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22
Waveguide bands for a system of macromolecules
V. M. Adamyan1, I. V. Blinova2, A.I. Popov2, I.Yu. Popov2
1 Odessa I.I. Mechnikov National University, Dvoryanskaya str., 2, Odessa, 65082, Ukraine 2 ITMO University, Kronverkskiy, 49, Saint Petersburg, 197101, Russia
PACS 72.80.Le; 03.65 Nk DOI 10.17586/2220-8054-2015-6-5-611-617
A system of parallel chain-type macromolecules (linear polymers) is considered. The spectrum of an electron in such a system is described. Waveguide bands are shown to be present, ensuring conductivity. Consideration is undertaken within the framework of a zero-range potentials model based on the theory of self-adjoint extensions of symmetric operators. Possible applications also discussed.
Keywords: spectrum, linear polymer, zero-range potential.
Received: 15 September 2015
Revised: 25 September 2015
1. Introduction
This paper deals with the theoretical investigation of a nanosystem which can be used for the creation of optically controlled magnetoresistance. The possibilities for this type of control were outlined in [1]. Layered structures having magnetic layers and a spacer filled with conducting polymer (linear molecules) with inserted banana-shaped photosensitive molecules were suggested to be used for this purpose. Linear polymeric molecules should ensure electronic conductivity between the magnetic layers. In the present paper, we investigate the electronic band structure for a system of linear molecules.
Consider an infinite periodic chain of impurity atoms placed along some line (let it be the axis OX) between two nanolayers (magnetic). We use the effective mass approximation, i.e. we take into account the influence of the nanolayers by changing the effective mass of the electron in the spacer (see, e.g., [2]). In chosen coordinate system, the position of n—th atom of the chain (n = 0, ±1, ±2,...) is given by a vector na = (na, 0, 0), where a is the chain period. A simple but less accurate quantum graph model of such systems (see, e.g., [3,4]) is used more often. To compute the energy spectrum of the chain, we use the zero-range potentials model. We mention that the spectrum for chain structures in R3 can be rather unusual (see, e.g., [5]), but in our periodic case it has a defined band structure. As for zero-range potentials, the background for the model is formed by the theory of self-adjoint extensions of symmetric operators (see, e.g., [6,7]). Namely, one starts from a self-adjoint operator - the Laplacian in L2(R3) with the domain W22(R3). Here W22 is the Sobolev space. Let us restrict the operator to the set of smooth functions which vanish at points pn = |r — na|; n = 0, ±1,.... The closure of this restricted operator is a symmetric operator with infinite deficiency indices. To construct a self-adjoint extension, it is more convenient to deal with the corresponding restriction of the adjoint operator. There are several ways to describe extensions, e.g., boundary triplets method ( [8,9], von Neumann formulas ( [10]), Krein resolvent formula ( [11,12]). We will use here a variant of the second
approach which allows one, in the case of semi-boundedness of the Hamiltonian, to present an element from the domain of the adjoint operator in the following form:
^ ——ifco|r—ma|
-(r) = - (*)+ £ Cm ■ 4-n-"T , (1)
4n |r — ma|
m=—oo
where —0 belongs to the domain of the Friedrichs extension of the initial symmetric operator, Cm is some constant, k0 = V^o, A0 is a regular value of the spectral parameter (particularly, one can choose real negative value of A0, Sk0 > 0). To explain the choice of extension, it is convenient to consider the simplest case of a single point-like potential (at r = 0). In this case, one has the symmetric operator with the deficiency indices (1,1). The domain of the adjoint operator (A*) consists of the following elements (compare with (1)):
-—ifco|r|
-(r) = -,(x) + C0——.
4n |r|
To construct a self-adjoint extension, one calculates the boundary form for elements — ,0 from the domain of the adjoint operator:
(A*-,0) - (-, A*0) = -0(0)Co - ¿o(0)Cf.
Evidently, one gets a self-adjoint extension (annihilation of this form) if there is the following relation between —0(0) and Cg :
-00(0) = b Ct, Sb = 0. (2)
Formally, the condition (2) takes the form of a "boundary condition" at the point:
d
lim
| r | —^0
b
|r| tf = 0. (3)
d |r|
One often rewrites condition (3) as a condition for the logarithmic derivative:
1 d
■^m dJl(|r| *) = b. (4)
|r|—o |r| tf d |r|
lim
Pn—0
b
Pntf = 0, (5)
For the case of the periodic chain, one has
_9pi
pn = |r — na| ; n = 0, ±1,... Below we will use this form of the condition.
