Linear, diatomic crystal: single-electron states and large-radius excitons

Linear, diatomic crystal: single-electron states and large-radius excitons

The large-radius exciton spectrum in a linear crystal with two atoms in the unit cell was obtained using the single-electron eigenfunctions and the band structure, which were found by the zero-range potentials method. The ground-state exciton binding energies for the linear crystal in vacuum appeare...

Повний опис

Збережено в:
Бібліографічні деталі
Дата:2009
Автори: Adamyan, V.M., Smyrnov, O.A.
Формат: Стаття
Мова:English
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
Назва видання:Физика низких температур
Теми:
Низкоразмерные и неупорядоченные системы
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/117140
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Linear, diatomic crystal: single-electron states and large-radius excitons / V.M. Adamyan, O.A. Smyrnov // Физика низких температур. — 2009. — Т. 35, № 5. — С. 503-509. — Бібліогр.: 21 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-117140
record_format dspace
spelling irk-123456789-1171402017-05-21T03:03:10Z Linear, diatomic crystal: single-electron states and large-radius excitons Adamyan, V.M. Smyrnov, O.A. Низкоразмерные и неупорядоченные системы The large-radius exciton spectrum in a linear crystal with two atoms in the unit cell was obtained using the single-electron eigenfunctions and the band structure, which were found by the zero-range potentials method. The ground-state exciton binding energies for the linear crystal in vacuum appeared to be larger than the corresponding energy gaps for any set of the crystal parameters. 2009 Article Linear, diatomic crystal: single-electron states and large-radius excitons / V.M. Adamyan, O.A. Smyrnov // Физика низких температур. — 2009. — Т. 35, № 5. — С. 503-509. — Бібліогр.: 21 назв. — англ. 0132-6414 PACS: 73.22.Dj, 73.22.Lp, 71.35.Cc http://dspace.nbuv.gov.ua/handle/123456789/117140 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Низкоразмерные и неупорядоченные системы
Низкоразмерные и неупорядоченные системы
spellingShingle Низкоразмерные и неупорядоченные системы
Низкоразмерные и неупорядоченные системы
Adamyan, V.M.
Smyrnov, O.A.
Linear, diatomic crystal: single-electron states and large-radius excitons
Физика низких температур
description The large-radius exciton spectrum in a linear crystal with two atoms in the unit cell was obtained using the single-electron eigenfunctions and the band structure, which were found by the zero-range potentials method. The ground-state exciton binding energies for the linear crystal in vacuum appeared to be larger than the corresponding energy gaps for any set of the crystal parameters.
format Article
author Adamyan, V.M.
Smyrnov, O.A.
author_facet Adamyan, V.M.
Smyrnov, O.A.
author_sort Adamyan, V.M.
title Linear, diatomic crystal: single-electron states and large-radius excitons
title_short Linear, diatomic crystal: single-electron states and large-radius excitons
title_full Linear, diatomic crystal: single-electron states and large-radius excitons
title_fullStr Linear, diatomic crystal: single-electron states and large-radius excitons
title_full_unstemmed Linear, diatomic crystal: single-electron states and large-radius excitons
title_sort linear, diatomic crystal: single-electron states and large-radius excitons
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2009
topic_facet Низкоразмерные и неупорядоченные системы
url http://dspace.nbuv.gov.ua/handle/123456789/117140
citation_txt Linear, diatomic crystal: single-electron states and large-radius excitons / V.M. Adamyan, O.A. Smyrnov // Физика низких температур. — 2009. — Т. 35, № 5. — С. 503-509. — Бібліогр.: 21 назв. — англ.
