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Thermodynamic and kinetic properties of phonons in cylindrical quantum dots
ARTICLE IN PRESS
Physica E 25 (2005) 479–491 www.elsevier.com/locate/physe
Thermodynamic and kinetic properties of phonons in cylindrical quantum dots Vjekoslav Sajferta,, Jovan P. SˇetrajWic´b, Stevo Jac´imovskib, Bratislav Togic´b a
Technical Faculty M. Pupin Zrenjanin, University of Novi Sad, Djure Djakovica bb, Zrenjanin 23000, Serbia and Montenegro b Department of Physics, Faculty of Sciences, University of Novi Sad, Serbia and Montenegro Received 25 April 2004; received in revised form 18 June 2004; accepted 31 July 2004 Available online 11 September 2004
Abstract The Green’s function technique, suitable for analyses of spatially deformed structures, is developed in this paper and applied to phonon system. The thermodynamic and kinetic phonon properties of cylindrical quantum dots are analysed using a developed method. As a consequence of the applied new method the configurational dependence of diffusion coefficient and dot’s density were included into calculations. Maximum of diffusion and minimum of density is located in central part of the cylindrical quantum dot. All thermodynamic and kinetic characteristics of quantum dot are exponentially small at low temperatures. The low phonons specific heat as well as the low thermal conductivity lead to conclusion that in cylindrical quantum dots exist more convenient conditions for appearance of electron superconductivity. r 2004 Elsevier B.V. All rights reserved. PACS: 73.21.La; 73.63.Kv; 8.67.Hc Keywords: Green’s function techniques; Diffusion; Spatial dependence; Broken symmetry; Nonhomogeneity; Crystal density
1. Introduction Analyses of nanostructures are very popular in several last years (see [1–6]). It is caused by the advance of technology of syntheses of nanostructures and also with wide possibilities of application of nanotubes in superconductive devices. The behaviour of some biosystems can be explained by some nanotube properties (see [1]). Corresponding author. Tel./fax: +38-122-55-4143.
E-mail address: [email protected] (V. Sajfert).
Investigations of phonon subsystem is necessary since phonons are always present in every structure and influence to behaviour of electrons, optical excitations, spin waves, etc. For that reason this analyses is devoted to phonon characteristics since they influence to the characteristics of other subsystems in the structure such as electrons subsystem, subsystem of ferroelectric excitations, subsystem of spin waves, etc. We shall consider the system of N linear molecular chains connecting molecules lying at the tops of M regular polygons whose planes are
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.07.008
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perpendicular to the chains (see Fig. 1). Molecules in chains are labelled with index m while molecules in discs (polygons) are labeled with index n. The numbers M and N are of the order 10. It will be assumed that phonons in chains as well as the phonons in discs are of longitudinally type, i.e., molecular displacements ðaÞ in chains are taken in chain direction, only, while molecular displacements in discs ðbÞ directed along lies connecting two nearest molecules of the polygon. Due to disc assumption the phonons in chains (in further axial phonons) and phonons in discs (in further disc phonons) are mutually independent. Consequently the phonon Hamiltonian of considered quantum dot can be written as follows: H ¼ H a þ H b; (1.1)
Notations in formula (1.2) and (1.3) are the following: a and b are molecular displacements, pa and pb are the corresponding momenta; M is the mass of the molecule and Ca and Cb are the stretch Hook’s constants. It should be pointed out that the symmetry break takes place in the direction of molecular chains, only. The simplest boundary conditions, consisting absence of layers labelled with m ¼ 1 and m ¼ M þ 1 will be assumed. The more real boundary conditions include the changes of boundary Hook’s constants with respect to those of internal layers. For the molecules lying in the discs the symmetry is not disturbed, but obvious cyclicity condition is valid: F n ¼ F Nþ1þn ;
(1.4)
where M X p2ma 1 þ Ca Ha ¼ 2M 4 m¼0
M X
½ðam amþ1 Þ2 þ ðam am1 Þ2
ð1:2Þ
m¼0
is Hamiltonian of mechanical oscillations along axis and N p2 X 1 nb þ Cb Hb ¼ 4 2M n¼0
N X
½ðbn bnþ1 Þ2 þ ðbn bn1 Þ2
where F denotes arbitrary physical characteristics of the system of disc molecules. In connection with this the reader can consult Refs. [7–9]. Thermodynamic and kinetic properties of cylindrical quantum dots will be analysed with the help of two-time temperature Green’s functions. This method is the most effective since, besides finding of energies of elementary excitations, it gives possibility for the calculation of mean values of operator products.
ð1:3Þ
n¼0
is Hamiltonian of mechanical oscillations in discs. b
a
2. Green’s functions of axial phonons Two types of Green functions will be found for axial phonons: one is of the type displacement–displacement, while another is of the type momentum–momentum. We shall look for Green’s function of the type displacement–displacement, first Anm ðtÞ hhan ðtÞjam ð0Þii ¼ yðtÞh½an ðtÞ; am ð0Þi:
α β Fig. 1. Schematic presentation of a cylindrical quantum dot.
ð2:1Þ
Differentiating this formula with respect to time we obtain d Anm ðtÞ ¼ yðtÞh½_an ðtÞ; am ð0Þi; dt
(2.2)
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where yðtÞ ¼
1; 0;
t40; to0;
(2.3)
is Heaviside’s step function. Taking into account equation of motion 1 p a_ n ¼ ½an ; H a ¼ ma i_ M we reduce (2.2) onto
(2.4)
d hhp ðtÞjam ð0Þii dt na ¼ i_dnm dðtÞC a ½hhanþ1 ðtÞjam ð0Þii þ hhan1 ðtÞjam ð0Þii hh2an ðtÞjam ð0Þii:
1pnpM 1; n ¼ 0;
AM1;m þ rAM;m ¼ FM;m ; n ¼ M:
ð2:11Þ (2.12) (2.13)
The solution of system (2.11)–(2.13) will be looked for in the form Mþ1 X
amm sinðn þ 1Þjm :
(2.14)
Introducing Eq. (2.14) into Eq. (2.11) we obtain ð2:6Þ
relation (2.7) becomes
Mþ1 X
ð2cos jm þ rÞamm sinðn þ 1Þjm ¼ Fn;m :
(2.15)
m¼1
After substitution A1;m and A0;m in Eq. (2.12) we require to left-hand side of Eq. (2.12) be analogous to Eq. (2.15). Since sin 2jm ¼ 2cos jm ; sin jm Eq. (2.12) goes over to the form which is analogous to Eq. (2.15), i.e. Mþ1 X
ð2cos jm þ rÞamm sin jm ¼ Fn;m :
(2.16)
m¼1
ð2:9Þ
After substitution Eq. (2.14) into Eq. (2.13) we reduce Eq. (2.13) to the form Mþ1 X
and M 2 o 2: Ca
¼ Fn;m ;
m¼1
d2 i_ Anm ðtÞ ¼ dnm dðtÞ 2 M dt Ca ½Anþ1;m þ An1;m 2An;m : ð2:7Þ þ M After time-frequency Fourier-transformation Z þ1 do eiot Anm ðoÞ; Anm ðtÞ ¼ 1 Z 1 þ1 do eiot ð2:8Þ dðtÞ ¼ 2p 1
r¼
Anþ1;m þ An1;m þ rAn;m
An;m ¼
Differentiating (2.5) with respect to time and substituting (2.6) we obtain the equation
Anþ1;m ðoÞ þ An1;m ðoÞ þ rAn;m ðoÞ ¼ Fn;m ; i_ 1 dn;m Fn;m ¼ 2p C a
layers 0 and M change Hook’s constants, but we shall assume that their changes are negligible. Taking into account all said before, becomes clear that Eq. (2.13) goes over to three systems of difference equations:
A1;m þ rA0;m ¼ F0;m ;
d 1 Anm ðtÞ ¼ hhpna ðtÞjam ð0Þii: (2.5) dt M Now we have to look for Green’s function hhpna ðtÞjam ð0Þii: Analogously to the upper calculation we obtain
481
ð2cos jm þ rÞamm sinðM þ 1Þjm ¼ FM;m :
m¼1
(2.10)
As it is seen for finding Green’s functions of the type displacement–displacement the system of M+1—difference-equation is obtained. Since the layers m ¼ 1 and M þ 1 are absent, it is clear that A1;m ðoÞ ¼ ANþ1;m ðoÞ ¼ 0: In boundary
(2.17) If the condition sin Mjm ¼ 2cos jm sinðM þ 1Þjm is satisfied.
