Tamm resonances and minibands in the models of atomic chains and superlattices
The spectrum of the modelled regular superlattice which includes contacts is investigated using the transfer-matrix formalism. It is shown that the separated pair resonances in the chain spectrum (which are distributed in the permitted and in the forbidden zones, respectively) appear due to the ex...
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Gerasimov, O.I. Khudyntsev, N.N. 2017-06-13T13:17:22Z 2017-06-13T13:17:22Z 2000 Tamm resonances and minibands in the models of atomic chains and superlattices / O.I. Gerasimov, N.N. Khudyntsev // Condensed Matter Physics. — 2000. — Т. 3, № 1(21). — С. 175-182. — Бібліогр.: 12 назв. — англ. 1607-324X DOI:10.5488/CMP.3.1.175 PACS: 68.65.+G, 73.90.Dx https://nasplib.isofts.kiev.ua/handle/123456789/121035 The spectrum of the modelled regular superlattice which includes contacts is investigated using the transfer-matrix formalism. It is shown that the separated pair resonances in the chain spectrum (which are distributed in the permitted and in the forbidden zones, respectively) appear due to the existence of a single contact. The eroding of the noted minizones after making correlations between the separate regular segments in the superlattice is demonstrated. Similar resonances and minizones which can be called the Tamm resonances and minizones are considered to be useful elements for describing the surface states and zone-band structure of real systems within the restricted geometry. У роботі за допомогою формалізму трансфер-матриці розв’язана проблема обчислення спектра регулярної супергратки та запропоновано досить простий метод опису контакту в її структурі. Показано, що існування одного контакту призводить до виникнення відокремлених парних резонансів у спектрі ланцюжка, а також продемонстровано розмиття таких резонансів у відповідні мінізони після поєднання окремих регулярних сегментів у супергратку. Означені резонанси та мінізони є таммівськими та можуть виступати зручними елементами опису поверхневих станів та зонної структури реальних низьковимірних систем. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Tamm resonances and minibands in the models of atomic chains and superlattices Таммівські резонанси та мінізони для модельного одновимірного потенціалу Article published earlier |
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| title |
Tamm resonances and minibands in the models of atomic chains and superlattices |
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Tamm resonances and minibands in the models of atomic chains and superlattices Gerasimov, O.I. Khudyntsev, N.N. |
| title_short |
Tamm resonances and minibands in the models of atomic chains and superlattices |
| title_full |
Tamm resonances and minibands in the models of atomic chains and superlattices |
| title_fullStr |
Tamm resonances and minibands in the models of atomic chains and superlattices |
| title_full_unstemmed |
Tamm resonances and minibands in the models of atomic chains and superlattices |
| title_sort |
tamm resonances and minibands in the models of atomic chains and superlattices |
| author |
Gerasimov, O.I. Khudyntsev, N.N. |
| author_facet |
Gerasimov, O.I. Khudyntsev, N.N. |
| publishDate |
2000 |
| language |
English |
| container_title |
Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
Таммівські резонанси та мінізони для модельного одновимірного потенціалу |
| description |
The spectrum of the modelled regular superlattice which includes contacts
is investigated using the transfer-matrix formalism. It is shown that the separated pair resonances in the chain spectrum (which are distributed in the
permitted and in the forbidden zones, respectively) appear due to the existence of a single contact. The eroding of the noted minizones after making
correlations between the separate regular segments in the superlattice is
demonstrated. Similar resonances and minizones which can be called the
Tamm resonances and minizones are considered to be useful elements
for describing the surface states and zone-band structure of real systems
within the restricted geometry.