series Физика низких температур
work_keys_str_mv AT adamyanvm lineardiatomiccrystalsingleelectronstatesandlargeradiusexcitons
AT smyrnovoa lineardiatomiccrystalsingleelectronstatesandlargeradiusexcitons
first_indexed 2025-07-08T11:42:53Z
last_indexed 2025-07-08T11:42:53Z
_version_ 1837078911322685440
fulltext Fizika Nizkikh Temperatur, 2009, v. 35, No. 5, p. 503–509 Linear, diatomic crystal: single-electron states and large-radius excitons V.M. Adamyan and O.A. Smyrnov Department of Theoretical Physics, Odessa I.I. Mechnikov National University 2 Dvoryanskaya Str., Odessa 65026, Ukraine E-mail: smyrnov@onu.edu.ua Received November 13, 2008, revised January 4, 2009 The large-radius exciton spectrum in a linear crystal with two atoms in the unit cell was obtained using the single-electron eigenfunctions and the band structure, which were found by the zero-range potentials method. The ground-state exciton binding energies for the linear crystal in vacuum appeared to be larger than the corresponding energy gaps for any set of the crystal parameters. PACS: 73.22.Dj Single particle states; 73.22.Lp Collective excitations; 71.35.Cc Intrinsic properties of excitons; optical absorption spectra. Keywords: quasione-dimensional semiconductors, zero-range potentials method, single-walled carbon nanotubes, exciton, exciton binding energy. 1. Introduction The study of the quasione-dimensional semiconduc- tors with the cylindrical symmetry became an urgent problem as soon as investigations of semiconducting nanotubes had been launched. One of the most important trends of research in this field is the study of optical spec- tra of such systems, which should include the exciton contributions [1–9]. Evidently, the quasione-dimensional large-radius exciton problem can be reduced to the 1D system of two quasi-particles with the potential having Coulomb attraction tail. Due to the parity of the interac- tion potential the exciton states should split into the odd and even series. In [10] we show that for the bare and screened Coulomb interaction potentials the binding en- ergy of even excitons in the ground state well exceeds the energy gap (in the same work we also discuss the factors, which prevent the collapse of single-electron states in iso- lated semiconducting single-walled carbon nanotubes (SWCNTs). But the electron-hole (e-h) interaction poten- tial and so the corresponding exciton binding energies may noticeably depend on the electron and hole charge distributions. So it is worth to ascertain whether the effect of seeming instability of single-electron states near the gap is inherent to the all quasione-dimensional semicon- ductors in vacuum or it maybe takes place only in SWCNTs for the specific localization of electrons (holes) at their surface and weak screening by the bound elec- trons. That is why we consider here the simplest model of the quasione-dimensional semiconductor with the cylin- drical symmetry, namely the linear crystal with two atoms in the unit cell. The electrons (holes) in this crystal are simply localized at its axis. The aim of this work is only a qualitative analysis of the mentioned effect. For study of electron structure of concerned 1D crystal we apply here the zero-range poten- tials (ZRPs) method [11,12] (see Sec. 2). The matter is that results on the band structure and single-electron states, obtained by this method for SWCNTs in [13,14], appeared to be in good accordance with the experimental data and results of ab initio calculations related to the band states. For certainty we use the linear crystal param- eters (the electron bare mass, lattice parameters) taken from works on nanotubes [13,14]. In Sec. 3 we obtain the e-h bare interaction potential and that screened by the crystal band electrons, and then the large-radius exciton spectrum for the linear crystal in vacuum. All these data are used in Sec. 4, where we present results of calcula- tions for the crystal with different lattice periods (it also means different band structures). As it turns out, the bind- ing energy of even excitons in the ground state well ex- ceeds ( � �2 5 times) the energy gap for the linear crystal in vacuum and the screening by the crystal band electrons is negligible. Note, that this result was obtained within the framework of exactly solvable ZRPs model with fea- sible parameters. Therefore, the mentioned instability ef- © V.M. Adamyan and O.A. Smyrnov, 2009 fect may take place not only for the considered simplest case, but, most likely, also for other quasione-dimen- sional isolated semiconductors in vacuum. 