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From (2.17) it follows that the parameter jm is playing role of wave vector is given by
we obtain from (2.20)
pm ; jm ¼ M þ2
Gm ðoÞ ¼ m ¼ 1; 2; 3; . . . ; M þ 1:
(2.18)
Index m characterizing wave vector in the direction of the chain cannot take values 0 and M þ 2 because it leads to sinðM þ 1Þjm ¼ 0: This is, as we have seen, forbidden. From the procedure quoted, it is clear that relation (2.15) Mþ1 X
ð2cos jm þ rÞamm sinðn þ 1Þjm ¼ Fn;m
m¼1
for jm ¼ pm=M þ 2 stands for every m, i.e., for m ¼ 0; 1; 2; 3; . . . ; M: Before defining the coefficient amm we shall represent Kronecker symbol in terms of sine functions. It can be easily shown that: Mþ1 X
sin2 ðm þ 1Þjm ¼
m¼1 Mþ1 X
M þ2 ; 2
sinðn þ 1Þjm sinðm þ 1Þjm ¼ 0;
nam:
m¼1
We obtain the following representation of Kronecker symbol: dn;m ¼
X 2 Mþ1 sinðn þ 1Þjm sinðm þ 1Þjm : M þ 2 m¼1 (2.19)
Taking into account the last result we can write (2.15) in the following form: Mþ1 X
ð2cos jm þ rÞamm sinðn þ 1Þjm
m¼1
¼
2 i_ 1 M þ 2 2p C a Mþ1 X sinðn þ 1Þjm sinðm þ 1Þjm :
ð2:20Þ
m¼1
Taking amm ðoÞ ¼
2 Gm ðoÞ sinðm þ 1Þjm M þ2
(2.21)
i_ 1 1 2p C a 2cos jm þ r
(2.22)
from which after substitution parameter r from (2.10) we obtain Gm ðoÞ ¼
1 i_ 1 2 M 2p o o2m
or, in terms of simple fractions i_ 1 1 1 Gm ðoÞ ¼ : 2p 2Mom o om o þ om Notations used in (2.34) are rffiffiffiffiffiffi Ca 2 jm 2 2 ; Oa ¼ om ¼ 4Oa sin ; 2 M pm ; m ¼ 1; 2; 3; . . . ; M þ 1: jm ¼ M þ2
(2.23)
(2.24)
ð2:25Þ
Formula (2.25) gives the dispersion law of phonons propagating in linear molecular chains. It should be pointed out that this dispersion in infinite (ideal molecular chain) is of the same form. The unique difference appears in the expression for dimensionless wave vector, which in infinite chain takes values 2pm M M jm ¼ ; m 2 þ 1; : (2.26) M 2 2 Comparing Eq. (2.35) to Eq. (2.36) the very important conclusion can be reduced: the minimum frequency in short molecular chain p (2.27) ðom Þmin ¼ 2Oa sin M þ2 is different from zero. In the infinite chain the minimal frequency is equal to zero. Since we deal with acoustic phonons, the above conclusion practically means that acoustic phonons in short chain possess energy gap, i.e., for excitation of central phonons in short chain the minimal, different form zero, energy must be inserted. This fact has positive repercussions to realization of superconductive effect in short chains (Fig. 2). In the infinite chain the phonon gap does not exist (see Fig. 3). pm From the value of 2sin Mþ2 quoted in the first row of the Table 1 it can be calculated that the
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Table 1 m
pm Mþ2
pm 2sin Mþ2
1 2 3 4
0.2856 0.5712 0.8568 1.1424
0.5634 1.0813 1.5115 2.2848
Using the relation Fig. 2. The histogram of axial phonon frequencies for four layer chain in accordance with Table 1.
1 1 ^ ¼P ipdðo o0 Þ; (2.29) o o0 id o o0 ^ denotes main value of integral and where P formula (2.34) we obtain _ dðo om Þ dðo þ om Þ _o I G ðoÞ ¼ : (2.30) 2Mom e_o y 1 ey 1 With help of spectral intensity one can find correlation functions in momentum space (the momenta p are characterizing by the integer m; mspace) as: Z þ1 do eiot I G ðoÞ: (2.31) hað0ÞaðtÞm i ¼ 1
Substituting Eq. (2.38) into Eq. (2.39) we obtain " # _ eitom eitom hað0ÞaðtÞm i ¼ : (2.32) 2Mom e_oy m 1 e_oy m 1 Fig. 3. Relative frequencies of phonons in infinite chain.
minimal energy for exciting of acoustical phonons is E min 102 eV what corresponds to temperature T 100 K: It means that at temperatures up to 100 K the short chain is quite frozen so that electrons propagate without friction, i.e. superconductively. This estimate is made for Debye’s energy of 200kB. After the discussion of short chain phonons dynamical properties we can go over to analysis of mean values characterizing these phonons. Spectral intensity of the function G m ðoÞ is I G ðoÞ ¼ d ! þ0
G m ðo þ idÞ G m ðo idÞ _o
ey 1 Y ¼ kB T:
Configurational correlation function can be obtained by operator application of Kronecker’s symbol to correlation function in momentum space (m-space). In this way, using formula (2.19) we obtain: hað0ÞaðtÞi ¼
X 2 Mþ1 hað0ÞaðtÞim M þ 2 m¼1 sinðn þ 1Þjm sinðm þ 1Þj
and this formula in accordance with (2.32) goes over to " # X 1
2 _ Mþ1 eitom eitom að0ÞaðtÞ ¼ M þ 2 2M m¼1 om e_oy m 1 e_oy m 1 sinðn þ 1Þjm sinðm þ 1Þjm :
; ð2:28Þ
ð2:33Þ
ð2:34Þ
For further analyses it is necessary to find momentum correlation function. To achieve this
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we have to find momentum Green’s function defined as follows:
The Green’s functions of the type displacement–displacement will be calculated first
Pnm ðtÞ hhpna ðtÞjpma ð0Þii
Bnm ðtÞ hhbn ðtÞjbm ð0Þii ¼ yðtÞh½bn ðtÞ; bm ð0Þi:
¼ yðtÞh½pna ðtÞ; pma ð0Þi
ð2:35Þ
Using the calculation procedure quite analogous to that which was used in the case of the Green’s function Anm we obtain i_ 1 1 1 G m ðoÞ ¼ ; (2.36) 2p 2Mom o om o þ om where the spectral intensity of this function is _C a dðo om Þ dðo þ om Þ I G ðoÞ ¼ o : (2.37) m _o _o 2O2a ey 1 ey 1 Using Eq. (2.37) we obtain in m-space hpma ð0Þpna ðtÞi ¼
"
#
X 1 _C a Mþ1 eitom eitom om _om _om 2 M þ 2 Oa m¼1 e y 1 e y 1 sinðn þ 1Þj sinðm þ 1Þj ð2:38Þ
from which the expression for the spatial momentum correlation function is the following: hpma ð0Þpna ðtÞi
" # X 1 _C a Mþ1 eitom eitom ¼ om _om _om M þ 2 O2a m¼1 e y 1 e y 1 sinðn þ 1Þj sinðm þ 1Þj ð2:39Þ
The relation obtained will be used for calculation of some thermodynamic and kinetic characteristics for cylindrical quantum dots, or more precise, for calculation of the contributions of axial phonons to thermodynamic and kinetic properties.
ð3:1Þ
Using the same procedure as in the case of Green’s function displacement–displacement for axial phonons we obtain: d 1 Bnm ðtÞ ¼ hhpmb ðtÞjbm ð0Þii: dt M
(3.2)
Since the calculation procedure is quite similar to that of the previous section the stages of calculation will not be quoted. We shall give the final results, only. The final result is: d hhp ðtÞjbm ð0Þii dt nb ¼ i_dnm dðtÞC a ½hhbnþ1 ðtÞjbm ð0Þii þ hhbn1 ðtÞjbm ð0Þii 2hhbn ðtÞjbm ð0Þii:
ð3:3Þ
Differentiating (3.2) with respect to time and substituting (3.3) we obtain: d2 i_ Bnm ðtÞ ¼ dnm dðtÞ M dt2 Cb ½Bnþ1;m þ Bn1;m 2Bn;m : ð3:4Þ þ M After Fourier-transformation time-frequency Z
þ1
do eiot Bnm ðoÞ
Bnm ðtÞ ¼
(3.5)
1
relation (3.4) becomes Bnþ1;m ðoÞ þ Bn1;m ðoÞ þ rBn;m ðoÞ ¼ f n;m ;
(3.6)
where 3. Green’s function of disc phonons Hamiltonian of subsystem disc phonons is given by formula (1.3). Phonon characteristics in discs will be examined by Green’s functions of the type displacement–displacement and momentum–momentum.