У роботі за допомогою формалізму трансфер-матриці розв’язана
проблема обчислення спектра регулярної супергратки та запропоновано досить простий метод опису контакту в її структурі. Показано, що існування одного контакту призводить до виникнення відокремлених парних резонансів у спектрі ланцюжка, а також продемонстровано розмиття таких резонансів у відповідні мінізони після
поєднання окремих регулярних сегментів у супергратку. Означені
резонанси та мінізони є таммівськими та можуть виступати зручними
елементами опису поверхневих станів та зонної структури реальних
низьковимірних систем.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121035 |
| citation_txt |
Tamm resonances and minibands in the models of atomic chains and superlattices / O.I. Gerasimov, N.N. Khudyntsev // Condensed Matter Physics. — 2000. — Т. 3, № 1(21). — С. 175-182. — Бібліогр.: 12 назв. — англ. |
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| first_indexed |
2025-11-26T20:20:32Z |
| last_indexed |
2025-11-26T20:20:32Z |
| _version_ |
1850773156432707584 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 1(21), pp. 175–182
Tamm resonances and minibands in the
models of atomic chains and
superlattices
O.I.Gerasimov, N.N.Khudyntsev
Odesa State Hydrometheorological Institute,
100 Lvivska Str., 270011 Odesa, Ukraine
Received June 5, 1998, in final form January 26, 2000
The spectrum of the modelled regular superlattice which includes contacts
is investigated using the transfer-matrix formalism. It is shown that the sep-
arated pair resonances in the chain spectrum (which are distributed in the
permitted and in the forbidden zones, respectively) appear due to the exis-
tence of a single contact. The eroding of the noted minizones after making
correlations between the separate regular segments in the superlattice is
demonstrated. Similar resonances and minizones which can be called the
Tamm resonances and minizones are considered to be useful elements
for describing the surface states and zone-band structure of real systems
within the restricted geometry.
Key words: contact, superlattice, minibands, spectrum
PACS: 68.65.+G, 73.90.Dx
1. Introduction
The modelling of the low-dimensional atom-molecular complexes, adatomic clus-
ters, chains, lattices and superlattices belongs to a number of traditionally and ac-
tively researched problems of the solid state theory [1]. The urgency of the research
is limited by the essentiality and the difficulties of a direct microscopic description
of the recently synthesized low-dimensional atom-molecular complexes and other
physical objects with the symmetry, for instance, different from the translational one
(icosahedra or quasicrystallic) [2]. This way, specifically, in [3], the one-dimensional
Kronig-Penny superlattice with a regular intrinsic structure determined by the com-
bination of zero-radius potentials [4] was studied. Then, the simple analytical model
is suggested [1] permitting to explore some general properties of superlattices, sep-
arate contacts (interfaces) being the model elements for describing semiconductor
and metallic films, quasicrystalls and other objects. Such a model can be used in
constructing more general theories. Using the transfer-matrix formalism this paper
c© O.I.Gerasimov, N.N.Khudyntsev 175
O.I.Gerasimov, N.N.Khudyntsev
considers the problem regarding the spectrum of a regular superlattice as well as
suggests a simple method of describing the contact in the superlattice structure. It
is shown that the existence of a single contact causes separate pair resonances in
the chain spectrum (which are distributed in the permitted and in the forbidden
zones, respectively). The eroding of the mentioned minizones after making a con-
nection between the separate regular segments in the superlattice is demonstrated
as well. Similar resonances and minizones can be called the Tamm’s ones, being
useful elements for describing the surface states and the zone structure of the real
low-dimensional systems.
2. Statement of the problem. Transfer–matrix formalism
Consider the sequence of nonstructured particles forming a one-dimensional su-
perlattice with the alternation of the regular segments which are characterized by
different periods as well as by different power constants. The potential energy of the
system is modeled using the point-like interactions:
U (x) =
1
2
+∞∑
n,n′=−∞
[Vaδ (x− na) + Vbδ (x− n′b)] , (1)
Figure 1. The schematic scene of the
superlattice potential with the transla-
tional symmetry (a,b are the segments
period, Va, Vb are the power constants).
where a, b are the lattice periods, Va, Vb
are the power constants of the centrums
of different segments. We also assume
that the mass (or the effective mass)
of the scattered particle and the Plank
constant are equal to 1. The descrip-
tion of one-dimensionality (quasi-one-
dimensionality) of the zond-particle mo-
tion is given, for instance, in [8]. The
summation in (1) is carried out over all
different indexes n and n′. Modelling of
the superlattices using the potential (1)
is demonstrated in figure 1. The con-
tacts of segments with different lattice
parameters are shown by means of the contrast horizontal lines. For simplicity, the
symmetry in the position of superlattice segments is suggested as a translational
(Na,Nb = const). The refusal from this assumption leads us to the problem of a
regular chain with defects, which was considered in [9]. In such a case with a su-
perlattice, a quasiperiod composed of two nearest segments (see figure 1) could be
considered.
The Shrödinger equation of the considered problem could be written in the fol-
lowing form: [
−
1
2
d2
dx2
+ U(x)− E
]
Ψ (x) = 0, (2)
176
Tamm resonances and minibands
where E and Ψ (x) are respectively the eigenvalue of the energy and of the wave
function of zond-particle in the field of the potential U (x).