2. Single-electron band structure and eigenfunctions of band electrons We have obtained the single-electron states in the lin- ear crystal using the ZRPs method [11,12]. The main point of this method is that the interaction of an electron with atoms or ions of a lattice is described instead of some periodic potentialV ( )r by the sum of Fermi pseudo-poten- tials [11] ( / ) ( )( / )1 � � � � � l l l l� � � (� l l� �| |r r , rl are the points of atoms location, � is a cer- tain fitting parameter) or equivalently by the set of bound- ary conditions imposed on the single-electron wave func- tion at points rl : lim ( )( ) ( )( ) � � � � � l d d l l l � � � � � � � 0 0r r . The electron wave functions satisfy at that the Schr�ding- er equation for a free particle for r r� l . Therefore we seek them for the linear crystal in the form: � � �� � �� ( , ) exp ( ) exp ( n A n B n n n A n A n n n B A B� � � � ��� � ��� � � � ) , �n B (1) where indices A and B denote two monatomic sublattices of the diatomic lattice, �n A n A� �| |r r and �n B n B� �| |r r , n numbers all the sublattices points, � � 2m Eb | | / �, E � 0 is the electron energy and mb is the bare mass. For certainty, following [13,14] we take from now on m mb e� 0 415. and the ZRP parameter � � 2m Eb | | /ion �, where E ion is the ionization energy of an isolated carbon atom. By [13,14], with these � and mb ZRPs method re- produces single-electron spectra of such quasione-dimen- sional structures as SWCNTs within an accuracy of exist- ing experiments. One can take infinite limits for the series in (1) even for the finite crystal, because terms of these se- ries decrease exponentially with increasing of n. According to the ZRPs method the wave functions (1) should satisfy the following boundary conditions at the all sublattices points: lim ( )( ) ( ( ) � � � � � � l i d d l i l i l i � � � �� � � � �� � 0 0r r , (2) here i A B� { , } according to each sublattice. Further we suppose that the linear crystal lies along the z-axis, thus r en A znd� and r en B znd a� �( ) , where e z is the z-axis unit vector, a is the distance between atoms in the unit cell of the crystal and d a� 2 is the distance be- tween the neighbour atoms in each sublattice. Note, that d a� 2 corresponds to the metallic monatomic crystal and for the case d a� 2 the smallest distance between atoms in the crystal is d a a� � . Substituting (1) to (2) and applying the Bloch theorem (A A iqdnn � exp ( ), B B iqdnn � exp ( ), q is the electron quasi-momentum) we get two equations for amplitudes A B, : AQ BQ AQ BQ 1 2 2 1 0 0 � � � � � � , ,* (3) where � �Q q d d qd1 1 2( , ) ln [ cos ] ,� � �� � �cosh (4) Q q nd a iqnd nd a n 2( , ) exp ( | | ) | | .� � � � � � � ��� � � (5) Setting d ja� : Q q a a x d iq x a j j 2 2 1 1 ( , ) exp [ ] exp [ ( )] exp [ ] exp � � � � � � � � � � � [ ( )]d iq x dx j� � � � � � � � � � � 0 1 (6) for each real j � 2. From (3) we get two equations, which define the band structure of the crystal: Q q Q q1 1 2 1 0( , ) | ( , )|� �� � , (7) Q q Q q1 2 2 2 0( , ) | ( , )|� �� � . (8) Equation (7) defines the conduction band and equation (8) defines the valence band (see Sec. 4, Fig. 1). So the electron and hole effective masses can be simply obtained from (7) and (8), respectively. Further, using the Hilbert identity for Green's function of the 3D Helmholtz equation, we obtain the normalized wave functions (1): � � �, ( ) ( , ) exp ( | | ) | | q z zn A q L nd iqnd nd r r e r e � � � � � � ��� � � � � � � � � � � � � � ��� � Q Q nd a iqnd nd a z zn 1 2 exp ( | ( ) | ) | ( ) | � r e r e� � � � � , (9) where L is the crystal length and A q( , )� is the normaliza- tion factor: 504 Fizika Nizkikh Temperatur, 2009, v. 35, No. 5 V.M. Adamyan and O.A. Smyrnov A q d d qd d y ( , ) cos – , / � � ! � � � � " � � �� � � �� 1 2 1 2 cosh sinh and � �y Q Q iqd a d a� � � �1 2 exp [ ]sinh sinh [ ] .