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

0.16 MB·English

by  Vadym M. Adamyan

Linear, diatomic crystal: single-electron states and
large-radius excitons
Vadym M. Adamyan∗, Oleksii A. Smyrnov†
9
Department of Theoretical Physics, Odessa I. I. Mechnikov National University,
0
0 2 Dvoryanskaya St., Odessa 65026, Ukraine
2
n January 2, 2009
a
J

2
Abstract

] Thelarge-radius excitonspectrum inalinear crystalwith twoatoms intheunitcellwas obtained using
r
e the single-electron eigenfunctions and the band structure, which were found by the zero-range potentials
h (ZRPs)method. Theground-stateexciton bindingenergies for thelinear crystalin vacuumappeared tobe
t larger than thecorresponding energy gaps for any set of the crystal parameters.
o
. PACS number(s): 73.22.Dj, 73.22.Lp, 71.35.Cc
t
a
m
1 Introduction
-
d
n
Thestudyofthequasione-dimensionalsemiconductorswiththecylindricalsymmetrybecameanurgentproblem
o
as soon as investigations of semiconducting nanotubes had been launched. One of the most important trends
c
[ of research in this field is the study of optical spectra of such systems, which should include the exciton

contributions [1]-[9]. Evidently, the quasione-dimensional large-radius exciton problem can be reduced to the
3
1D system of two quasi-particles with the potential having Coulomb attraction tail. Due to the parity of the
v
9 interaction potential the exciton states should split into the odd and even series. In [10] we show that for the
4 bare and screened Coulomb interaction potentials the binding energy of even excitons in the ground state well
4 exceeds the energy gap (in the same work we also discuss the factors, which prevent the collapse of single-
0
electron states in isolated semiconducting single-walled carbon nanotubes (SWCNTs). But the electron-hole
.
4 (e-h) interaction potential and so the corresponding exciton binding energies may noticeably depend on the
0
electron and hole charge distributions. So it is worth to ascertain whether the effect of seeming instability of
8
single-electron states near the gap is inherent to the all quasione-dimensional semiconductors in vacuum or it
0
: maybe takes place only in SWCNTs for the specific localization of electrons (holes) at their surface and weak
v
screeningbythe boundelectrons. Thatiswhyweconsiderherethe simplestmodelofthe quasione-dimensional
i
X semiconductor with the cylindrical symmetry, namely the linear crystal with two atoms in the unit cell. The
r electrons (holes) in this crystal are simply localized at its axis.
a
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 potentials (ZRPs) method [11],[12] (see section 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 parameters (the electron bare mass, lattice
parameters) taken from works on nanotubes [13],[14]. In section 3 we obtain the e-h bare interaction potential
andthatscreenedbythecrystalbandelectrons,andthenthelarge-radiusexcitonspectrumforthelinearcrystal
in vacuum. All these data are used in section 4, where we present results of calculations for the crystal with
different lattice periods (it also means different band structures). As it turns out, the binding energy of even
excitonsinthegroundstatewellexceeds( 2 5times)theenergygapforthelinearcrystalinvacuumandthe
∼ −
screening by the crystal band electrons is negligible. Note, that this result was obtained within the framework
of exactly solvable ZRPs model with feasible parameters. Therefore, the mentioned instability effect may take
place not only for the considered simplest case, but, most likely, also for other quasione-dimensional isolated
semiconductors in vacuum.
