc ) k z ( nz −m z ) + J ]
Gnr ,nl ;mr ,ml (ω ) = ∑
G
e
,
(6)
;
n
m
N kr l l
r
where N = N x N y N z , k ≡ {k x , k y , k z } and:
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XVI National Symposium on Condensed Matter Physics, Sokobanja 2004




J =




a a k z (nl − ml ) ,
a k z (na − 1) + a
a
nl − ml < na
a/b
nl − ml = na
kz ,
a a k z (na − 1) + a a / b k z + ab k z (nl − ml − na ) ,
na < nl − ml < na + nb
a a k z (na − 1) + a a / b k z + ab k z (nb − 1) + ab / c k z ,
nl − ml = na + nb
a k z (na − 1) + ab k z (nb − 1) + a a / b k z + ab / c k z + a c k z (nl − ml − na − nb ) , na + nb < nl − ml < na + nb + nc
a a k z (na − 1) + ab k z (nb − 1) + a a / b k z + ab / c k z + a c k z (nc − 1) + a a / c k z ,
nl − ml = na + nb + nc
Applying this to equations of motion of each layer inside the basic motive of the
superlattice, we obtain the system of na + nb + nc of nonhomogeneous algebraic-difference equations with the same number of undetermined Green's functions. They can be
expressed as follows: Gm = Dm D where Dm is the determinant of the variable and
D the determinant of the system. Poles of Green's functions by which the phonon dispersion law is determined [11–14], can be obtained on condition that the determinant
of the system is equal to zero. As the equation that D = 0 , in general is not
analytically solvable, we have made here the numerical method of approach for certain
cases. Different combinations of atom numbers na , nb and nc have been examined,
and also the changes of relations of Hooke's elastic constants between and inside
crystalline films. During further analyze, the superlattices made of films (with na , nb
and nc layers) of the same atom types ( M a = M b = M c ≡ M , Ca = Cb = Cc ≡ C and
Ω a2 = Ω b2 = = Ω c2 ≡ Ω 2 = C M ) have been examined in two different cases:
1. When the connection between atoms within layers is stronger than the connection
between atoms on boundary surfaces of films.
2. When the connection between atoms within the layers is weaker than the one
between atoms on the boundary surfaces of films.
These two cases are graphically presented on Fig.2, where one can see reduced
phonon frequencies ( ω Ω ) on ordinate and reduced wave vectors along z − direction
a~k z (na + nb + nc ) π on abscisses. Only the centre of the first Brillouin zone was taken
into consideration ( k x = k y = 0 ). Numbers of atoms in the relative layers are shown in
brackets: (n a , nb , nc ) . On the basis of these analysis we have come to the following
conclusion:
1. As the result of breaking of the translational invariance along z − direction, the
energy zone (which is continual as in unbounded crystals) passes into subzones
separated by forbidden energy zones.
2. Since the length of the motive, repeating itself along z − direction of the
superlattice is greater than the distance between atoms, z − component of the wave
vector must be redefined.
3. With the increase of energy, the density of phonon states in all examined cases
becomes higher.
4. In case of the symmetric superlattice ( na = nb = nc ) with the identical atoms,
energy levels form groups made from three dispersion branches in the boundary as
well as in the center of the first Brillouin zone.
a
z
z
z
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XVI National Symposium on Condensed Matter Physics, Sokobanja 2004
FIGURE 2. Energy spectra of phonons in superlattices with three layers
5. In case of a loose connection between layers of the superlattice, it comes to
shifting of energy levels inside the bulk zone ( ω Ω = 2 ) disregarding the total
number of atoms inside the basic motive of the superlattice. If the connection
between the layers of the superlattice is stronger, energy levels are being shifted
above bulk zone.
6. The phonon dispersion law is invariant to permutation of film-thickness inside of
one basic superlattice motive.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
D.Raković: Physical Basics and Characteristics of Electrotechnical Materials, Faculty of Electrical
Engineering, Belgrade 1995.
Z.Ikonić and V.Milanović: Semiconducting Quantum Microstructures, University of Belgrade, Belgrade 1997.
D.Popov, S.K.Jaćimovski, B.S.Tošić and J.P.Šetrajčić, Physica A 317, 129 (2003).
S.G.Davison and M.Steslicka: Basic Theory of Surface States, Clarendon, Oxford 1996.
M.G.Cottam, D.R.Tilley: Introduction to Surface and Superlattice Excitations, Univ. Press, Cambridge 1989.
M.Prutton: Introduction to Surface Physics, Clarendon Press, Oxford 1995.
C.Kittel: Quantum Theory of Solids, Wiley, New York 1963.
B.S.Tošić: Statistical Physics, Institute of Physics, Novi Sad 1978.
G.Rickayzen: Green's Functions and Condensed Matter, Acad.Press, London 1980.
E.N.Economou: Green’s Functions in Quantum Physics, Springer, Berlin 1979.
S.K.Jaćimovski, J.P.Šetrajčić, B.S.Tošić and V.D.Sajfert, Materials Science Forum . . . , 33 (2004).
J.P.Šetrajčić, S.K.Jaćimovski, D.Raković and D.I.Ilić, Proceedings 2nd WSEAS International Conference on
Nanoelectronics and ElectroMagnetic Compatibility, (ICONEMC) 1, 1091 (2002).
J.P.Šetrajčić, I.D.Vragović, D.Lj.Mirjanić, S.K.Jaćimovski, Bal.Phys.Lett. 5, 418 (1998).
S.Lazarev, D.Mirjanić, M.Pantić, B.Tošić, J.P.Šetrajčić, Phys.Chem.Sol. 60, 849 (1999).
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