For the model potential (1), the zond-particle motion can be considered as a free
motion and the wave function can be described by superposition of two plane waves:
Ψ (x) = A exp (ikx) +B exp (−ikx) , (3)
where k2 = 2E. The coefficients A and B could be extracted from the boundary
conditions.
The natural boundary conditions in the point d of n-atomic centrum for the wave
function and their first derivative could be chosen in the following way:
Ψ (d+ 0) = Ψ (d− 0) ,
dΨ (d+ 0)
dx
=
dΨ (d− 0)
dx
+ VnΨ (d− 0) . (4)
The boundary conditions (4) follow from the original structure of equation (2)
and also from the general properties of the zero-radius potential (see [4]). In the
matrix form, the boundary conditions (4) could be rewritten as:
T̂n
(
An
Bn
)
=
(
An+1
Bn+1
)
, (5)
where T̂n is the transfer-matrix being the square unimodular matrix of the size 2×2
in the Kelly form [6]:
T̂n =
(
α∗
n β∗
n
βn αn
)
. (6)
Within the framework of the transfer-matrix formalism, the problem regarding
the spectrum of a one-dimensional system of point-like centrums could be solved
exactly. A general approach to using the matrix form for the boundary conditions
for their connection with the energy spectrum as well as some characteristics of
particle transmission in the field of different potentials is given in [6]. This formalism
also makes it possible to determine some transport characteristics for the system
studied (for example, the Landauer resistance coefficient [9] which represents the
relation between the quantum-mechanical coefficients of transmission and reflection,
respectively). Following the general approach described in [6,9], the expressions for
the spectrum E and the resistance coefficient ρ could be extracted in the form of the
following relations (the conversion from the boundary conditions (4) to the following
expressions have also been considered in [9]):
E =
Q2
2
, cosQa =
1
2
Sp T̂n, (7)
ρ =
Ktran
Krefl
= |βn|
2 , (8)
where Q is the quasi-wave vector of the zond-particle, Ktran, Krefl are the reflection
and transmission quantum-mechanical coefficients. The expressions for αn, βn are
given in [9].
177
O.I.Gerasimov, N.N.Khudyntsev
Two types of a zond-particle motion in the field of the system considered would
be analyzed below: the resonance and nonresonance tunnelling [8]. Completely real
and completely imaginary values of the quasi-wave vector Q correspond to these
cases respectively. Now consider the pure resonance tunnelling (otherwise, we can
put Q → iQ to obtain correlations). If Na, Nb > 1, the diagonal representation of
transfer-matrix T̂n could be written in the following form:
T̂nd = Ŝ
−1
n T̂nŜn =
(
λn+ 0
0 λn−
)
, (9)
where λn± are the eigenvalues of the matrix (6), Ŝn =
(
Sn+ Sn−
1 1
)
is the diago-
nalized matrix, Sn± = (λn± − αn)β
−1
n .
For instance, in the case of regular chain with the period a (Va=Vb=V, d=na):
λa± = exp (±iqa) , cos qa = cos ka+
V
k
sin ka,
αa =
(
1 +
iV
k
)
exp (−ika) , βa =
iVa
k
exp (ika) . (10)
The chain containing isotopic and (or) shift defects was considered in detail in
[9]. Within the next chapters, the suggested transfer-matrix formalism will be used
for describing the complex superlattice (see figure 1).
3. A model of contact within the superlattice. The spectrum
and resistance of a single contact
Consider the superlattice within the quasi-period which includes Na atoms in
the segment with the period a and Nb atoms in the segment with the period b. The
respective transfer-matrix has the following form:
T̂c =
(
T̂a
)Na
(
T̂b
)Nb
, (11)
where T̂a and T̂b are the transfer-matrix (6) with the elements (10) for the regular
chains with periods a and b, respectively.
We are interested in the diagonal representation of the matrix T̂c (only the nu-
merical solution of the problem could be provided within all other representations).