� � 3. Exciton spectrum and eigenfunctions. Bare and screened e–h interaction Using the same arguments as in the 3D case one can show (see, for example [10]), that the wave equation for the Fourier transform # of envelope function in the wave packet from products of the electron and hole Bloch func- tions, which represents a two-particle state of large-ra- dius rest exciton in a (quasi)one-dimensional semicon- ductor with period d, is reduced to the following 1D Schr�dinger equation: � # � # � # � � �� � � � � 2 2$ ' ' ( ) ( ) ( ) ( ), , ,z V z z z E E zg� � exc (10) where $ is the e–h reduced effective mass and V z( ) is the e–h interaction potential: V z e x x y y z z z EE dd ( ) (( ) ( ) ( ) ) / � � � � � � � � % 2 1 2 2 1 2 2 1 2 2 1 2 33 % � % � �| ( )| | ( )| , (; , / ; , /u u d d E E zc d v d d � ! � !r r r r1 2 2 2 1 2 3 2 0 d ). Here u c v q, ; , ( )� r are the Bloch amplitudes of the Bloch wave functions � �c v q c v qiqz u, ; , , ; ,( ) exp ( ) ( )r r� of the conduction and valence band electrons of the linear crys- tal, respectively. Using the actual localization of the Bloch amplitudes at the crystal axis, after several Fourier transformations and simplifications we adduce the e-h in- teraction potential to the following form: V z e r r d J k J kr r k k r d r r1 2 4 2 1 2 2 2 1 1 2 1 4 0 1 , ( ) ( ) ( / ) (| � � % % � � z d z z k r d z k r d z | | | | | ) exp | | exp | � � � � � �� � � � & ' ( ) * + � � � � 2 1 1 | exp | | , & ' ( ) * + � � & ' ( ) * + � � ��2 1 k r z dk (11) where J is the Bessel function of the first kind and r1 (r2) is the radius of the electron (hole) wave functions trans- verse localization r r u z dz dD E c v q L D D1 2 2 2 0 2 2 22 2 , , ; ,| ( , )|� � � � � � � � � � � � r r 1 2/ , where r2D is the transverse component of the radius-vec- tor, q d� ! / and � �� � !� 1 2, / d correspond to the conduc- tion and valence bands edges at the energy gap (according to (7) and (8), respectively). Equation (10) with the poten- tial given by (11) defines the spectrum of large-radius exciton in the linear, diatomic crystal if the screening ef- fect by the crystal electrons is ignored. Actually, the screening of the potential (11) by the band electrons is in- significant. Indeed, following the Lindhard method (so-called RPA), to obtain the e–h interaction potential ,( )r , screened by the electrons of linear lattice, let us consider the Poisson equation: � -, ! � �( ) ( ( ) ( ))r r r� �4 ext ind (12) where r is the radius-vector, �ext ( )r is the density of extra- neous charge and � ind ( )r is the charge density induced by the extraneous charge. By (12) the screened e–h interaction potential may be written as: , ! � �( ) ( ( ' ) ( ' )) ( , ' ) ' ,r r r r r r� � 4 3 ext ind E G d (13) where G( , ' ) / ( | ' | )r r r r� �1 4! is Green's function of the 3D Poisson equation. Let E q0( ) and � �, ,( ) exp ( ) ( )q qiqz u0 0 r r� be the band energies and corresponding Bloch wave functions of the crystal electrons and E q( ), �, ( )q r be those in the pres- ence of the extraneous charge. Then � � � ind ( ) [ ( ( ))| ( )| ( ( ))| ( )| ], ,r r r� � �e f E q f E qq q q 2 0 0 2� , (14) where f is the Fermi–Dirac function. Using the trans- verse localization of the Bloch wave functions near the crystal axis, we get in the linear in , approximation: � � ind ( ' , ' ) ( , ; , ', ' ; , 'z e E u zD g q qq q v q E L Dr r2 2 0 2 1 2 2 � � � ) * ( , ) ( ) exp [ ( ' )] * ; , ; , % % � % % u z d z iz q q dz u c q D D v � � , 1 2 2 2r r q D c q Dz u z iz q q' ; ,( ' , ' ) ( ' , ' ) exp [ ' ( ' )] ,r r2 21� � (15) where E E q E qg q q c v; , ' ( ) ( ' )� � . Here and further ,( )z is the e–h interaction potential averaged in E2 over the