f n;m ¼ r¼
i_ 1 dn;m ; 2p C b
M 2 o 2: Cb
ð3:7Þ
At this stage the analogy with previous case is finished. System of difference equations (3.6) can
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be written in the following way:
Values of parameter jn will be determined form cyclicity condition:
Bnþ1;m þ Bn1;m þ rBn;m ¼ f n;m ; 1pnpN 1; A1;m þ B1;m þ rB0;m ¼ f 0;m ;
ð3:8Þ n ¼ 0;
BNþ1;m þ BN1;m þ rBN;m ¼ f M;m ;
(3.9)
B1;m ¼ BN;m a0;
(3.11)
BNþ1;m ¼ B0;m a0:
(3.12)
It enables us to look for the solution of system (3.8)–(3.10) in the form bn;m einjn :
(3.13)
n¼1
Introducing Eq. (3.13) into Eq. (3.8) we obtain N þ1 X
ð2cos jn þ rÞbnm einjn ¼ f n;m ;
n¼1
n ¼ 1; 2; 3; . . . ; N 1:
ð3:14Þ
After substitution Eq. (3.13) into Eq. (3.9) it follows: N þ1 X
n ¼ 0:
(3.15)
n¼1
The condition Eq. (3.19) is fulfilled if eiðNþ1Þjn ¼ 1; i.e., if 2pn ; n ¼ 0; 1; 2; 3; . . . ; N þ 1: (3.20) N þ1 It can be easily shown that Kronecker’s symbol is given by jn ¼
dn;m ¼
þ1 X 1 N eiðnmÞjn ; N þ 1 n¼1
2pn : N þ1 If we take that
jn ¼
1 g ðoÞeimjn (3.22) N þ1 n then, with respect to Eq. (3.7), the system of equations (3.17) goes over to þ1 X 1 N g ðoÞeiðnmÞjn N þ 1 n¼1 n
X 1 i_ 1 Nþ1 eiðnmÞjn N þ 1 2p C b n¼1
from which it follows gn ðoÞ ¼
ð2cos jn þ rÞbnm eiNjn ¼ f N;m ;
n ¼ N:
(3.16)
n¼1
As it is seen Eqs. (3.14)–(3.16) have absolutely identical form and reduce to N þ1 X
i_ 1 1 : 2p C b 2cos jn þ r
(3.23)
Substituting, r from Eq. (3.7) into Eq. (3.23) we obtain gn ðoÞ ¼
1 i_ 1 ; 2 M 2p o o2n
(3.24)
jn 2
(3.25)
where
ð2cos jn þ rÞbnm einjn ¼ f n;m ;
n¼1
n ¼ 0; 1; 2; 3; . . . ; N:
ð3:21Þ
bn;m ¼
Including Eq. (3.13) into Eq. (3.10) we obtain N þ1 X
(3.19)
n¼1
¼ ð2cos jn þ rÞbnm ei0jn ¼ f 0;m ;
(3.18)
which, with accordance with Eq. (3.13), becomes ½1 eiðNþ1Þjn bnm einjn ¼ 0:
In the case of axial phonons the Green’s functions with indices –1 and M+1 were equal to zero because the layers with indices –1 and M+1 do not exist. In disc cyclic symmetry (1.4) takes place and consequently, the following is valid:
Bn;m ¼
Bn;m ¼ BNþ1þn;m
N þ1 X
n ¼ N: (3.10)
N þ1 X
485
ð3:17Þ
o2n ¼ 4O2b sin2
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Table 2
and rffiffiffiffiffiffi Cb Ob ¼ ; M
2pn : jn ¼ N þ1
(3.26)
In Fig. 4 the dispersion law of disc phonons will be quoted in histogram form (Table 2). As it can be seen the minimal frequency of disc phonons is equal to zero. Consequently the exciting of disc phonons does not require inserting the external energy (see Table 4). In this sense phonons in the discs behave as phonons in infinite linear chain. It is clear from the fact that the histogram in Fig. 4 obtained from Table 2 is the discrete equivalent of continual curve given in Fig. 3. Now we can go over to finding the mean values of products of operators corresponding to disc phonons. The spectral intensity of the function gn ðoÞ is given by _ dðo on Þ dðo þ on Þ _o I g ðoÞ ¼ : (3.27) 2Mon e_o y 1 ey 1 Using this formula we can find the correlation function in n-space " # _ eiton eiton : (3.28) hbð0ÞbðtÞin ¼ 2Mom e_oy n 1 e_oy n 1
n
2pn Nþ1
2pn 2 sin Nþ1
0 1 2 3
0 0.62832 1.25664 1.88495
0 0.61803 1.17557 1.61803
Finally, the configurational correlation function is given by the formula hbm ð0Þbn ðtÞi ¼ jn ¼
þ1 X 1 N hb ð0Þbn ðtÞin eiðnmÞjn N þ 1 n¼1 m
2pn N þ1
ð3:29Þ
which, with respect to (3.29) becomes hbm ð0Þbn ðtÞi 1 _ ¼ N þ 1 2M " # Nþ1 X 1 eiton eiton _on eiðnmÞjn ; _on o e y 1 n¼1 n e y 1 jn ¼
2pn : N þ1
ð3:30Þ
Now we shall calculate Green’s function of the type momentum–momentum Pnm ðtÞ hhpnb ðtÞjpmb ð0Þii ¼ yðtÞh½pnb ðtÞ; pmb ð0Þi:
ð3:31Þ
Using quite similar procedure as in the case of displacement–displacement Green’s function we obtain the following relevant formula: gn ðoÞ ¼
i_ C b 1 ; 2p O2b o2 o2n
(3.32)
where gn ðoÞ is the Fourier transform of the function Pn;m ; i.e.:
Pn;m ¼ Fig. 4. The histogram of axial phonon frequencies in accordance with Table 2.
þ1 X 1 N g ðoÞeinjn ; N þ 1 n¼1 n
n ¼ 0; 1; 2; . . . ; N
ð3:33Þ
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the set of Eqs. (3.35)–(3.37) goes over to gn ðoÞ ¼
i_ o2n 1 M 2 : 2p Ob 2cos jn þ r
4. Thermodynamic properties of cylindrical quantum dot (3.34)
After substitution the parameter r into Eq. (3.34) the last formula becomes gn ðoÞ ¼
i_ C b 1 2 2 2p Ob o o2n
(3.35)
or, after resolving to simple fractions i_ C b 1 1 o gn ðoÞ ¼ n 2p O2b o on o þ on
(3.36)
Using Eq. (3.36) we obtain spectral intensity of the Green’s function g; _C b dðo on Þ dðo þ on Þ _o on (3.37) I g ðoÞ ¼ _o 2O2b ey 1 ey 1 from which it follows: hpb ð0Þpb ðtÞin ¼
"
It should be noticed in the beginning, that thermodynamical properties will be investigated at low temperatures, when e_o=y b1: In this case the second term in correlation functions (2.34) and (3.30) approximately goes over to eiot : This term does not depend on temperature and obviously comes from zero oscillations. As it was said earlier, the contributions of zero oscillations will not be taken into account, and consequently this term will be omitted in further analyses. In the first term of formulae (2.35) and (3.30) Bose’s distribution 1=e_o=y 1 will be substituted with Boltzmann’s one e_o=y : It means that in further the following expressions for configurational correlation functions will be used: hað0ÞaðtÞi ¼
X 1 _o _ 1 Mþ1 e Y itom M M þ 2 m¼1 om sinðn þ 1Þjm sinðm þ 1Þjm ;
#
_C b eiton eiton o : n _on _on 2O2b e y 1 e y 1
487
ð3:38Þ
hbm ð0Þbn ðtÞi ¼
The configurational correlation function is given by
_ 1 2M N þ 1 N þ1 X 1 _oiton þiðnmÞjn e Y : o n¼1 n
ð4:1Þ
ð4:2Þ
The same requirements for momentum correlation functions give
hpmb ð0Þpnb ðtÞi 1 _C b ¼ N þ 1 O2b " # Nþ1 X eiton eiton _on on _on eiðnmÞjn : e y 1 e y 1 n¼1
hpma ð0Þpna ðtÞi ¼
X _C a 1 Mþ1 _o om e Y itom 2 M þ2 Oa m¼1 sinðn þ 1Þj sinðm þ 1Þj
ð3:39Þ Comparing the properties of axial phonons with disc ones we can notice two essential differences. Axial phonons are standing waves while disc phonons are plane waves. The main consequence of noticed difference is that probability current of axial phonons in dot is equal to zero, while the probability current of disc phonons is non/zero quantity. The exciting of axial phonons requires inserting of relatively high energy quant, while disc phonons appear without inserting of external energy.
ð4:3Þ
and hpmb ð0Þpnb ðtÞi ¼
þ1 X _C b 1 N _o on e Y iton þiðnmÞjn : O2b N þ 1 n¼1
ð4:4Þ
Besides mentioned, the following approximations will be used, too: (a) The approximation of small wave vectors on
pOa ; M þ2
om
2pOb : N þ1
(4.5)
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(b) All finite geometrical progression whose quotient contains factor eC=Y will be approximately considered as infinite ones. (c) Mathematical forms of the type ex =ðex 1Þ2 and lnð1 þ ex Þ will be approximately taken equal C to ex ; if x contains factor Y : In accordance with approximations quoted the internal energy of axial phonons in cylindrical quantum dot is given by U a ¼ hH a i ¼
M 1 X 1 hp2 i þ C a 2M m¼0 ma 4
M X ðha2m i þ ha2m1 i þ ha2mþ1 i m¼0
2ham am1 i 2ham am1 iÞ:
ð4:6Þ
Using formula (4.1), taken for t ¼ 0 m and formula (4.3) taken for t ¼ 0; too, we find U a ¼ hH a i
p M þ2 þ 4ðM þ 2Þ p p 1 p sin2 sin2 : 2ðM þ 2Þ 2p M þ2
¼ _Oa e
p_Oa ðMþ2 ÞY
ð4:7Þ
Internal energy of disc phonons is given by: U b ¼ hH b i ¼
M 1 X 1 hp2 i þ C b 2M m¼0 nb 4
M X ðhb2n i þ hb2n1 i þ hb2nþ1 i
CV ¼
@hHi @hHi @Y ¼ ; @T @Y @T
Y ¼ kB T; the following result: pkB _Oa 2 p _Oa p e Mþ2 Y CV ¼ 4ðM þ 2Þ M þ2 Y Mþ2 p 1 p sin2 sin2 þ p 2ðM þ 2Þ 2p M þ2 2 _O 2p 2pkB _Ob b þ eNþ1 Y N þ1 Y p N þ1 2 p þ sin : ð4:12Þ 2ðN þ 1Þ 2p N þ1 As it is seen from the results obtained, the internal energy as well as the specific heat decrease exponentially at low temperatures. It means that cylindrical qunatum dots behave as a rigid body at low temperature, i.e., population of mechanical oscillation is very small. This makes suitable conditions for appearance of high temperature superconductivity. At the end of this section the specific heat of axial phonons will be discussed. The general formula for the specific heat is very clumsy and
c
ð4:8Þ
d
Using (4.2) for t ¼ 0 and (4.4) for t ¼ 0 we obtain 2p_Oa p ðNþ1ÞY U b ¼ hH b i ¼ _Ob e 2ðN þ 1Þ N þ1 2 p sin þ : ð4:9Þ 2p 2ðN þ 1Þ The total internal energy of cylindrical quantum dot is hHi ¼ hH a i þ hH b i:
(4.11)
CV
m¼0
2hbn bn1 i 2hbn bn1 iÞ:
dance with general formula
(4.10)
Specific heat of cylindrical quantum dot can be found by differentiating the internal energy with respect to temperature. So we obtain in accor-
b O
a
Θ
Fig. 5. Full line represents specific heat of nanostructure. The part a b corresponds to extremely small temperatures while part b c corresponds to the interval of the middle temperatures. Dashed line represents specific heat of one-dimensional ideal structure.