∗E-mail: [email protected]
†E-mail: [email protected]
1
2 Single-electron band structure and eigenfunctions of band elec-
trons
We have obtained the single-electron states in the linear crystal using the zero-range potentials (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 potential V(r) by the sum of Fermi pseudo-potentials [11]
(1/α) δ(ρ )(∂/∂ρ)ρ (ρ = r r , r are the points of atoms location, α is a certain fitting parameter) or
equivalenltlylby the sletlof bloun|da−ry cl|ondlitions imposed on the single-electron wave function ψ at points r :
l
P
d
lim (ρ ψ)(r)+α(ρ ψ)(r) =0.
l l
ρl→0(cid:26)dρl (cid:27)
The electron wave functions satisfy at that the Schr¨odinger equation for a free particle for r = r . Therefore
l
6
we seek them for the linear crystal in the form:
∞ ∞
exp( κρA) exp( κρB)
ψ ρA,ρB = A − n + B − n , (2.1)
n n n ρA n ρB
n=−∞ n n=−∞ n
(cid:0) (cid:1) X X
where indices A and B denote two monatomic sublattices of the diatomic lattice, ρA = r rA and ρB =
n − n n
r rB , n numbers all the sublattices points, κ = 2m E /~, E < 0 is the electron energy and m is the
− n b| | (cid:12) (cid:12) b
bare mass. For certainty, following [13],[14] we take from now on m 0.415m and t(cid:12)he ZRP(cid:12) parameter
b e
α(cid:12)(cid:12) = 2(cid:12)(cid:12)mb Eion /~, where Eion is the ionization enerpgy of an isolated car≃bon atom. By [13],[14], with these α
| |
and m ZRPs method reproduces single-electron spectra of such quasione-dimensional structures as SWCNTs
b
p
withinanaccuracyofexistingexperiments. Onecantakeinfinite limits forthe seriesin(2.1)evenforthe finite
crystal, because terms of these series decrease exponentially with increasing of n.
Accordingto the ZRPsmethod the wavefunctions (2.1) shouldsatisfy the followingboundaryconditionsat
the all sublattices points:
d
lim ρiψ (r)+α ρiψ (r) =0, (2.2)
ρil→0(cid:26)dρil l l (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
here i= A,B according to each sublattice.
Furth{er we}suppose that the linear crystal lies along the z-axis, thus rA = nde and rB = (nd+a)e ,
n z n z
where e is the z-axis unit vector, a is the distance between atoms in the unit cell of the crystal and d > 2a
z
is the distance between the neighbour atoms in each sublattice. Note, that d=2a corresponds to the metallic
monatomic crystal and for the case d<2a the smallest distance between atoms in the crystal is d a<a.
−
Substituting (2.1) to (2.2) and applying the Bloch theorem (A = Aexp(iqdn), B = Bexp(iqdn), q is the
n n
electron quasi-momentum) we get two equations for amplitudes A,B:
AQ +BQ =0,
1∗ 2 (2.3)
AQ +BQ =0,
(cid:26) 2 1
where
1
Q (κ,q)=α ln(2[coshκd cosqd]), (2.4)
1
− d −
∞
exp( κnd+a +iqnd)
Q (κ,q)= − | | . (2.5)
2
nd+a
n=−∞ | |
X
Setting d=ja:
1
1 exp[ κa] xj−2exp[κa]
Q (κ,q)= − + dx (2.6)
2 a 1 xjexp[d(iq κ)] exp[d(iq+κ)] xj
Z0 (cid:18) − − − (cid:19)
for each real j >2.
From (2.3) we get two equations, which define the band structure of the crystal:
Q (κ ,q) Q (κ ,q) =0, (2.7)
1 1 2 1
−| |
Q (κ ,q)+ Q (κ ,q) =0. (2.8)
1 2 2 2
| |
2
Equation(2.7)definesthe conductionbandandequation(2.8)definesthevalenceband(seesection4,figure1).
So the electron and hole effective masses can be simply obtained from (2.7) and (2.8), respectively.
Further, using the Hilbert identity for Green’s function of the 3D Helmholtz equation, we obtain the nor-
malized wave functions (2.1):
A(κ,q) ∞ exp( κr nde +iqnd)
ψ (r)= − | − z|
κ,q √L n=−∞ |r −ndez|
X (2.9)
Q ∞ exp( κr (nd+a)e +iqnd)
1 z
− | − | ,
−Q2 n=−∞ |r −(nd+a)ez| !