As it was shown in [6,9], the diagonal representation of the matrix (6) with different
parameters doesn’t exist. Using matrix Ŝa (as a diagonalizing one), using (9) and
(10) we obtain:
T̂cd = Ŝ
−1
a T̂cŜa =
(
T̂ad
)Na
T̂q
(
T̂b
)Nb
T̂p, (12)
where T̂a,bd are the transfer-matrixes within the diagonal representation, and
T̂q = Ŝ
−1
a Ŝb, T̂p = Ŝ
−1
b Ŝa. (13)
178
Tamm resonances and minibands
In this representation, the contact of two regular segments could be considered
as the last one, just like the regular chain that includes two defects in their structure
could be described by the non-diagonal transfer-matrixes T̂q and T̂p (where matrix
T̂p characterizes the next contact from the left to the right). Thus, the transfer-
matrix which corresponds to a single contact is T̂q, namely:
T̂q =
1
λa+ − λa−
(
λb+ − λa− + αa − αb λb− − λa− + αa − αb
λa+ − λb+ + αb − αa λa+ − λb− + αb − αa
)
. (14)
Note that the expression for the matrix T̂p could be obtained from (14) by
changing the periods a and b. Finally, for describing the zone structure of the segment
which includes the contact we obtain:
Ec =
Q2
c
2
, cosQca = 1 +
λb+ − λb−
λa+ − λa−
, (15)
where Qc is the quasi-wave vector of zond-particle in the field of the segment with
the contact.
Figure 2. The dependence of SpTq (ver-
tical axis) from the wave vector ka (hor-
izontal axis): Va = 0.4, b/a = 4.
The results of the numerical calcu-
lations using the formulas (15) are pre-
sented in figure 2. We should take into
account the deformation of the zone
structure of the lattice consisting of two
regular segments in comparison with the
structures of zones of separate regular
segments. The positions of a boundary
limit of the permitted and of the for-
bidden zones are shifted, and the sep-
arate resonances (interfaces) have ap-
peared which correspond to the forbid-
den levels in the permitted zones and
(or) to the permitted levels in the for-
bidden zones, respectively. Those reso-
nances (which have a Tamm’s nature)
are split into pairs near the point of the zone boundary q = πn/a (n is the in-
teger number). Note that the same effect of splitting the separate resonances was
also found in the spectrum of the regular chains that include the isotopic- and (or)
shift-type defects [11].
Now consider the resistance coefficient (see chapter 2) which is limited by the
contact of two regular segments. Using the substitution of a nondiagonal element of
the transfer-matrix (14) into (8) one can obtain:
ρc =
1
sin2 qa
[
sin2 ω +
(
1 +
W 2
k2
)
sin2 κ+ 2 sinκ sinω
(
cosϕ+
W
k
sinω
)]
, (16)
where
ω = q (a− b) /2, ϕ = (q + k) (a+ b) /2, κ = k(a− b)/2, W = 2VaVb(Va + Vb)
−1.
179
O.I.Gerasimov, N.N.Khudyntsev
Figure 3. Coefficient of resistivity ρc (vertical axis) as a function of ka (a) and
of 1− b/a (b) (horizontal axis): Va = 2, ka = 2.45.
The coefficient ρc as a function of ka and 1 − b/a which was calculated using
formula (16) is shown in figures 3a,b. Besides the simplicity of calculation we put
Va = Vb. The coordinate dependence for the resistance coefficient is not present in
the homogeneous chain model. Asimptotically, the resistance coefficient increases
near the edges of the zones as a function of the parameter 1− b/a. Note that at the
determined values of V and b/a, the decrease of the resistance coefficient up to zero
could be observed which corresponds to the regime of zond-particle pure tunnelling
(without dissipation). The mentioned properties of the contact can be considered as
general properties for any one-dimensional structure having an arbitrary superlattice
symmetry.
4. Spectrum of the superlattice having a translational symme-
try
As it was stressed in chapter 1, the problem of determining the spectrum of
the superlattice having a translational symmetry reduces to the similar problem of
a regular chain with the change of the chain periods to the superlattice quasi-one
scale. Within the framework of the transfer-matrix formalism, we need to calculate
the trace of the matrix (11). Finding the solutions of the equation
∣∣∣T̂c − ΛÎ
∣∣∣ = 0, (17)
where Λ is the eigenvalue of the matrix T̂c, Î is the unity matrix, in accordance with
(7) the expression for superlattice spectrum could be expressed in the following form:
Es =
Q2
s
2
, (18)
cosQsa =
(Sb+ − Sa−) (Sa+ − Sb−)G+ + (Sb+ − Sa+) (Sb− − Sa−)G−
(Sa+ − Sa−) (Sb+ − Sb−)
,
where G± = cos q (Naa±Nbb) .