re- gion of the Bloch wave functions transverse localization and over the lattice period d along the crystal axis. Linear, diatomic crystal: single-electron states and large-radius excitons Fizika Nizkikh Temperatur, 2009, v. 35, No. 5 505 Due to the periodicity of the Bloch amplitudes � ind may be written as: � , � ind ( ' ) ( , ' ; ) ( ' ) * ; , ', ' ; r � � � % % �e N L C q q d E q q u g q qq q v 2 2 , ' ; ,( ' ) ( ' ) exp [ ' ( ' )] ,q c qu iz q qr r�1 � (16) where C q q d u z u z d E d c q D v q D D( , ' ; ) * ( , ) ( , ); , ; , '� 2 1 2 0 2 2 2� �r r r dz and N is the number of unit cells in the crystal. Further, after several transformations we obtain from (13) and (16) the one-dimensional Fourier transform of the potential ,: , , . . ! ( ) ( ) ( ) , ( ) | ( , ; )| ; , k k k k e N C q q k d Eg q q k � � � � � 0 2 2 2 2 1 2 � ! ! / / ~ ( ) sin( / ) d d dqK k kd kd 0 2 2 (17) where , 0 is the Fourier transform of the averaged electro- static potential induced by �ext and ~ ( )K k0 is the modified Bessel function of the second kind averaged over r2D and r' 2D in the region of the Bloch wave functions transverse localization in E2, namely ~ ( ) ( ) (| | | ' | ) 'K k r r K k d d EE D D D rr 0 1 2 2 0 2 2 2 1 2 2 2 1 � � ! r r r r 2 2 20 0 2 D r D iE r ri , ( ) ( ) .� / / % / /0 ! In the long-wave limit we get: | ( , ; )| | ( ; )| , ( ; ) * ; , C q q k d U q d k U q d u k c q E d � 1 � 0 2 2 2 0 1 2 � ( , ) ( , ) .; ,z q u z d dzD v q D Dr r r2 2 22 � � � (18) Using of the Schr�dinger equation for the orthogonal Bloch wave functions �, ( )q r yields U q d i E m z zg q q b c q E d D v q ( ; ) ( , ) ; , ; , * ; , � � � � � 2 0 21 2 2 � �r ( , ) .z d dzD Dr r2 2 (19) Hence, in the long-wave limit the screened quasione- dimensional electrostatic potential induced by a charge e0, distributed with the density: � ! ext ( , ) ( [ / ] [ / ]) ( [ ] [ z e R d z d z d r r D D r2 0 2 2 2 2� � � � % % � 2 2 2 2 2 0 0 0 0 0 1 D R R x x x x x x � � � � � � � � ]), , [ ] , , , , 2 in accordance with (17) and (19), is given by the expres- sion , ! ( ) ( ) sin ( ) ~ ( ) cos ( ) z e r d /k kd/ r K k/r kz/r � 8 1 20 1 2 2 2 1 0 1 1 1 21 1 0 1 0 � � g kr /d kd/ r K k/r dk d ( ) sin ( ) ~ ( ) (20) with g e r m E z d b g q qd d c q� � � � � � � � � � � 3 3 3 3 � � 2 1 2 3 1 1! ! ! � ; ,/ / ; , 3 33 3 3 3 3 3 �v q dq; , . 2 2 (21) According to equation (9) the dimensionless screening parameter gd may be also written as: g e dr m A q A q q d b c v� � � �� � � �� � 16 1 2 2 1 2 2 2 2 1 2� ( , ) ( , ) ( ( ) � � � � ( )) ( , ) ( , ) ( , ) ( , ) / q Q q Q q Q* q Q q d 5 0 1 1 1 2 2 1 2 2 1 ! � � � � % % � � � � � � � � � 3 3 3 3 � % % Q q d Q q Q* q Q* 1 1 1 2 1 2 1 2 1 2 � � � � � � � � ( , ) ( , ) ( , ) �( , ) ( , ) ( , ) ( , ) .� � � � � � � � q d Q q Q q Q q d dq1 2 2 2 2 2 1 2 (22) Note, that �1 and �2 are the implicit functions of q defined by (7) and (8), respectively. It appears, that gd calculated according to (22) for d varying in the interval [ . , ]21 3a a are about 10 6� . 4. Discussion. Stabilization of single-electron states Using Eqs. (7) and (8) we obtained the band structure (see Fig. 1) and the electrons and holes effective masses for the linear crystal of dimers for different values of the ratio j d a� / of its period d and the distance a between at- oms in dimers. Besides, using wave equation (10) and po- tentials (11) and (20) we found the large-radius exciton energy spectrum in the crystal for the bare e–h interaction and e–h interaction screened by the bound electrons of the crystal. We present here results for the crystal with j 456789:;. Contrary to the single-band metallic crystal with j � 2, the crystals with j � 2 are semiconductors with band gaps varying from zero to the difference between the electron levels in an isolated dimer. Particularly, the crys- 506 Fizika Nizkikh Temperatur, 2009, v. 35, No. 5 V.M. Adamyan and O.A. Smyrnov tals with j � 2 001. and realistic values of a and � (as in nanotubes and some 1D polymer chains) are narrow-gap semiconductors, in which excitons may possess