X
where L is the crystal length and A(κ,q) is the normalization factor:
1/2
1 κdcoshκd cosqd
A(κ,q)= − ,
2 π sinhκd y
(cid:18) −ℜ (cid:19)
and
Q
1
y = (exp[ iqd]sinhκa+sinhκ[d a]).
Q − −
2
3 Exciton spectrum and eigenfunctions. Bare and screened e-h in-
teraction
Using the same arguments as in the 3D case one can show (see, for example [10]), that the wave equation
for the Fourier transform φ of envelope function in the wave packet from products of the electron and hole
Bloch functions, which represents a two-particle state of large-radius rest exciton in a (quasi)one-dimensional
semiconductor with period d, is reduced to the following 1D Schr¨odinger equation:
~2
′′
φ (z)+V(z)φ(z)= φ(z), =E E , <z < , (3.1)
exc g
− 2µ E E − −∞ ∞
where µ is the e-h reduced effective mass and V(z) is the e-h interaction potential:
e2
V(z)=
− ((x x )2+(y y )2+(z+z z )2)1/2
EZd3EZd3 1− 2 1− 2 1− 2
u (r )2 u (r )2dr dr ,
c;κ,π/d 1 v;κ,π/d 2 1 2
×| | | |
Ed =E (0<z <d).
3 2×
Here u (r) are the Bloch amplitudes of the Bloch wave functions ψ (r) = exp(iqz)u (r) of the
c,v;κ,q c,v;κ,q c,v;κ,q
conduction and valence band electrons of the linear crystal, respectively. Using the actual localization of the
Bloch amplitudes at the crystal axis, after several Fourier transformations and simplifications we adduce the
e-h interaction potential to the following form:
∞
4e2r2 J (k)J (kr /r ) k
V (z)= 1 1 1 2 1 (d z + d+z 2z )
r1,r2 − r d2 k4 r | − | | |− | |
2 Z0 (cid:18) 1 (3.2)
k k k
+exp d z +exp d+z 2exp z dk,
−r | − | −r | | − −r | |
(cid:20) 1 (cid:21) (cid:20) 1 (cid:21) (cid:20) 1 (cid:21)(cid:19)
where J is the Bessel function of the first kind and r (r ) is the radius of the electron (hole) wave functions
1 2
transverse localization
1/2
L
r = 2 r2 u (z,r )2dzdr ,
1,2  2D | c,v;κ,q 2D | 2D
Z Z
E2 0
 
where r is the transverse component of the radius-vector, q = π/d and κ = κ (π/d) correspond to the
2D 1,2
conductionandvalencebandsedgesattheenergygap(accordingto(2.7)and(2.8),respectively). Equation(3.1)
3
with the potential given by (3.2) defines the spectrum of large-radius exciton in the linear, diatomic crystal if
thescreeningeffectbythecrystalelectronsisignored. Actually,thescreeningofthepotential(3.2)bytheband
electrons is insignificant.
Indeed,followingtheLindhardmethod(so-calledRPA),toobtainthee-hinteractionpotentialϕ(r),screened
by the electrons of linear lattice, let us consider the Poisson equation:
∆ϕ(r)=4π ρext(r)+ρind(r) , (3.3)
−
wherer istheradius-vector,ρext(r)isthedensityofe(cid:0)xtraneouschargea(cid:1)ndρind(r)isthechargedensityinduced
by the extraneous charge.
By (3.3) the screened e-h interaction potential may be written as:
ϕ(r)=4π ρext(r′)+ρind(r′) G(r,r′)dr′, (3.4)
Z
E3 (cid:0) (cid:1)
where G(r,r′)=1/(4π r r′ ) is Green’s function of the 3D Poisson equation.