180
Tamm resonances and minibands
Figure 4. The energy spectrum Es of the su-
perlattice as a function of ka.
The respective pictures for the
spectrum (obtained by calculations
using the expression (18)) are shown
in figure 4. The complex character
of (18) is demonstrated at the de-
termined values of the potential and
the lattice period parameters. Tak-
ing into account the correlation be-
tween separate interfaces, the multi-
plication of resonance factors of the
kind (sin qa sin qb)−1 occurs. This ef-
fect can be interpreted as the ero-
sion of splitting Tamm resonances
(interface states) in some minizones.
In the case of disturbance of translational symmetry of the superlattice by the
“point-like” defects, the change of zone structure and Tamm minizones would be
provided. Thus the considered model gives the exact solution to the problem of cal-
culating the spectrum and introduced transport characteristics of one-dimensional
superlattices (including contacts). In spite of their model character, the obtained
results should be taken into account for describing the real physical objects having
the structure similar to atomic chains and superlattices [10–12].
References
1. Esaki L. (ed.) Highlights in Condensed Matter Physics and Future Prospects. //
NATO ASI Series B, 1991, vol. 285, p. 1–390.
2. Levitov L.S., Riner G. Icosahedral symmetry of quasicrystals. // Letters to JETP,
1988, vol. 47, No. 12, p. 658–667 (in Russian).
3. Tikhodeev S.G. Tamm minizones in the theory of superlattices. // JETP, 1991, vol. 99,
No. 6, p. 1871–1880 (in Russian).
4. Demkov Yu.N., Ostrovsky V.N. Method of Zero-radius Ranged Potentials in Physics.
Leningrad, Pub. Leningrad State University, 1975 (in Russian).
5. Kelly M.J., Weisbush C. (ed.) The Physics and Fabrication of Microstructures and
Microdevices. Berlin, Spr. Verlag, 1986.
6. Ziman J.M. Models of Disorder. The Theoretical Physics of Homogeneously Disordered
Systems. Cambrige, Cambrige University Press, 1979.
7. Tamm I.Ye. About the question of surface resonances in the theory of solid bodies
zone structure. // JETP, 1933, vol. 3, p. 34–49 (in Russian).
8. Lifshits I.L., Kirpichenkov V.Ya. About the tunnel effects in the disordered systems.
– In: Physics of Real Crystals and Disordered Systems. Moscow, Nauka Pub., 1987,
p. 272–300 (in Russian).
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erties of atomic chains. Preprint of the Institute for Theoretical Physics, ITP–91–95,
Kyiv, 1991.
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O.I.Gerasimov, N.N.Khudyntsev
10. Gerasimov O.I., Khudyntsev N.N. Tamm’s resonances and minibands within the model
one-dimensional potential. – In: Thes. of Workshop on Cond.Matt.Phys., Lviv, 1998,
p. 69.
11. Gerasimov O.I., Alexeev A.E., Idomsky V.M., Khudyntsev N.N. Conductance prop-
erties of dry granular systems within restricted geometry. – In: Thes. of XX IUPAP
International conference on statistical physics, Paris, 1998, p. 13.
12. Gerasimov O.I., Alexeev A.E., Idomsky V.M., Khudyntsev N.N. Conductance prop-
erties of dry granular systems within restricted geometry. – In: Thes. of NATO ASI,
Leiden, 1998.
Таммівські резонанси та мінізони для модельного
одновимірного потенціалу
Герасимов О.І., Худинцев М.М.
Одеський державний гідрометеорологічний інститут,
270011 Одеса, вул. Львівська, 100
Отримано 5 червня 1998 р., в остаточному вигляді – 26 січня
2000 р.
У роботі за допомогою формалізму трансфер-матриці розв’язана
проблема обчислення спектра регулярної супергратки та запропо-
новано досить простий метод опису контакту в її структурі. Показа-
но, що існування одного контакту призводить до виникнення відо-
кремлених парних резонансів у спектрі ланцюжка, а також проде-
монстровано розмиття таких резонансів у відповідні мінізони після
поєднання окремих регулярних сегментів у супергратку. Означені
резонанси та мінізони є таммівськими та можуть виступати зручними
елементами опису поверхневих станів та зонної структури реальних
низьковимірних систем.
Ключові слова: контакт, супергратка, мінізона, спектр
PACS: 68.65.+G, 73.90Dx
182
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