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therefore the leading term of the formula will be quoted: 2 p _Oa p_ 1 C V ¼ kB Oa 2 eMþ2 Y : (4.13) M þ2 Y This formula is graphically presented in Fig. 5 (full line) together with the specific heat of an infinite chain (dashed line in Fig. 5). It should be pointed out that, at low temperature, the specific heat of the dot is exponentially small. In the middle temperature range the formula for specific heat of cylindrical quantum nanodot is Y C V ¼ 2kB ðM þ 2Þ : p_Oa
(4.14)
As it is seen specific heat linearly increases with temperature. It should be pointed out that specific heat in bulk structure behaves in the same way as specific heat of nanostructures in middle temperature range but in the interval Y 2 ½0; YM ; where YM YD (YD is Debye’s temperature). Specific heat of disc phonons should not be graphically presented since its behaviour is very similar to the behaviour of axial phonons.
5. The kinetic parameters of cylindrical quantum dots Here we shall look for diffusion tensor of quantum dots and for tensor of thermal conductivity. In connection with this we note in the beginning that the diffusion of axial phonons is independent of diffusion processes in subsystem of the disc phonons. Calculating the diffusion tensor of axial phonons and the diffusion tensor of disc phonons we can conclude that the sum of the two mentioned tensors cannot be considered as a diffusion tensor of the whole cylindrical dot. Eventually, the arithmetical mean of two mentioned diffusion tensors can be (only by definition) considered as the diffusion tensor of the whole cylindrical quantum dot. Heat conductivity is given by the formula l ¼ D C V r;
(5.1)
where D is diffusion tensor and r is density of cylindrical quantum dot. Earlier it was shown that
489
the specific heat of the whole quantum dot is the sum of specific heat of axial phonons and specific heat of the disc phonons. On the other hand the density r is the characteristic of the whole cylindrical quantum dot and cannot be separated into parts corresponding to axial phonons and the one corresponding to disc phonons. We are of the opinion that the introducing of two thermal conductivities in cylindrical quantum dot is the most realistic. These conductivities are la ¼ Da C Va r
(5.2)
and lb ¼ Db C Vb r:
(5.3)
From these formulae it is seen that thermal conductivity for every one of the phonon subsystems depends, through the density, on characteristics of the other subsystem. It means that thermal conductivity la for axial phonons and lb for the disc phonons are not independent. In the previous section, the specific heat of cylindrical quantum dot was found. From formula (4.12) it is quite clear what contributions give one phonon subsystem and what contributions coming from the other one. The density of the ‘‘frozen’’ cylindrical quantum dot is given by m0 r0 ¼ ; (5.4) V0 where V0 ¼
p ðM þ 1Þab2 : p 4 sin2 Nþ1
(5.5)
In Eq. (5.5) the mass of cylindrical quantum dot it is denoted with m0 : With a and b are denoted the ‘‘lattice constants’’ (a is distance between two neighbour discs, while b is the chord length between neighbour molecules in discs). During the heating, the ‘‘lattice constant’’ a gets correction qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi am ¼ ha2m ð0Þi: (5.6) The correction of constant b is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi bn ¼ hb2n ð0Þi:
(5.7)
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Consequently for T40 we have a ! a þ am ;
in the Section 4, we obtain from formula (4.3), (5.8)
b ! b þ bn : It can be easily concluded that the corrected density, up to linear terms in a and b; has the form am bn 2 rðYÞ ¼ r0 1 : (5.9) a b With respect to formulae (4.1) and (4.2), taken for t ¼ 0 and m=n, we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffi p_Oa _ pðn þ 1Þ 2ðMþ2ÞY (5.10) e am ¼ sin M þ 2 pMOa and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p_Ob _ bm ¼ e ðNþ1ÞY : 4pMOb
(5.11)
It is seen, from Eqs. (5.9)–(5.11), that the density depends on temperature and on spatial coordinates. The last is the result of the application of the Green’s functions method, developed here, which is suitable for analysis of spatially deformed structures. The numerical calculations have shown that maximum of decrease of the density can be expected in the middle of the cylindrical quantum dot. At the ends of cylindrical quantum dot density changes are minimal. Now we shall look for the diffusion tensor of axial phonons and diffusion tensor of disc phonons. In Refs. [10,11] the diffusion tensor is defined as follows: Z 1 Dnm ¼ dt et hvm ð0Þvn ðtÞi; 0
! 0þ;
ð5:12Þ
where v are velocities of phonon propagation, i.e., v ¼ Mp : Correlation function of the type velocity/velocity for axial phonons will be found with the help of formula (4.3). Using the approximation quoted
DðaÞ nm ðYÞ ¼
12 _ p_Oa e ðMþ2ÞY i MM þ 2 sinðn þ 1Þj sinðm þ 1Þj:
ð5:13Þ
As it is seen, the diffusion tensor of axial phonons depends on temperature and on spatial coordinates. For three-layer structure the following proportionality for diffusion tensor is valid: 0 1 0:500 0:707 0:500 B C aÞ Dðnm @ 0:707 1:000 0:707 A: (5.14) 0:500
0:707
0:500
It is clear from this formula that diffusion is maximal in the middle of the structure and decreases in the direction of boundaries. The diffusion of disc phonons can be calculated using formula (4.4). After long but elementary calculations, in which approximations quoted in Section 4 are used, we obtain DðbÞ nm ðYÞ ¼
1 _ i 2MðN þ 1Þ 2p_O
e
b 2p ðNþ1ÞY eiðnmÞNþ1 :
ð5:15Þ
It should be pointed out that in practical calculations diffusion tensors are real quantities. Consequently, instead of complex values (5.13) and (5.15) their absolute values have to be used. Those absolute values are 2 _ ðaÞ p_Oa e ðMþ2ÞY D~ nm ðYÞ ¼ DðaÞ nm ðYÞ ¼ MM þ 2 p sinðm þ 1Þ sinðn þ 1Þ M þ2 p : ð5:16Þ Mþ2 and ðbÞ D~ nm ðYÞ ¼ DðbÞ nm ðYÞ 2p_Ob _ e ðNþ1ÞY : ¼ ð5:17Þ 2MðN þ 1Þ The most important conclusion of the analyses of this section is the fact that in the direction of broken symmetry, diffusion is characterized by tensor depending on temperature and on spatial coordinates. For disc phonons (the symmetry is not disturbed in discs) the diffusion tensor is losing
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tensor characteristics and goes over to the function depending on the temperature, only. The thermal conductivity characteristics la and lb will not be quoted because the formulae are clumsy. Independently on the basis of formulae (4.12), (5.9), (5.10), (5.11), (5.13) and (5.15), it is clear that thermal conductivities are exponentially small at low temperature. It means that the cylindrical quantum dots are bad heat conductors and can be used for thermoisolation.
6. Conclusion The exposed method for calculation of Green’s function of spatially deformed structure appears here, first. The usual approach in analyses of spatially deformed structures, consisting in calculation of spatially diagonal Green’s functions in the broken symmetry direction cannot reproduce the spatial dependence of such structures [12–19]. In this approach the spectral dependence of physical characteristics of deformed structures was lost. The new method presented here does not suffer this failure. The spatial dependence of density and of diffusion coefficient of axial phonons in quantum dots were analysed in this paper. It turned out that the maximum of density change as well as the maximum of diffusion occur in the middle of quantum dots, while on the boundaries they are minimal. It should be also denoted that specific heat as well as the thermal conductivity of quantum dots are exponentially small at low temperatures. We are of opinion that from cylindrical quantum dots superconductive microchips can be synthetized as well as the microelements of isolation systems.
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Acknowledgement This work was supported by the Serbian Ministry of Science and Technology: Grant No. 1895.
References [1] V. Sajfert, R. ajic´, M. C`etkovic´, B. Togic´, Nanotoechnology 14 (2003) 358. [2] G. Cao, Nanostructures and Nanomaterials, Imperial College Physics, London, 2004. [3] X.Y. Kong, Z.L. Wang, Nano Lett. 3 (2003) 1625. [4] A. Mews, A.V. Kadavanich, U. Banin, A.P. Alivisatos, Phys. Rev. B 53 (1996) 13242. [5] H. Dai, J. Gong, H. Kim, D. Lee, Nanotechnology 13 (2002) 674. [6] K. Nikolic´, A. MacKinnon, Conductance fluctuations of narrow disordered quantum wires, http://www.arxiv.org/ list/cond-mat, 9405055. [7] A.C. Scott, Phys. Scripta 29 (1984) 279. [8] A.S. Davydov, Solitons in Molecular Systems, Naukova Dumka, Kiev, 1988. [9] B.S. Togic´, Lj.D. Magkovic´, M.J. Sˇkrinjar, D. Kapor, G. Knemevic´, J. Phys. Condens. Matter. 3 (1991) 7619. [10] R.J. Kubo, Phys. Soc. Jpn. 12 (1957) 570. [11] V.M. Agranovich, M.D. Galanin, Migration of Electron Energy Excitations in Condensed Matter, Nauka, Moscow, 1978. [12] J.K. Wesselinova, Phys. State Sol. B 223 (2001) 737. [13] J.K. Wesselinova, Phys. State Sol. B 229 (2002) 1329. [14] M. Tkach, V. Holovatsky, O. Voitsekhivska, M. Mikhalyova, Phys. State Sol. B 203 (1997) 373. [15] M.V. Tkach, J. Phys. Stud. (Ukraine) 3 (1999) 377. [16] N.V. Tkach, Fiz. Tverd. Tela 39 (1997) 1109. [17] D. Popov, S.K. Jac´imovski, B.S. Togic´, J.P. SˇetrajWic´, Physica A 317 (2003) 129. [18] H. Ibach, H. Lu¨th, Solid State Physics, An Introduction to Principles of Materials Science, Springer, New York, 2003. [19] G. Strobl, Condensed Matter Physics: Crystals, Liquid Crystals and Polymers, Springer, New York, 2004.