Let E0(q) and ψ0 (|r)−= e|xp(iqz)u0 (r) be the band energies and corresponding Bloch wave functions of
κ,q κ,q
the crystal electrons and E(q), ψ (r) be those in the presence of the extraneous charge. Then
κ,q
ρind(r)= e f(E(q))ψ (r)2 f(E0(q))ψ0 (r)2 , (3.5)
− | κ,q | − | κ,q |
q
X(cid:2) (cid:3)
where f is the Fermi-Dirac function. Using the transverse localization of the Bloch wave functions near the
crystal axis, we get in the linear in ϕ approximation:
L
1
ρind(z′,r′2D)=−e2Xq,q′ Eg;q,q′ Z0 EZ2 uv;κ2,q′(z,r2D)u∗c;κ1,q(z,r2D)dr2Dϕ(z)exp[iz(q′−q)]dz (3.6)
×u∗v;κ2,q′(z′,r′2D)uc;κ1,q(z′,r′2D)exp[iz′(q−q′)],
′
where Eg;q,q′ = Ec(q) Ev(q ). Here and further ϕ(z) is the e-h interaction potential averaged in E2 over the
−
region of the Bloch wave functions transverse localization and over the lattice period d along the crystal axis.
Due to the periodicity of the Bloch amplitudes ρind may be written as:
e2N C(q,q′;d)
ρind(r′)= ϕ(q q′)
− L q,q′ Eg;q,q′ − (3.7)
X
×u∗v;κ2,q′(r′)uc;κ1,q(r′)exp[iz′(q−q′)],
where
d
C(q,q′;d)= u∗c;κ1,q(z,r2D)uv;κ2,q′(z,r2D)dr2Ddz
Z Z
0 E2
and N is the number of unit cells in the crystal.
Further,after severaltransformationswe obtainfrom(3.4) and(3.7) the one-dimensionalFouriertransform
of the potential ϕ:
ϕ (k)
0
ϕ(k)= ,
ε(k)
π/d (3.8)
e2N2 C(q,q k;d)2 2sin(kd/2)
ε(k)=1+ | − | dqK (k) ,
2π2 −πZ/d Eg;q,q−k 0 kd
e
where ϕ is the Fourier transform of the averaged electrostatic potential induced by ρext and K (k) is the
0 0
modifiedBesselfunctionofthesecondkindaveragedoverr andr′ intheregionoftheBlochwavefunctions
2D 2D
transverse localization in E2, namely e
1
K (k)= K (k r r′ )dr dr′ ,
0 (πr r )2 0 | || 2D− 2D| 2D 2D
1 2
EZr21EZr22
e
Eri =(0 r r ) (0 β 2π).
2 ≤ 2D ≤ i × ≤ ≤
4
In the long-wave limit we get:
C(q,q k;d)2 U(q;d)2k2,
| − |k→0 ≈| |
d
∂ (3.9)
U(q;d)= u∗ (z,r ) u (z,r )dr dz.