binding energies about �10 meV, but the crystals with j � 2 1. are already wide-gap ( �1eV) semiconductors with strongly bound e–h pairs, and the crystals with j � 3 are almost flat band semiconductors, but their electrons and holes at the energy gaps (q d� ! / ) still have the finite effective masses (these electrons and holes form the excitons in the crystals). For certainty, the distance a we have chosen equal to the graphite in-plane parameter 0 142. nm. The ZRP interaction parameter � � �11 01 1. nm corresponds to the ionization energy of an isolated carbon atom ( .E ion eV�11 255 ). As one can see from table 1 the obtained from (22) dimensionless screening parameter gd �� 1 for the all considered values of j. So, it turns out that the screening of the e–h interaction potential by the band electrons in the linear, diatomic crystal may be ignored. This result could be expected since the considered model of linear crystal is close to that of the electron gas confined to a cy- lindrical well. In the latter case, for which the separation of the angular variables takes place, the states with differ- ent quantum numbers m of the angular momentum play the role of electron bands. Accordingly, the matrix ele- ment | | / | |< � � = c vz from (21) for the direct transitions between bands with different m appears to be identically equal to zero. This is why only the binding energies of excitons with unscreened interaction potential are listed in table 1. To obtain estimates of the main linear crystal charac- teristics we considered several limiting cases. In the case of d a�� (or j �� 1) and a � const equations (7) and (8) can be reduced to � � �� � �1 2 1 2 0, ,exp [ ] /� a a , thus bands be- come flat (�1 2, do not depend on q) and the band gap tends to the finite value ( / )( )� 2 2 2 1 22mb � �� (for a � = 0.142 nm it is about 6.3 eV), hence the reduced effective mass and exciton binding energy tend to infinity, while the exciton r a d i u s r eexc � � 2 2/ $ t e n d s t o z e r o . T h e r e f o r e , the large-radius exciton theory is actually appropriate ( )r aexc �� for excitons in the linear, diatomic crystal only when its period d runs the interval ( , . )2 2 4a a (e.g., rexc is � 9a for j � 21. and � 2a for j � 2 4. ). If d � const, but a 0, the conduction band moves to the region of posi- tive energies and at some critical value of a disappears, while the valence band shifts correspondingly to the re- gion of deep negative energies. Table 1 shows that the ground-state exciton binding energies for the linear, diatomic crystals with any value of the ratio j are greater than the corresponding energy gaps. Note, that according to the same calculations, but with the bare mass m mb e� , the ground-state exciton binding en- ergy for the linear crystal in vacuum appears significantly greater than the energy gap. We should note also, that the ground-state binding energies of excitons in the linear crystal with different periods d in vacuum, calculated us- ing the 1D analogue of the Ohno potential [15] instead of potential (11), remain greater than the corresponding energy gaps. Particularly, for d a� 2 3. calculations with the 1D unscreened Ohno potential with the energy parameter U taken from [16] (U �11 3. eV) and [17] (U �16 eV) give the ground-state exciton binding ener- gies �0;even � 5 90. eV and �0 763; .even eV� , respectively, while Eg � 3 31. eV (see table 1). Thus, all calculations Linear, diatomic crystal: single-electron states and large-radius excitons Fizika Nizkikh Temperatur, 2009, v. 35, No. 5 507 Table 1. Band gaps Eg and reduced effective masses $ according to (7), (8); radii of the electrons and holes transverse localization r1 and r2, respectively; screening parameter gd according to (22) and the exciton binding energies � of the even and odd series for the linear, diatomic crystal according to equation (10) with potential (11) for different values of the ratio j d a� / . j Eg , eV $ ( )me r 1 , nm r 2 , nm gd ( )10 6� �0; ,even eV �1; ,odd eV �0;even /Eg 2.1 1.4422 0.041 