Physica E 25 (2005) 479–491 www.elsevier.com/locate/physe
Thermodynamic and kinetic properties of phonons in cylindrical quantum dots Vjekoslav Sajferta,, Jovan P. SˇetrajWic´b, Stevo Jac´imovskib, Bratislav Togic´b a
Technical Faculty M. Pupin Zrenjanin, University of Novi Sad, Djure Djakovica bb, Zrenjanin 23000, Serbia and Montenegro b Department of Physics, Faculty of Sciences, University of Novi Sad, Serbia and Montenegro Received 25 April 2004; received in revised form 18 June 2004; accepted 31 July 2004 Available online 11 September 2004
Abstract The Green’s function technique, suitable for analyses of spatially deformed structures, is developed in this paper and applied to phonon system. The thermodynamic and kinetic phonon properties of cylindrical quantum dots are analysed using a developed method. As a consequence of the applied new method the configurational dependence of diffusion coefficient and dot’s density were included into calculations. Maximum of diffusion and minimum of density is located in central part of the cylindrical quantum dot. All thermodynamic and kinetic characteristics of quantum dot are exponentially small at low temperatures. The low phonons specific heat as well as the low thermal conductivity lead to conclusion that in cylindrical quantum dots exist more convenient conditions for appearance of electron superconductivity. r 2004 Elsevier B.V. All rights reserved. PACS: 73.21.La; 73.63.Kv; 8.67.Hc Keywords: Green’s function techniques; Diffusion; Spatial dependence; Broken symmetry; Nonhomogeneity; Crystal density
1. Introduction Analyses of nanostructures are very popular in several last years (see [1–6]). It is caused by the advance of technology of syntheses of nanostructures and also with wide possibilities of application of nanotubes in superconductive devices. The behaviour of some biosystems can be explained by some nanotube properties (see [1]). Corresponding author. Tel./fax: +38-122-55-4143.
E-mail address: [email protected] (V. Sajfert).
Investigations of phonon subsystem is necessary since phonons are always present in every structure and influence to behaviour of electrons, optical excitations, spin waves, etc. For that reason this analyses is devoted to phonon characteristics since they influence to the characteristics of other subsystems in the structure such as electrons subsystem, subsystem of ferroelectric excitations, subsystem of spin waves, etc. We shall consider the system of N linear molecular chains connecting molecules lying at the tops of M regular polygons whose planes are
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.07.008
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perpendicular to the chains (see Fig. 1). Molecules in chains are labelled with index m while molecules in discs (polygons) are labeled with index n. The numbers M and N are of the order 10. It will be assumed that phonons in chains as well as the phonons in discs are of longitudinally type, i.e., molecular displacements ðaÞ in chains are taken in chain direction, only, while molecular displacements in discs ðbÞ directed along lies connecting two nearest molecules of the polygon. Due to disc assumption the phonons in chains (in further axial phonons) and phonons in discs (in further disc phonons) are mutually independent. Consequently the phonon Hamiltonian of considered quantum dot can be written as follows: H ¼ H a þ H b; (1.1)
Notations in formula (1.2) and (1.3) are the following: a and b are molecular displacements, pa and pb are the corresponding momenta; M is the mass of the molecule and Ca and Cb are the stretch Hook’s constants. It should be pointed out that the symmetry break takes place in the direction of molecular chains, only. The simplest boundary conditions, consisting absence of layers labelled with m ¼ 1 and m ¼ M þ 1 will be assumed. The more real boundary conditions include the changes of boundary Hook’s constants with respect to those of internal layers. For the molecules lying in the discs the symmetry is not disturbed, but obvious cyclicity condition is valid: F n ¼ F Nþ1þn ;
(1.4)
where M X p2ma 1 þ Ca Ha ¼ 2M 4 m¼0
M X
½ðam amþ1 Þ2 þ ðam am1 Þ2
ð1:2Þ
m¼0
is Hamiltonian of mechanical oscillations along axis and N p2 X 1 nb þ Cb Hb ¼ 4 2M n¼0
N X
½ðbn bnþ1 Þ2 þ ðbn bn1 Þ2
where F denotes arbitrary physical characteristics of the system of disc molecules. In connection with this the reader can consult Refs. [7–9]. Thermodynamic and kinetic properties of cylindrical quantum dots will be analysed with the help of two-time temperature Green’s functions. This method is the most effective since, besides finding of energies of elementary excitations, it gives possibility for the calculation of mean values of operator products.
ð1:3Þ
n¼0
is Hamiltonian of mechanical oscillations in discs. b
a
2. Green’s functions of axial phonons Two types of Green functions will be found for axial phonons: one is of the type displacement–displacement, while another is of the type momentum–momentum. We shall look for Green’s function of the type displacement–displacement, first Anm ðtÞ hhan ðtÞjam ð0Þii ¼ yðtÞh½an ðtÞ; am ð0Þi:
α β Fig. 1. Schematic presentation of a cylindrical quantum dot.
ð2:1Þ
Differentiating this formula with respect to time we obtain d Anm ðtÞ ¼ yðtÞh½_an ðtÞ; am ð0Þi; dt
(2.2)
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where yðtÞ ¼
1; 0;
t40; to0;
(2.3)
is Heaviside’s step function. Taking into account equation of motion 1 p a_ n ¼ ½an ; H a ¼ ma i_ M we reduce (2.2) onto
(2.4)
d hhp ðtÞjam ð0Þii dt na ¼ i_dnm dðtÞC a ½hhanþ1 ðtÞjam ð0Þii þ hhan1 ðtÞjam ð0Þii hh2an ðtÞjam ð0Þii:
1pnpM 1; n ¼ 0;
AM1;m þ rAM;m ¼ FM;m ; n ¼ M:
ð2:11Þ (2.12) (2.13)
The solution of system (2.11)–(2.13) will be looked for in the form Mþ1 X
amm sinðn þ 1Þjm :
(2.14)
Introducing Eq. (2.14) into Eq. (2.11) we obtain ð2:6Þ
relation (2.7) becomes
Mþ1 X
ð2cos jm þ rÞamm sinðn þ 1Þjm ¼ Fn;m :
(2.15)
m¼1
After substitution A1;m and A0;m in Eq. (2.12) we require to left-hand side of Eq. (2.12) be analogous to Eq. (2.15). Since sin 2jm ¼ 2cos jm ; sin jm Eq. (2.12) goes over to the form which is analogous to Eq. (2.15), i.e. Mþ1 X
ð2cos jm þ rÞamm sin jm ¼ Fn;m :
(2.16)
m¼1
ð2:9Þ
After substitution Eq. (2.14) into Eq. (2.13) we reduce Eq. (2.13) to the form Mþ1 X
and M 2 o 2: Ca
¼ Fn;m ;
m¼1
d2 i_ Anm ðtÞ ¼ dnm dðtÞ 2 M dt Ca ½Anþ1;m þ An1;m 2An;m : ð2:7Þ þ M After time-frequency Fourier-transformation Z þ1 do eiot Anm ðoÞ; Anm ðtÞ ¼ 1 Z 1 þ1 do eiot ð2:8Þ dðtÞ ¼ 2p 1
r¼
Anþ1;m þ An1;m þ rAn;m
An;m ¼
Differentiating (2.5) with respect to time and substituting (2.6) we obtain the equation
Anþ1;m ðoÞ þ An1;m ðoÞ þ rAn;m ðoÞ ¼ Fn;m ; i_ 1 dn;m Fn;m ¼ 2p C a
layers 0 and M change Hook’s constants, but we shall assume that their changes are negligible. Taking into account all said before, becomes clear that Eq. (2.13) goes over to three systems of difference equations:
A1;m þ rA0;m ¼ F0;m ;
d 1 Anm ðtÞ ¼ hhpna ðtÞjam ð0Þii: (2.5) dt M Now we have to look for Green’s function hhpna ðtÞjam ð0Þii: Analogously to the upper calculation we obtain
481
ð2cos jm þ rÞamm sinðM þ 1Þjm ¼ FM;m :
m¼1
(2.10)
As it is seen for finding Green’s functions of the type displacement–displacement the system of M+1—difference-equation is obtained. Since the layers m ¼ 1 and M þ 1 are absent, it is clear that A1;m ðoÞ ¼ ANþ1;m ðoÞ ¼ 0: In boundary
(2.17) If the condition sin Mjm ¼ 2cos jm sinðM þ 1Þjm is satisfied.