c;κ1,q 2D ∂q v;κ2,q 2D 2D
Z Z
0 E2
Using of the Schr¨odinger equation for the orthogonalBloch wave functions ψ (r) yields
κ,q
d
i~2 ∂
U(q;d)= ψ∗ (z,r ) ψ (z,r )dr dz. (3.10)
E m c;κ1,q 2D ∂z v;κ2,q 2D 2D
g;q,q b Z Z
0 E2
Hence, in the long-wavelimit the screened quasione-dimensionalelectrostatic potential induced by a charge
e , distributed with the density:
0
e
ρext(z,r )= 0 (Θ[z+d/2] Θ[z d/2])(Θ[r ] Θ[r R]), R>0,
2D πR2d − − 2D − 2D−
0, x<x ,
Θ[x x ]= 0
− 0 1, x>x0,
(cid:26)
in accordance with (3.8) and (3.10), is given by the expression
∞
8e r (1/k2)sin2(kd/2r )K (k/r )cos(kz/r )
0 1 1 0 1 1
ϕ(z)= dk (3.11)
πd2 1+g (kr /d)sin(kd/2r )K (k/r )
Z d 1 1 0 1
0 e
with e
π/d
e~2 2 1 ∂ 2
g = ψ ψ dq. (3.12)
d πr m E3 c;κ1,q ∂z v;κ2,q
(cid:18) 1 b(cid:19) −πZ/d g;q,q (cid:12)(cid:12)(cid:28) (cid:12)(cid:12) (cid:12)(cid:12) (cid:29)(cid:12)(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
According to equation (2.9) the dimensionless screen(cid:12)ing param(cid:12) ete(cid:12)r g may(cid:12)be also written as:
d
π/d
16e 2 A2(κ ,q)A2(κ ,q)
g = m c 1 v 2
d ~dr b (κ2(q) κ2(q))5
(cid:18) 1(cid:19) Z0 2 − 1
κ2
Q (κ ,q)Q (κ ,q)
1 1 1 2
×(cid:12) 1+ Q∗(κ ,q)Q (κ ,q) Q1(κ,q)dκ (3.13)
(cid:12)(cid:18) 2 1 2 2 (cid:19)Z
(cid:12) κ1
(cid:12)(cid:12) κ2 κ2 2
(cid:12)Q1(κ1,q) ∗ Q1(κ2,q)
Q (κ,q)dκ Q (κ,q)dκ dq.
− Q∗(κ ,q) 2 − Q (κ ,q) 2 (cid:12)
2 1 Z 2 2 Z (cid:12)
κ1 κ1 (cid:12)
(cid:12)
(cid:12)
Note, that κ1 and κ2 are the implicit functions of q defined by (2.7) and (2.8), respectivel(cid:12)y.
It appears, that g calculated according to (3.13) for d varying in the interval [2.1a,3a] are about 10−6.
d
4 Discussion. Stabilization of single-electron states
Usingequations(2.7)and(2.8)weobtainedthebandstructure(seefigure1)andtheelectronsandholeseffective
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 atoms in dimers. Besides, using wave equation (3.1) and potentials (3.2) and (3.11) we found the
large-radiusexciton 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 [2.1,3]. Contrary to the
∈
single-bandmetalliccrystalwithj =2,thecrystalswithj >2aresemiconductorswithbandgapsvaryingfrom
zero to the difference between the electron levels in an isolated dimer. Particularly,the crystals with j =2.001
and realistic values of a and α (as in nanotubes and some 1D polymer chains) are narrow-gapsemiconductors,
5
-4
-6
-8
V
e -10
E,
-12
-14
-16
0.0 0.2 0.4 0.6 0.8 1.0
q
Figure 1: The band structure of the linear crystal with parameters: j =2.1 (dashed line), j =2.5 (dot-dashed
line)andj =3(solidline);qinunitsofπ/d. Theupperandlowerbandscorrespondtoequation(2.7)and(2.8),
respectively.
Table 1: Band gaps E and reduced effective masses µ according to (2.7), (2.8); radii of the electrons and
g
holes transverselocalizationr andr , respectively;screeningparameterg accordingto (3.13) andthe exciton
1 2 d
binding energies of the even and odd series for the linear, diatomic crystal according to equation (3.1) with
E
potential (3.2) for different values of the ratio j =d/a.
j E (eV) µ (m ) r (nm) r (nm) g (10−6) (eV) (eV) /E
g e 1 2 d 0;even 1;odd 0;even g
E E E
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
in which excitons may possess binding energies about 10 meV, but the crystals with j = 2.1 are already
∼
wide-gap ( 1 eV) 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 = π/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 α = 11.01 nm−1
corresponds to the ionization energy of an isolated carbon atom (E =11.255 eV).