2. Two chains
Consider the spectral problem for two parallel chains of zero-range potentials in R3. We assume that centers of the chains are shifted by vector g = (gi,g2, 0). The first chain has centers at points na = naei, the second chain - at points na + g = (na + g1) ei + $2e2; n = 0, ±1, ±2,...; e1, e2 are unit vectors of axis X and axis Y, correspondingly. The Green function for the operator with point-like potentials has the form:
G (r, r', E) = ^exp,[is |r — r/|] + 4n |r — r'|
f --11 [is |r — na — g|
|c(i) exP [is |r - na|] + C(2) exP[i | n ir _ na i n lr
|r — na| n |r — na — g|
n=
Coefficients CT, CT are determined for the system given by conditions (5), which gives one the following system after the Fourier transform:
D (k,s)ei1) + Q (k,s)¿2) = Vk (r')
q (k,s)ek1) + d (k,s) ek2) = Vk (r')
where
Hence,
where
tj = E cn')eikna; j = 1, 2,
n
1 exp [is |r — na| + ikna]
Vk (r) = — ^-^na-
n
d (k, s) = ^ exp[is|;n;a+iknal + «s — b
n=0
Q (k s) = V expNnia-g|+ikna]
^ V > / |na-g|
t(1) = 1 i Vk(rQ+XVk(r/-g) | Vk(r/)-XVk(r/-g) ?k 2 1 D(k.s) + I0(k.s)l +
sk 2 1 D(k,s)+|Q(k,s)| D(k,s)-|Q(k,s)|
^(2) = X J Vk^Q+XVk^-g) Vk(r')-xik(r;-g' sk 2 D(k,s)+|Q(k,s)| D(k,s)-|Q(k,s)|
Q (k,s)
X = X (k,s)
|Q (k, s)|
Returning from tk^ to coefficients C^ using formulas:
+n/a
C« = ^ I tki)e-iknadk
2n
-n/a
+n / a
G (r, r',E) = Go (r — r',E) — a J dk ^
one obtains the following expression for the Green function:
+n/a
exp [—ik (n — n') a] nn, D (k,s) + |Q (k,s)| X
—n/a '
X [Go (r — na, E) + xGo (r — na — g, E)] [Go (na — r', E) + xGo (na + g — r', E)]
+n/a
exp [—ik (n — n') a] . n n, D (k,s) — |Q (k,s)|
—n/a '
X [G0 (r — na, E) — xG0 (r — na — g, E)] [G0 (na — r', E) — xG0 (na + g — r', E)] (6) where
—a dk ^^ „ V—;—\ ^ ,, ' s J, x
Go (r, E)
4nr
Expression (6) shows that two localized waveguide bands can appear near the chains. The dispersion law for one band is determined by roots of D (k, s) + |Q (k, s)| of the second band - by D (k, s) — |Q (k, s)|. The corresponding wave functions have the forms:
<£± (k, r) =
—1/2
d
— (D (k,s) ±|Q (k,s)|)
E=£± (k)
eisr
exp (—ikna) [G0 (r — na, E) ± xG0 (r — na — g, E)]
] E=e± (k)
where e± (k) is determined from the equations:
D k
12^E±\ )
±
Q k,
2^E±
h2
0
(7)
Consider the particular case for which the distance between the chains g2 is greater than the distance between neighboring centers of one chain. Let us find the approximate expression for function Q (k, s) if s = ix, X is real. Using the formula:
"xA _ 1 y eipAdp = W P2 + A2 ;
one transforms the expression for Q (k,ix) to the following form:
1
Q (k,iX) = - Y] e
n=
ikna
eiÇxdC
C2 + (na — gi)2 +
1 /e
— oo
£
e
ikna
^co C2 + (na — gi)2 + gi
dC
If 0 ^ ka < n then:
ikna
n 2aa
cth
^ a2 + (na — P)2
nil r rn
— (P — ia) — i exp [ik (P + ia)] — cth — (P — ia) — i exp [ik (P — ia)] aa
for a > 0. However, for a ^ 0:
ikna
n exp (ikp)
for a ^ 0.
Hence,
Q (k,ix)
aa
3ikgi
^ 02 + (na — P)2 exp (—ka) — exp (ka) exp
— — (a + P) a
x/C^+gf
dC
oikgi r pinX92
?-kg2\/ 1+n2 _ e-2ngi/ae(k-2n/a)g2\J 1+n2
dn
(8)
The last integral in (8) can be transformed to the integral over the curve which comprises the half-axis (i,iœ) in upper half-plane. Thus,
Q (k,ix)
2eikgi
e
Sxg2
a J v^2—1 i
cos (kg2) v^2—I — e-2ngi/a cos ( ^ — M g^v^2—T
a
a
g2
^ { Vx + T^^ + ^fe2 — ^Jx + Vx2 +2(2n/a — k)2 (9)
Vx2 + 2k2 Vx2 + 2 (2n/a — k)2
Substituting (9) into (7) one determines the value of the waveguide band splitting caused by the existence of the second chain. Taking into account the smallness of |Q (, ix)| for sufficiently remote chains, one finds that the level e0 (k) of single chain is split in the following manner:
(k) = ,0 (k) ± ^rf^2!^)!