0.070 0.0611 0.7235 –6.90 –0.5939 4.7845 2.3 3.3146 0.125 0.080 0.0569 2.4716 –8.9992 –2.0631 2.715 2.5 4.403 0.2199 0.088 0.0549 2.7036 –9.5352 –3.4812 2.1656 3 5.6281 0.5665 0.0994 0.0551 1.1206 –9.6588 –5.8421 1.7162 –4 –6 –8 –10 –12 –14 –16 0 0.2 0.4 0.6 0.8 1.0 q E , eV Fig. 1. The band structure of the linear crystal with parame- ters: j � 2 1. (dashed line), j � 2 5. (dot-dashed line) and j � 3 (solid line); q in units of ! / d. The upper and lower bands cor- respond to equation (7) and (8), respectively. made on the base of solvable zero-range potentials model indicate the instability of the single-electron states in the vicinity of the energy gap with respect to the formation of excitons. This might be a shortage of this model, but it is worth mentioning that results obtained on one-particle states in real 1D systems, like SWCNTs [13,14], within its framework agree with existing experimental data in limits of accuracy of the latter. The stability of single-electron states of 1D semicon- ductors with respect to the exciton formation in vacuum can be explained by bringing multi-particle contributions into the picture. With the advent of some number of excitons in the quasione-dimensional crystal the addi- tional screening appears, which is caused by a rather great polarizability of excitons in the longitudinal electric field. This collective contribution of born excitons into the crystal permittivity returns the lowest exciton binding energy �0;even into the energy gap and so blocks further spontaneous transitions to the exciton states. To show this let us consider the model of linear crystal immersed into the gas of excitons with dielectric constant .exc confined to the region of linear crystal carriers transverse localiza- tion, namely: cylinder with radius r1 and axis coinciding with that of linear crystal (from now on, for estimates, we assume that electron and hole have the same transverse localization radius). In this case it is easy to show that the e–h interaction potential is given by: , ! . ( ) sin ( / ) cos ( / ) z e r d kd r kz r k � % % � � 16 2 1 2 2 1 2 2 1 1 4 0 exc K k I k k K k I k K k I k dk1 1 0 1 1 0 ( ) ( ) ( ( ) ( ) ( ) ( )) , .exc � � � �� � � �� (23) where I i and K i are the modified Bessel functions of the order i of the first and second kind, respectively. Further, like in [18], we use the known elementary relation be- tween the permittivity of exciton gas and its polarizability � in the direction of linear crystal . !� �exc � � � < = ��1 4 2 2 0 2 0 , | | | | ,e n k kk > >r � � where n is the bulk concentration of excitons, >0 and �0 are the exciton eigenfunction and binding energy, which correspond to the ground state, and >k and �k are those, which correspond to the all excited states of exciton. Thus, the upper and lower limits for � are: 2 2 2 2 0 1 0 1 2 2 0 1 0 2e n e n k k � � � �� < = / / � < = � � �| | | | | | | |> > > >r r� e n2 0 1 0 2 0 � �� < =| | | | ,> >r where >1 and �1 correspond to the lowest excited exciton state. Hence, one can obtain the upper and lower limits for n: . ! exc even odd� � # 3 3 33 3 3 33 �� � � 1 4 2 0 1 2 2 0 2� �; ; | ( )| e z z dz 1 0 1 2 0 1 1 4 2 / / � � # # 3 3�� � n n e z z z dz , ( ) ( ) ; ;. ! exc even odd� � 33 3 3 33 �2 , (24) where each # is the component of Fourier transform of the corresponding exciton envelope function, it depends only on the distance z between the electron and hole. At that, #0 is the even solution of (10) with potential (23), which corresponds to the exciton ground state and satisfies the boundary condition # �' ( )0 0, and #1 is the odd solution of the same equation, which corresponds to the lowest ex- cited exciton state and satisfies the boundary condition # �( )0 0. Varying .exc in (23) substituted into the wave equation (10) one can match �0;even to the forbidden band width. Further, �1;odd can be obtained from the same equation with the fixed .exc and with the corresponding boundary condition. All these magnitudes allow to calculate from (24) the rough upper and lower limits for the critical con- centration of born excitons nc , which is sufficient to re- turn �0;even into the energy gap. Further, knowing nc we can calculate the shift of the forbidden band edges, which move apart due to the transformation of some single-elec- tron states into excitons. This results in the enhancement of energy gap � ! $ E n g c � ( ~ )� 2 2 (25) like in [19] and [20]. Here ~n n rc c� ! 