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From (2.17) it follows that the parameter jm is playing role of wave vector is given by
we obtain from (2.20)
pm ; jm ¼ M þ2
Gm ðoÞ ¼ m ¼ 1; 2; 3; . . . ; M þ 1:
(2.18)
Index m characterizing wave vector in the direction of the chain cannot take values 0 and M þ 2 because it leads to sinðM þ 1Þjm ¼ 0: This is, as we have seen, forbidden. From the procedure quoted, it is clear that relation (2.15) Mþ1 X
ð2cos jm þ rÞamm sinðn þ 1Þjm ¼ Fn;m
m¼1
for jm ¼ pm=M þ 2 stands for every m, i.e., for m ¼ 0; 1; 2; 3; . . . ; M: Before defining the coefficient amm we shall represent Kronecker symbol in terms of sine functions. It can be easily shown that: Mþ1 X
sin2 ðm þ 1Þjm ¼
m¼1 Mþ1 X
M þ2 ; 2
sinðn þ 1Þjm sinðm þ 1Þjm ¼ 0;
nam:
m¼1
We obtain the following representation of Kronecker symbol: dn;m ¼
X 2 Mþ1 sinðn þ 1Þjm sinðm þ 1Þjm : M þ 2 m¼1 (2.19)
Taking into account the last result we can write (2.15) in the following form: Mþ1 X
ð2cos jm þ rÞamm sinðn þ 1Þjm
m¼1
¼
2 i_ 1 M þ 2 2p C a Mþ1 X sinðn þ 1Þjm sinðm þ 1Þjm :
ð2:20Þ
m¼1
Taking amm ðoÞ ¼
2 Gm ðoÞ sinðm þ 1Þjm M þ2
(2.21)
i_ 1 1 2p C a 2cos jm þ r
(2.22)
from which after substitution parameter r from (2.10) we obtain Gm ðoÞ ¼
1 i_ 1 2 M 2p o o2m
or, in terms of simple fractions i_ 1 1 1 Gm ðoÞ ¼ : 2p 2Mom o om o þ om Notations used in (2.34) are rffiffiffiffiffiffi Ca 2 jm 2 2 ; Oa ¼ om ¼ 4Oa sin ; 2 M pm ; m ¼ 1; 2; 3; . . . ; M þ 1: jm ¼ M þ2
(2.23)
(2.24)
ð2:25Þ
Formula (2.25) gives the dispersion law of phonons propagating in linear molecular chains. It should be pointed out that this dispersion in infinite (ideal molecular chain) is of the same form. The unique difference appears in the expression for dimensionless wave vector, which in infinite chain takes values 2pm M M jm ¼ ; m 2 þ 1; : (2.26) M 2 2 Comparing Eq. (2.35) to Eq. (2.36) the very important conclusion can be reduced: the minimum frequency in short molecular chain p (2.27) ðom Þmin ¼ 2Oa sin M þ2 is different from zero. In the infinite chain the minimal frequency is equal to zero. Since we deal with acoustic phonons, the above conclusion practically means that acoustic phonons in short chain possess energy gap, i.e., for excitation of central phonons in short chain the minimal, different form zero, energy must be inserted. This fact has positive repercussions to realization of superconductive effect in short chains (Fig. 2). In the infinite chain the phonon gap does not exist (see Fig. 3). pm From the value of 2sin Mþ2 quoted in the first row of the Table 1 it can be calculated that the
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Table 1 m
pm Mþ2
pm 2sin Mþ2
1 2 3 4
0.2856 0.5712 0.8568 1.1424
0.5634 1.0813 1.5115 2.2848
Using the relation Fig. 2. The histogram of axial phonon frequencies for four layer chain in accordance with Table 1.
1 1 ^ ¼P ipdðo o0 Þ; (2.29) o o0 id o o0 ^ denotes main value of integral and where P formula (2.34) we obtain _ dðo om Þ dðo þ om Þ _o I G ðoÞ ¼ : (2.30) 2Mom e_o y 1 ey 1 With help of spectral intensity one can find correlation functions in momentum space (the momenta p are characterizing by the integer m; mspace) as: Z þ1 do eiot I G ðoÞ: (2.31) hað0ÞaðtÞm i ¼ 1
Substituting Eq. (2.38) into Eq. (2.39) we obtain " # _ eitom eitom hað0ÞaðtÞm i ¼ : (2.32) 2Mom e_oy m 1 e_oy m 1 Fig. 3. Relative frequencies of phonons in infinite chain.
minimal energy for exciting of acoustical phonons is E min 102 eV what corresponds to temperature T 100 K: It means that at temperatures up to 100 K the short chain is quite frozen so that electrons propagate without friction, i.e. superconductively. This estimate is made for Debye’s energy of 200kB. After the discussion of short chain phonons dynamical properties we can go over to analysis of mean values characterizing these phonons. Spectral intensity of the function G m ðoÞ is I G ðoÞ ¼ d ! þ0
G m ðo þ idÞ G m ðo idÞ _o
ey 1 Y ¼ kB T:
Configurational correlation function can be obtained by operator application of Kronecker’s symbol to correlation function in momentum space (m-space). In this way, using formula (2.19) we obtain: hað0ÞaðtÞi ¼
X 2 Mþ1 hað0ÞaðtÞim M þ 2 m¼1 sinðn þ 1Þjm sinðm þ 1Þj
and this formula in accordance with (2.32) goes over to " # X 1
2 _ Mþ1 eitom eitom að0ÞaðtÞ ¼ M þ 2 2M m¼1 om e_oy m 1 e_oy m 1 sinðn þ 1Þjm sinðm þ 1Þjm :
; ð2:28Þ
ð2:33Þ
ð2:34Þ
For further analyses it is necessary to find momentum correlation function. To achieve this
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we have to find momentum Green’s function defined as follows:
The Green’s functions of the type displacement–displacement will be calculated first
Pnm ðtÞ hhpna ðtÞjpma ð0Þii
Bnm ðtÞ hhbn ðtÞjbm ð0Þii ¼ yðtÞh½bn ðtÞ; bm ð0Þi:
¼ yðtÞh½pna ðtÞ; pma ð0Þi
ð2:35Þ
Using the calculation procedure quite analogous to that which was used in the case of the Green’s function Anm we obtain i_ 1 1 1 G m ðoÞ ¼ ; (2.36) 2p 2Mom o om o þ om where the spectral intensity of this function is _C a dðo om Þ dðo þ om Þ I G ðoÞ ¼ o : (2.37) m _o _o 2O2a ey 1 ey 1 Using Eq. (2.37) we obtain in m-space hpma ð0Þpna ðtÞi ¼
"
#
X 1 _C a Mþ1 eitom eitom om _om _om 2 M þ 2 Oa m¼1 e y 1 e y 1 sinðn þ 1Þj sinðm þ 1Þj ð2:38Þ
from which the expression for the spatial momentum correlation function is the following: hpma ð0Þpna ðtÞi
" # X 1 _C a Mþ1 eitom eitom ¼ om _om _om M þ 2 O2a m¼1 e y 1 e y 1 sinðn þ 1Þj sinðm þ 1Þj ð2:39Þ
The relation obtained will be used for calculation of some thermodynamic and kinetic characteristics for cylindrical quantum dots, or more precise, for calculation of the contributions of axial phonons to thermodynamic and kinetic properties.
ð3:1Þ
Using the same procedure as in the case of Green’s function displacement–displacement for axial phonons we obtain: d 1 Bnm ðtÞ ¼ hhpmb ðtÞjbm ð0Þii: dt M
(3.2)
Since the calculation procedure is quite similar to that of the previous section the stages of calculation will not be quoted. We shall give the final results, only. The final result is: d hhp ðtÞjbm ð0Þii dt nb ¼ i_dnm dðtÞC a ½hhbnþ1 ðtÞjbm ð0Þii þ hhbn1 ðtÞjbm ð0Þii 2hhbn ðtÞjbm ð0Þii:
ð3:3Þ
Differentiating (3.2) with respect to time and substituting (3.3) we obtain: d2 i_ Bnm ðtÞ ¼ dnm dðtÞ M dt2 Cb ½Bnþ1;m þ Bn1;m 2Bn;m : ð3:4Þ þ M After Fourier-transformation time-frequency Z
þ1
do eiot Bnm ðoÞ
Bnm ðtÞ ¼
(3.5)
1
relation (3.4) becomes Bnþ1;m ðoÞ þ Bn1;m ðoÞ þ rBn;m ðoÞ ¼ f n;m ;
(3.6)
where 3. Green’s function of disc phonons Hamiltonian of subsystem disc phonons is given by formula (1.3). Phonon characteristics in discs will be examined by Green’s functions of the type displacement–displacement and momentum–momentum.