ion
As one can see from table 1 the obtained from (3.13) dimensionless screening parameter g 1 for the all
d
≪
consideredvalues of j. So, it turns out that the screening of the e-h interactionpotential 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 cylindrical well. In the latter case, for which the
separation of the angular variables takes place, the states with different quantum numbers m of the angular
momentumplay the roleofelectronbands. Accordingly,the matrixelement ψ ∂/∂z ψ from(3.12)forthe
c v
|h | | i|
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 characteristics we considered several limiting cases. In the
caseofd a(orj 1)anda=constequations(2.7)and(2.8)canbereducedtoα κ exp[ κ a]/a=0,
1,2 1,2
thus band≫s become≫flat(κ do notdepend onq)andthe bandgaptends to the finit−e valu∓e (~2/−2m )(κ2 κ2)
1,2 b 2− 1
(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 radius r ~2/µe2 tends to zero. Therefore, the large-radius exciton theory is
exc
∼
actually appropriate (r a) for excitons in the linear, diatomic crystal only when its period d runs the
exc
≫
interval (2a,2.4a) (e.g., r is 9a for j =2.1 and 2a for j =2.4). If d=const, but a 0, the conduction
exc
∼ ∼ →
band movesto the regionofpositive energies andatsome criticalvalue ofa disappears,while the valence band
shifts correspondingly to the region of deep negative energies.
Table1showsthatthe ground-stateexcitonbindingenergiesforthe linear,diatomiccrystalswithanyvalue
of the ratio j are greater than the corresponding energy gaps. Note, that according to the same calculations,
but with the bare mass m = m , the ground-state exciton binding energy for the linear crystal in vacuum
b e
6
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 using the 1D analogue of the
Ohnopotential[15]insteadofpotential(3.2), remaingreaterthanthe correspondingenergygaps. Particularly,
for d=2.3a 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 energies = 5.90 eV and
0;even
E
= 7.63 eV, respectively, while E = 3.31 eV (see table 1). Thus, all calculations made on the base of
0;even g
E
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.
Thestabilityofsingle-electronstatesof1Dsemiconductorswithrespecttothe excitonformationinvacuum
can be explained by bringing multi-particle contributionsinto the picture. With the adventof some number of
excitonsinthe quasione-dimensionalcrystalthe additionalscreeningappears,whichiscausedby arathergreat
polarizability of excitons in the longitudinal electric field. This collective contribution ofborn excitons into the
crystal permittivity returns the lowest exciton binding energy into the energy gap and so blocks further
0;even
E
spontaneoustransitionsto the excitonstates. To showthis let us considerthe model oflinearcrystalimmersed
into the gas of excitons with dielectric constant ε confined to the region of linear crystal carriers transverse
exc
localization, namely: cylinder with radius r and axis coinciding with that of linear crystal (from now on, for
1
estimates,weassumethatelectronandholehavethe sametransverselocalizationradius). Inthiscaseitiseasy
to show that the e-h interaction potential is given by:
∞
16e2r sin2(kd/2r )cos(kz/r )
1 1 1
ϕ(z)=
πd2 ε k4
exc
Z0 (4.1)
2K (k)I (k)
1 1
1 dk,
× − k(ε K (k)I (k)+K (k)I (k))
(cid:18) exc 0 1 1 0 (cid:19)
where I and K are the modified Bessel functions of the order i of the first and second kind, respectively.