1 2 is the linear critical concentration of excitons, and r1 is the radius of electron wave functions transverse localization at the linear crys- tal axis. In accordance with (24) the described model yields ~nc � �180 1$m ( � 3% of the atoms number in the crystal) and � �400 1$m ( � 7%) for the linear crystal with j � 2 1. and j � 2 3. , respectively, while by (25) the corresponding �Eg are � 300 meV and � 500 meV in the same order. Here, however, we should mention that for SWCNTs both the measured in [19,20] and estimated in the same manner [18] values of �E Eg g/ appeared to be two–four times less. Note, finally, that the instability of the single-electron states weakens or disappears for linear crystals immersed into dielectric media. As it was shown in [9] by the exam- ple of the poly-para-phenylenevinylene 1D chain or in [16,17,21] by the example of SWCNTs the environmental effect may substantially decrease the excitons binding en- ergies. Indeed, for the linear crystal in media even with 508 Fizika Nizkikh Temperatur, 2009, v. 35, No. 5 V.M. Adamyan and O.A. Smyrnov permittivity about . � 2–3 (e.g., like in [16] or [17]) the ground-state exciton binding energy becomes smaller than the energy gap. This work was supported by the Ministry of Education and Science of Ukraine, Grant#0106U001673. 1. M.J. O'Connell, S.M. Bachilo, C.B. Huffman, V.C. Moore, M.S. Strano, E.H. Haroz, K.L. Rialon, P.J. Boul, W.H. Noon, C. Kittrell, J. Ma, R.H. Hauge, R.B. Weisman, and R.E. Smalley, Science 297, 593 (2002). 2. S.M. Bachilo, M.S. Strano, C. Kittrell, R.H. Hauge, R.E. Smalley, and R.B. Weisman, Science 298, 2361 (2002). 3. M. Ichida, S. Mizuno, Y. Saito, H. Kataura, Y. Achiba, and A. Nakamura, Phys. Rev. B65, 241407(R) (2002). 4. F. Wang, G. Dukovic, L.E. Brus, and T.F. Heinz, Science 308, 838 (2005). 5. E. Chang, G. Bussi, A. Ruini, and E. Molinari, Phys. Rev. Lett. 92, 196401 (2004). 6. T. Ando, J. Phys. Soc. Jpn. 66, 1066 (1997). 7. A. Jorio, G. Dresselhaus, and M.S. Dresselhaus (eds.), Carbon Nanotubes. Advanced Topics in the Synthesis, Structure, Properties and Applications, Springer-Verlag, Berlin, Heidelberg (2008). 8. S. Abe, J. Photopolym. Sci. Technol. 6, 247 (1993). 9. A. Ruini, M.J. Caldas, G. Bussi, and E. Molinari, Phys. Rev. Lett. 88, 206403 (2002). 10. V.M. Adamyan and O.A. Smyrnov, J. Phys. A: Math. Theor. 40, 10519 (2007). 11. S. Albeverio, F. Gesztesy, R. H�egh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics. Texts and Monographs in Physics, Springer, New York (1988). 12. Yu.N. Demkov and V.N. Ostrovskii, Zero-Range Poten- tials and Their Applications in Atomic Physics, Plenum, New York (1988). 13. S.V. Tishchenko, Fiz. Nizk. Temp. 32, 1256 (2006) [Low Temp. Phys. 32, 953 (2006)]. 14. V. Adamyan and S. Tishchenko, J. Phys.: Condens. Matter 19, 186206 (2007). 15. K. Ohno, Theor. Chim. Acta 2, 219 (1964). 16. J. Jiang, R. Saito, Ge.G. Samsonidze, A. Jorio, S.G. Chou, G. Dresselhaus, and M.S. Dresselhaus, Phys. Rev. B75, 035407 (2007). 17. R.B. Capaz, C.D. Spataru, S. Ismail-Beigi, and S.G. Louie, Phys. Rev. B74, 121401(R) (2006). 18. V.M. Adamyan, O.A. Smyrnov, and S.V. Tishchenko, J. Phys.: Conf. Series 129, 012012 (2008). 19. Y. Ohno, S. Iwasaki, Y. Murakami, S. Kishimoto, S. Maruyama, and T. Mizutani, Phys. Rev. B73, 235427 (2006). 20. O. Kiowski, S. Lebedkin, F. Hennrich, S. Malik, H. R�sner, K. Arnold, C. S�rgers, and M.M. Kappes, Phys. Rev. B75, 075421 (2007). 21. V. Perebeinos, J. Tersoff, and P. Avouris, Phys. Rev. Lett. 92, 257402 (2004). Linear, diatomic crystal: single-electron states and large-radius excitons Fizika Nizkikh Temperatur, 2009, v. 35, No. 5 509

Схожі ресурси