f n;m ¼ r¼
i_ 1 dn;m ; 2p C b
M 2 o 2: Cb
ð3:7Þ
At this stage the analogy with previous case is finished. System of difference equations (3.6) can
ARTICLE IN PRESS V. Sajfert et al. / Physica E 25 (2005) 479–491
be written in the following way:
Values of parameter jn will be determined form cyclicity condition:
Bnþ1;m þ Bn1;m þ rBn;m ¼ f n;m ; 1pnpN 1; A1;m þ B1;m þ rB0;m ¼ f 0;m ;
ð3:8Þ n ¼ 0;
BNþ1;m þ BN1;m þ rBN;m ¼ f M;m ;
(3.9)
B1;m ¼ BN;m a0;
(3.11)
BNþ1;m ¼ B0;m a0:
(3.12)
It enables us to look for the solution of system (3.8)–(3.10) in the form bn;m einjn :
(3.13)
n¼1
Introducing Eq. (3.13) into Eq. (3.8) we obtain N þ1 X
ð2cos jn þ rÞbnm einjn ¼ f n;m ;
n¼1
n ¼ 1; 2; 3; . . . ; N 1:
ð3:14Þ
After substitution Eq. (3.13) into Eq. (3.9) it follows: N þ1 X
n ¼ 0:
(3.15)
n¼1
The condition Eq. (3.19) is fulfilled if eiðNþ1Þjn ¼ 1; i.e., if 2pn ; n ¼ 0; 1; 2; 3; . . . ; N þ 1: (3.20) N þ1 It can be easily shown that Kronecker’s symbol is given by jn ¼
dn;m ¼
þ1 X 1 N eiðnmÞjn ; N þ 1 n¼1
2pn : N þ1 If we take that
jn ¼
1 g ðoÞeimjn (3.22) N þ1 n then, with respect to Eq. (3.7), the system of equations (3.17) goes over to þ1 X 1 N g ðoÞeiðnmÞjn N þ 1 n¼1 n
X 1 i_ 1 Nþ1 eiðnmÞjn N þ 1 2p C b n¼1
from which it follows gn ðoÞ ¼
ð2cos jn þ rÞbnm eiNjn ¼ f N;m ;
n ¼ N:
(3.16)
n¼1
As it is seen Eqs. (3.14)–(3.16) have absolutely identical form and reduce to N þ1 X
i_ 1 1 : 2p C b 2cos jn þ r
(3.23)
Substituting, r from Eq. (3.7) into Eq. (3.23) we obtain gn ðoÞ ¼
1 i_ 1 ; 2 M 2p o o2n
(3.24)
jn 2
(3.25)
where
ð2cos jn þ rÞbnm einjn ¼ f n;m ;
n¼1
n ¼ 0; 1; 2; 3; . . . ; N:
ð3:21Þ
bn;m ¼
Including Eq. (3.13) into Eq. (3.10) we obtain N þ1 X
(3.19)
n¼1
¼ ð2cos jn þ rÞbnm ei0jn ¼ f 0;m ;
(3.18)
which, with accordance with Eq. (3.13), becomes ½1 eiðNþ1Þjn bnm einjn ¼ 0:
In the case of axial phonons the Green’s functions with indices –1 and M+1 were equal to zero because the layers with indices –1 and M+1 do not exist. In disc cyclic symmetry (1.4) takes place and consequently, the following is valid:
Bn;m ¼
Bn;m ¼ BNþ1þn;m
N þ1 X
n ¼ N: (3.10)
N þ1 X
485
ð3:17Þ
o2n ¼ 4O2b sin2
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486
Table 2
and rffiffiffiffiffiffi Cb Ob ¼ ; M
2pn : jn ¼ N þ1
(3.26)
In Fig. 4 the dispersion law of disc phonons will be quoted in histogram form (Table 2). As it can be seen the minimal frequency of disc phonons is equal to zero. Consequently the exciting of disc phonons does not require inserting the external energy (see Table 4). In this sense phonons in the discs behave as phonons in infinite linear chain. It is clear from the fact that the histogram in Fig. 4 obtained from Table 2 is the discrete equivalent of continual curve given in Fig. 3. Now we can go over to finding the mean values of products of operators corresponding to disc phonons. The spectral intensity of the function gn ðoÞ is given by _ dðo on Þ dðo þ on Þ _o I g ðoÞ ¼ : (3.27) 2Mon e_o y 1 ey 1 Using this formula we can find the correlation function in n-space " # _ eiton eiton : (3.28) hbð0ÞbðtÞin ¼ 2Mom e_oy n 1 e_oy n 1
n
2pn Nþ1
2pn 2 sin Nþ1
0 1 2 3
0 0.62832 1.25664 1.88495
0 0.61803 1.17557 1.61803
Finally, the configurational correlation function is given by the formula hbm ð0Þbn ðtÞi ¼ jn ¼
þ1 X 1 N hb ð0Þbn ðtÞin eiðnmÞjn N þ 1 n¼1 m
2pn N þ1
ð3:29Þ
which, with respect to (3.29) becomes hbm ð0Þbn ðtÞi 1 _ ¼ N þ 1 2M " # Nþ1 X 1 eiton eiton _on eiðnmÞjn ; _on o e y 1 n¼1 n e y 1 jn ¼
2pn : N þ1
ð3:30Þ
Now we shall calculate Green’s function of the type momentum–momentum Pnm ðtÞ hhpnb ðtÞjpmb ð0Þii ¼ yðtÞh½pnb ðtÞ; pmb ð0Þi:
ð3:31Þ
Using quite similar procedure as in the case of displacement–displacement Green’s function we obtain the following relevant formula: gn ðoÞ ¼
i_ C b 1 ; 2p O2b o2 o2n
(3.32)
where gn ðoÞ is the Fourier transform of the function Pn;m ; i.e.:
Pn;m ¼ Fig. 4. The histogram of axial phonon frequencies in accordance with Table 2.
þ1 X 1 N g ðoÞeinjn ; N þ 1 n¼1 n
n ¼ 0; 1; 2; . . . ; N
ð3:33Þ
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the set of Eqs. (3.35)–(3.37) goes over to gn ðoÞ ¼
i_ o2n 1 M 2 : 2p Ob 2cos jn þ r
4. Thermodynamic properties of cylindrical quantum dot (3.34)
After substitution the parameter r into Eq. (3.34) the last formula becomes gn ðoÞ ¼
i_ C b 1 2 2 2p Ob o o2n
(3.35)
or, after resolving to simple fractions i_ C b 1 1 o gn ðoÞ ¼ n 2p O2b o on o þ on
(3.36)
Using Eq. (3.36) we obtain spectral intensity of the Green’s function g; _C b dðo on Þ dðo þ on Þ _o on (3.37) I g ðoÞ ¼ _o 2O2b ey 1 ey 1 from which it follows: hpb ð0Þpb ðtÞin ¼
"
It should be noticed in the beginning, that thermodynamical properties will be investigated at low temperatures, when e_o=y b1: In this case the second term in correlation functions (2.34) and (3.30) approximately goes over to eiot : This term does not depend on temperature and obviously comes from zero oscillations. As it was said earlier, the contributions of zero oscillations will not be taken into account, and consequently this term will be omitted in further analyses. In the first term of formulae (2.35) and (3.30) Bose’s distribution 1=e_o=y 1 will be substituted with Boltzmann’s one e_o=y : It means that in further the following expressions for configurational correlation functions will be used: hað0ÞaðtÞi ¼
X 1 _o _ 1 Mþ1 e Y itom M M þ 2 m¼1 om sinðn þ 1Þjm sinðm þ 1Þjm ;
#
_C b eiton eiton o : n _on _on 2O2b e y 1 e y 1
487
ð3:38Þ
hbm ð0Þbn ðtÞi ¼
The configurational correlation function is given by
_ 1 2M N þ 1 N þ1 X 1 _oiton þiðnmÞjn e Y : o n¼1 n
ð4:1Þ
ð4:2Þ
The same requirements for momentum correlation functions give
hpmb ð0Þpnb ðtÞi 1 _C b ¼ N þ 1 O2b " # Nþ1 X eiton eiton _on on _on eiðnmÞjn : e y 1 e y 1 n¼1
hpma ð0Þpna ðtÞi ¼
X _C a 1 Mþ1 _o om e Y itom 2 M þ2 Oa m¼1 sinðn þ 1Þj sinðm þ 1Þj
ð3:39Þ Comparing the properties of axial phonons with disc ones we can notice two essential differences. Axial phonons are standing waves while disc phonons are plane waves. The main consequence of noticed difference is that probability current of axial phonons in dot is equal to zero, while the probability current of disc phonons is non/zero quantity. The exciting of axial phonons requires inserting of relatively high energy quant, while disc phonons appear without inserting of external energy.
ð4:3Þ
and hpmb ð0Þpnb ðtÞi ¼
þ1 X _C b 1 N _o on e Y iton þiðnmÞjn : O2b N þ 1 n¼1
ð4:4Þ
Besides mentioned, the following approximations will be used, too: (a) The approximation of small wave vectors on
pOa ; M þ2
om
2pOb : N þ1
(4.5)
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(b) All finite geometrical progression whose quotient contains factor eC=Y will be approximately considered as infinite ones. (c) Mathematical forms of the type ex =ðex 1Þ2 and lnð1 þ ex Þ will be approximately taken equal C to ex ; if x contains factor Y : In accordance with approximations quoted the internal energy of axial phonons in cylindrical quantum dot is given by U a ¼ hH a i ¼
M 1 X 1 hp2 i þ C a 2M m¼0 ma 4
M X ðha2m i þ ha2m1 i þ ha2mþ1 i m¼0
2ham am1 i 2ham am1 iÞ:
ð4:6Þ
Using formula (4.1), taken for t ¼ 0 m and formula (4.3) taken for t ¼ 0; too, we find U a ¼ hH a i
p M þ2 þ 4ðM þ 2Þ p p 1 p sin2 sin2 : 2ðM þ 2Þ 2p M þ2
¼ _Oa e
p_Oa ðMþ2 ÞY
ð4:7Þ
Internal energy of disc phonons is given by: U b ¼ hH b i ¼
M 1 X 1 hp2 i þ C b 2M m¼0 nb 4
M X ðhb2n i þ hb2n1 i þ hb2nþ1 i
CV ¼
@hHi @hHi @Y ¼ ; @T @Y @T
Y ¼ kB T; the following result: pkB _Oa 2 p _Oa p e Mþ2 Y CV ¼ 4ðM þ 2Þ M þ2 Y Mþ2 p 1 p sin2 sin2 þ p 2ðM þ 2Þ 2p M þ2 2 _O 2p 2pkB _Ob b þ eNþ1 Y N þ1 Y p N þ1 2 p þ sin : ð4:12Þ 2ðN þ 1Þ 2p N þ1 As it is seen from the results obtained, the internal energy as well as the specific heat decrease exponentially at low temperatures. It means that cylindrical qunatum dots behave as a rigid body at low temperature, i.e., population of mechanical oscillation is very small. This makes suitable conditions for appearance of high temperature superconductivity. At the end of this section the specific heat of axial phonons will be discussed. The general formula for the specific heat is very clumsy and
c
ð4:8Þ
d
Using (4.2) for t ¼ 0 and (4.4) for t ¼ 0 we obtain 2p_Oa p ðNþ1ÞY U b ¼ hH b i ¼ _Ob e 2ðN þ 1Þ N þ1 2 p sin þ : ð4:9Þ 2p 2ðN þ 1Þ The total internal energy of cylindrical quantum dot is hHi ¼ hH a i þ hH b i:
(4.11)
CV
m¼0
2hbn bn1 i 2hbn bn1 iÞ:
dance with general formula
(4.10)
Specific heat of cylindrical quantum dot can be found by differentiating the internal energy with respect to temperature. So we obtain in accor-
b O
a
Θ
Fig. 5. Full line represents specific heat of nanostructure. The part a b corresponds to extremely small temperatures while part b c corresponds to the interval of the middle temperatures. Dashed line represents specific heat of one-dimensional ideal structure.