(d + cos (ka))2 — 1
Q(k,-V2^ ko (k)|
A gap in the waveguide band can appear if the top of the band (k), i.e. (n/a) is greater than the bottom (0) of the band (k), i.e. if e_ (n/a) < (0) .
3. The case of many chains
We now consider the system of M — 1 (M = 2, 3,...) parallel identical chains. Let the position of centers be given by vectors nax + ma2, n,m— integer. In the framework of the zero-range potential approach, the wave function of the electron is ^ (r):
r d 1
lim ---b Pnm^ = 0, (10)
|_ dpnm J
where pnm = |r — nax — ma21. We will seek the perturbed Green function G (r, r',E) for SE > 0 in the following form:
G (r, r', E) = G0 (r — r', E) + 4n ^ CnmG0 (r — nai — ma2, E),
n,m
where G0(r,E) is the Green function for free space. Conditions (10) gives one a system for coefficients Cnm:
(■ h\n , V^ n exp [is |(n — n') ai + (m — m') 4|] _ / a
(is — b) Cnm + / Cn'm/ 1 / , / <\ 1 — ^nm (r ) ,
, , |(n — n') ai + (m — m') a21
n' =n,m'=m
where
, ^ 1 exp [is |r' — na1 — ma2|]
^nm (r )
4n |r' — nai — ma2| The prime sign near the sum means that it is sufficient to satisfy only one from the conditions n' = n or m = m!. ,,To simplify formulas for Cnm, we assume that |Cn,m±p| ^ |Cn,m±1| for p > 1. Then, after the Fourier transform along the chain, one obtains:
D (k, s) tm (k) + D1 (k, s) [tm+1 (k) + tm-1 (k)] = ^m (k, r') , (11)
where
tm (k) = Cnm exp (ikna)
n
1d
D (k, s) = — ln d
a cos (a1s) - cos (a2k)
exp [is |nai + a2| + ikna1] , .
D1 (k,s)= >. i . i , (12) ' |nai + a21
n=-œ
nm (k, r') = -^ Pnm (r') exp (ikna).
n
The dispersion equation is obtained from the solvability condition for the following homogeneous difference equation:
Dim + Di (£m+1 + £m-l) = 0. (13)
The solution of (13) has the form:
im (k) = Si (k) cos (k2ma2) + S2 (k) sin (k2ma2).
The range for k2 and value of S1 (for M — 1 chains) are determined from the boundary conditions Cn0 = CnM = 0. Consequently, £o (k) = iM (k) = 0, S1 (k) = 0 and
k2 = n!/ (Ma2); ! = 1, 2,..., M — 1.
If one inserts the solution:
im (k) = S (k) sin (im!/M), into (11), the dispersion equation is obtained:
D (k,s) + 2D1 (k, s) cos (n!/M) = 0. (14)
Waveguide bands are given by the solution s = ix, X > 0, of (14). Taking into account (12), one reduces (14) to the following form:
cosh (a1x) = cos (a1k) + d1 (x,!), (15)
where d1 (x,!) = dexp [2a1D1 cos (n!) /M], or, correspondingly, to the form:
2
h2
E <k' = — 2/ia^ ln
d1 (x,!) + cos (a1k) + ^[d1 (x,!) + cos (a1k)]2 — 1
where E1 (k) = — ^"X". Note that for small k,x, one can replace the sum for D1 from (12) by the integral:
D f exp[—x|vai + a21 + ikva1]dv = 1 / /—r+—A, J |vai + a2| a1 V /
—oo
where K0 (z) is the Macdonald function of zero order. Due to the integer parameter !, equation (15) determines, generally speaking, M — 1 waveguide bands. The conductivity is determined by such bands which are empty or partially filled, i.e. contain the Fermi level EF. If each atom of the chain has one "free" electron, then the Fermi level EF is determined from the equation:
n (M — 1) ^-1 [ dk
M -1 Ep
7_1 v
2a1 dE
(16)
1=1 Eo
Taking into account (15), one rewrites (16) in the form:
M— 1
n - 1 £ / f (xa1) — d1 (x,!) = dx, (17)
2a1 M — 1
1 1=1 Xo
)/1 — [ch (xa1) — d1]2
2
where d1 (x, l) = ^^Xi'^. If one ignores the difference between d1 (x, l) and d, then equation
(17) gives one the following value of the Fermi level:
e0 = —ln2 Id + Vd2—!
which corresponds to the case of single chain.
Acknowledgments
This work was partially financially supported by the Government of the Russian
Federation (grant 074-U01), by Ministry of Science and Education of the Russian Federation
(GOSZADANIE 2014/190, Projects No 14.Z50.31.0031 and No. 1.754.2014/K), by grant
MK-5001.2015.1 of the President of the Russian Federation, and by grant of St. Petersburg.
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