i i
Further, like in [18], we use the known elementary relation between the permittivity of exciton gas and its
polarizability α in the direction of linear crystal
Ψ r Ψ 2
ε =1+4πα, α=2e2n |h 0| | ki| ,
exc
0 k
k E −E
X
where n is the bulk concentration of excitons, Ψ and are the exciton eigenfunction and binding energy,
0 0
E
which correspond to the ground state, and Ψ and are those, which correspond to the all excited states of
k k
E
exciton. Thus, the upper and lower limits for α are:
2e2n 2e2n 2e2n
Ψ r Ψ 2 α Ψ r Ψ 2 = Ψ r2 Ψ ,
0 1 0 k 0 0
|h | | i| ≤ ≤ |h | | i| |h | | i|
0 1 0 1 0 1
E −E E −E k E −E
X
where Ψ and correspond to the lowest excited exciton state. Hence, one can obtain the upper and lower
1 1
E
limits for n:
∞ −1
ε 1
exc− E0;even−E1;odd z2 φ (z)2dz n
4π 2e2 (cid:12) | 0 | (cid:12) ≤
(cid:12)−Z∞ (cid:12)
(cid:12) (cid:12) (4.2)
(cid:12)(cid:12) (cid:12)(cid:12) ∞ −2
(cid:12) (cid:12) εexc 1 0;even 1;odd
− E −E zφ (z)φ (z)dz ,
≤ 4π 2e2 (cid:12) 0 1 (cid:12)
(cid:12)−Z∞ (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where eachφ is the component of Fourier transformof the correspondingexci(cid:12)ton envelope funct(cid:12)ion, it depends
(cid:12) (cid:12)
onlyonthedistancezbetweentheelectronandhole. Atthat,φ istheevensolutionof(3.1)withpotential(4.1),
0
′
whichcorrespondstothe excitongroundstateandsatisfiestheboundaryconditionφ(0)=0,andφ isthe odd
1
solution of the same equation, which correspondsto the lowestexcited excitonstate and satisfies the boundary
condition φ(0)=0.
Varying ε in (4.1) substituted into the wave equation (3.1) one can match to the forbidden band
exc 0;even
E
width. Further, can be obtained from the same equation with the fixed ε and with the corresponding
1;odd exc
E
7
boundary condition. All these magnitudes allow to calculate from (4.2) the rough upper and lower limits for
the critical concentration of born excitons n , which is sufficient to return into the energy gap. Further,
c 0;even
E
knowingn we cancalculatethe shiftofthe forbiddenbandedges,whichmoveapartdue to the transformation
c
of some single-electron states into excitons. This results in the enhancement of energy gap
(π~n )2
c
δE (4.3)
g
≃ 2µ
e
like in [19] and [20]. Here n = n πr2 is the linear critical concentration of excitons, and r is the radius of
c c 1 1
electron wave functions transverse localization at the linear crystal axis.
In accordance with (4.2) the described model yields n 180 µm−1 ( 3% of the atoms number in the
e c
crystal)and 400µm−1 ( 7%)for the linear crystalwithj∼=2.1andj =∼2.3,respectively,while by (4.3)the
∼ ∼
corresponding δE are 300 meV and 500 meV in the same order. Here, however, we should mention that
g e
∼ ∼
for SWCNTs both the measuredin [19],[20] and estimated in the same manner [18] values of δE /E appeared
g g
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 example 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 energies. Indeed, for the linear crystal in media even with permittivity about ε 2 3 (e.g.,
∼ −
like in [16] or [17]) the ground-state exciton binding energy becomes smaller than the energy gap.
Acknowledgments
This work was supported by the Ministry of Education and Science of Ukraine, Grant #0106U001673.
References
[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, 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. Japan 66, 1066 (1997).
[7] A. Jorio, G. Dresselhaus, and M.S. Dresselhaus (editors), 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. Høegh-Krohn,andH.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 Potentials and Their Applications in Atomic Physics,
Plenum, New York (1988).
[13] S.V. Tishchenko, Low Temp. Phys. 32, 953 (2006).
[14] V. Adamyan and S. Tishchenko, J. Phys.: Condens. Matter 19, 186206 (2007).
8
[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,andT.Mizutani,Phys. Rev.B73,235427
(2006).
[20] O. Kiowski, S. Lebedkin, F. Hennrich, S. Malik, H. Ro¨sner, K. Arnold, C. Su¨rgers, and M.M. Kappes,
Phys. Rev. B75, 075421 (2007).
[21] V. Perebeinos, J. Tersoff, and P. Avouris, Phys. Rev. Lett. 92, 257402(2004).
9