ARTICLE IN PRESS V. Sajfert et al. / Physica E 25 (2005) 479–491
therefore the leading term of the formula will be quoted: 2 p _Oa p_ 1 C V ¼ kB Oa 2 eMþ2 Y : (4.13) M þ2 Y This formula is graphically presented in Fig. 5 (full line) together with the specific heat of an infinite chain (dashed line in Fig. 5). It should be pointed out that, at low temperature, the specific heat of the dot is exponentially small. In the middle temperature range the formula for specific heat of cylindrical quantum nanodot is Y C V ¼ 2kB ðM þ 2Þ : p_Oa
(4.14)
As it is seen specific heat linearly increases with temperature. It should be pointed out that specific heat in bulk structure behaves in the same way as specific heat of nanostructures in middle temperature range but in the interval Y 2 ½0; YM ; where YM YD (YD is Debye’s temperature). Specific heat of disc phonons should not be graphically presented since its behaviour is very similar to the behaviour of axial phonons.
5. The kinetic parameters of cylindrical quantum dots Here we shall look for diffusion tensor of quantum dots and for tensor of thermal conductivity. In connection with this we note in the beginning that the diffusion of axial phonons is independent of diffusion processes in subsystem of the disc phonons. Calculating the diffusion tensor of axial phonons and the diffusion tensor of disc phonons we can conclude that the sum of the two mentioned tensors cannot be considered as a diffusion tensor of the whole cylindrical dot. Eventually, the arithmetical mean of two mentioned diffusion tensors can be (only by definition) considered as the diffusion tensor of the whole cylindrical quantum dot. Heat conductivity is given by the formula l ¼ D C V r;
(5.1)
where D is diffusion tensor and r is density of cylindrical quantum dot. Earlier it was shown that
489
the specific heat of the whole quantum dot is the sum of specific heat of axial phonons and specific heat of the disc phonons. On the other hand the density r is the characteristic of the whole cylindrical quantum dot and cannot be separated into parts corresponding to axial phonons and the one corresponding to disc phonons. We are of the opinion that the introducing of two thermal conductivities in cylindrical quantum dot is the most realistic. These conductivities are la ¼ Da C Va r
(5.2)
and lb ¼ Db C Vb r:
(5.3)
From these formulae it is seen that thermal conductivity for every one of the phonon subsystems depends, through the density, on characteristics of the other subsystem. It means that thermal conductivity la for axial phonons and lb for the disc phonons are not independent. In the previous section, the specific heat of cylindrical quantum dot was found. From formula (4.12) it is quite clear what contributions give one phonon subsystem and what contributions coming from the other one. The density of the ‘‘frozen’’ cylindrical quantum dot is given by m0 r0 ¼ ; (5.4) V0 where V0 ¼
p ðM þ 1Þab2 : p 4 sin2 Nþ1
(5.5)
In Eq. (5.5) the mass of cylindrical quantum dot it is denoted with m0 : With a and b are denoted the ‘‘lattice constants’’ (a is distance between two neighbour discs, while b is the chord length between neighbour molecules in discs). During the heating, the ‘‘lattice constant’’ a gets correction qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi am ¼ ha2m ð0Þi: (5.6) The correction of constant b is qffiffiffiffiffiffiffiffiffiffiffiffiffiffi bn ¼ hb2n ð0Þi:
(5.7)
ARTICLE IN PRESS V. Sajfert et al. / Physica E 25 (2005) 479–491
490
Consequently for T40 we have a ! a þ am ;
in the Section 4, we obtain from formula (4.3), (5.8)
b ! b þ bn : It can be easily concluded that the corrected density, up to linear terms in a and b; has the form am bn 2 rðYÞ ¼ r0 1 : (5.9) a b With respect to formulae (4.1) and (4.2), taken for t ¼ 0 and m=n, we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffi p_Oa _ pðn þ 1Þ 2ðMþ2ÞY (5.10) e am ¼ sin M þ 2 pMOa and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p_Ob _ bm ¼ e ðNþ1ÞY : 4pMOb
(5.11)
It is seen, from Eqs. (5.9)–(5.11), that the density depends on temperature and on spatial coordinates. The last is the result of the application of the Green’s functions method, developed here, which is suitable for analysis of spatially deformed structures. The numerical calculations have shown that maximum of decrease of the density can be expected in the middle of the cylindrical quantum dot. At the ends of cylindrical quantum dot density changes are minimal. Now we shall look for the diffusion tensor of axial phonons and diffusion tensor of disc phonons. In Refs. [10,11] the diffusion tensor is defined as follows: Z 1 Dnm ¼ dt et hvm ð0Þvn ðtÞi; 0
! 0þ;
ð5:12Þ
where v are velocities of phonon propagation, i.e., v ¼ Mp : Correlation function of the type velocity/velocity for axial phonons will be found with the help of formula (4.3). Using the approximation quoted
DðaÞ nm ðYÞ ¼
12 _ p_Oa e ðMþ2ÞY i MM þ 2 sinðn þ 1Þj sinðm þ 1Þj:
ð5:13Þ
As it is seen, the diffusion tensor of axial phonons depends on temperature and on spatial coordinates. For three-layer structure the following proportionality for diffusion tensor is valid: 0 1 0:500 0:707 0:500 B C aÞ Dðnm @ 0:707 1:000 0:707 A: (5.14) 0:500
0:707
0:500
It is clear from this formula that diffusion is maximal in the middle of the structure and decreases in the direction of boundaries. The diffusion of disc phonons can be calculated using formula (4.4). After long but elementary calculations, in which approximations quoted in Section 4 are used, we obtain DðbÞ nm ðYÞ ¼
1 _ i 2MðN þ 1Þ 2p_O
e
b 2p ðNþ1ÞY eiðnmÞNþ1 :
ð5:15Þ
It should be pointed out that in practical calculations diffusion tensors are real quantities. Consequently, instead of complex values (5.13) and (5.15) their absolute values have to be used. Those absolute values are 2 _ ðaÞ p_Oa e ðMþ2ÞY D~ nm ðYÞ ¼ DðaÞ nm ðYÞ ¼ MM þ 2 p sinðm þ 1Þ sinðn þ 1Þ M þ2 p : ð5:16Þ Mþ2 and ðbÞ D~ nm ðYÞ ¼ DðbÞ nm ðYÞ 2p_Ob _ e ðNþ1ÞY : ¼ ð5:17Þ 2MðN þ 1Þ The most important conclusion of the analyses of this section is the fact that in the direction of broken symmetry, diffusion is characterized by tensor depending on temperature and on spatial coordinates. For disc phonons (the symmetry is not disturbed in discs) the diffusion tensor is losing
ARTICLE IN PRESS V. Sajfert et al. / Physica E 25 (2005) 479–491
tensor characteristics and goes over to the function depending on the temperature, only. The thermal conductivity characteristics la and lb will not be quoted because the formulae are clumsy. Independently on the basis of formulae (4.12), (5.9), (5.10), (5.11), (5.13) and (5.15), it is clear that thermal conductivities are exponentially small at low temperature. It means that the cylindrical quantum dots are bad heat conductors and can be used for thermoisolation.
6. Conclusion The exposed method for calculation of Green’s function of spatially deformed structure appears here, first. The usual approach in analyses of spatially deformed structures, consisting in calculation of spatially diagonal Green’s functions in the broken symmetry direction cannot reproduce the spatial dependence of such structures [12–19]. In this approach the spectral dependence of physical characteristics of deformed structures was lost. The new method presented here does not suffer this failure. The spatial dependence of density and of diffusion coefficient of axial phonons in quantum dots were analysed in this paper. It turned out that the maximum of density change as well as the maximum of diffusion occur in the middle of quantum dots, while on the boundaries they are minimal. It should be also denoted that specific heat as well as the thermal conductivity of quantum dots are exponentially small at low temperatures. We are of opinion that from cylindrical quantum dots superconductive microchips can be synthetized as well as the microelements of isolation systems.
491
Acknowledgement This work was supported by the Serbian Ministry of Science and Technology: Grant No. 1895.
References [1] V. Sajfert, R. ajic´, M. C`etkovic´, B. Togic´, Nanotoechnology 14 (2003) 358. [2] G. Cao, Nanostructures and Nanomaterials, Imperial College Physics, London, 2004. [3] X.Y. Kong, Z.L. Wang, Nano Lett. 3 (2003) 1625. [4] A. Mews, A.V. Kadavanich, U. Banin, A.P. Alivisatos, Phys. Rev. B 53 (1996) 13242. [5] H. Dai, J. Gong, H. Kim, D. Lee, Nanotechnology 13 (2002) 674. [6] K. Nikolic´, A. MacKinnon, Conductance fluctuations of narrow disordered quantum wires, http://www.arxiv.org/ list/cond-mat, 9405055. [7] A.C. Scott, Phys. Scripta 29 (1984) 279. [8] A.S. Davydov, Solitons in Molecular Systems, Naukova Dumka, Kiev, 1988. [9] B.S. Togic´, Lj.D. Magkovic´, M.J. Sˇkrinjar, D. Kapor, G. Knemevic´, J. Phys. Condens. Matter. 3 (1991) 7619. [10] R.J. Kubo, Phys. Soc. Jpn. 12 (1957) 570. [11] V.M. Agranovich, M.D. Galanin, Migration of Electron Energy Excitations in Condensed Matter, Nauka, Moscow, 1978. [12] J.K. Wesselinova, Phys. State Sol. B 223 (2001) 737. [13] J.K. Wesselinova, Phys. State Sol. B 229 (2002) 1329. [14] M. Tkach, V. Holovatsky, O. Voitsekhivska, M. Mikhalyova, Phys. State Sol. B 203 (1997) 373. [15] M.V. Tkach, J. Phys. Stud. (Ukraine) 3 (1999) 377. [16] N.V. Tkach, Fiz. Tverd. Tela 39 (1997) 1109. [17] D. Popov, S.K. Jac´imovski, B.S. Togic´, J.P. SˇetrajWic´, Physica A 317 (2003) 129. [18] H. Ibach, H. Lu¨th, Solid State Physics, An Introduction to Principles of Materials Science, Springer, New York, 2003. [19] G. Strobl, Condensed Matter Physics: Crystals, Liquid Crystals and Polymers, Springer, New York, 2004.