Electrostatic screening and Friedel oscillations in semiconducting nanotubes
In 3D and 2D electronic systems the singular contribution to the static permittivity ε (Kohn singularity) is a small correction to the regular part of ε but it results in the leading term in asymptotic behavior of the screened potential (Friedel oscillations). In the present letter we show that fo...
Gespeichert in:
| Datum: | 2008 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2008
|
| Schriftenreihe: | Физика низких температур |
| Schlagworte: | |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/117552 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Electrostatic screening and Friedel oscillations in semiconducting nanotubes / A.V. Chaplik, L.I. Magarill, R.Z. Vitlina // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1094–1097. — Бібліогр.: 8 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-117552 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1175522025-02-09T17:38:16Z Electrostatic screening and Friedel oscillations in semiconducting nanotubes Chaplik, A.V. Magarill, L.I. Vitlina, R.Z. Carbon nanotubes, quantum wires and Luttinger liquid In 3D and 2D electronic systems the singular contribution to the static permittivity ε (Kohn singularity) is a small correction to the regular part of ε but it results in the leading term in asymptotic behavior of the screened potential (Friedel oscillations). In the present letter we show that for nanotubes quite different results are valid: ε becomes infinitely large at the singular point and the Friedel oscillations do not play the dominant role in the screening at the large distances. Moreover, the zero and highest cylindrical harmonics of the effective potential are screened by quite different mechanisms. This work has been supported by the RFBR, by the RF President grant for scientific schools, as well as by the Programs of the Russian Academy of Sciences. 2008 Article Electrostatic screening and Friedel oscillations in semiconducting nanotubes / A.V. Chaplik, L.I. Magarill, R.Z. Vitlina // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1094–1097. — Бібліогр.: 8 назв. — англ. 0132-6414 PACS: 73.63.Fg https://nasplib.isofts.kiev.ua/handle/123456789/117552 en Физика низких температур application/pdf 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 |
Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid |
| spellingShingle |
Carbon nanotubes, quantum wires and Luttinger liquid Carbon nanotubes, quantum wires and Luttinger liquid Chaplik, A.V. Magarill, L.I. Vitlina, R.Z. Electrostatic screening and Friedel oscillations in semiconducting nanotubes Физика низких температур |
| description |
In 3D and 2D electronic systems the singular contribution to the static permittivity ε (Kohn singularity)
is a small correction to the regular part of ε but it results in the leading term in asymptotic behavior of the
screened potential (Friedel oscillations). In the present letter we show that for nanotubes quite different results
are valid: ε becomes infinitely large at the singular point and the Friedel oscillations do not play the
dominant role in the screening at the large distances. Moreover, the zero and highest cylindrical harmonics
of the effective potential are screened by quite different mechanisms. |
| format |
Article |
| author |
Chaplik, A.V. Magarill, L.I. Vitlina, R.Z. |
| author_facet |
Chaplik, A.V. Magarill, L.I. Vitlina, R.Z. |
| author_sort |
Chaplik, A.V. |
| title |
Electrostatic screening and Friedel oscillations in semiconducting nanotubes |
| title_short |
Electrostatic screening and Friedel oscillations in semiconducting nanotubes |
| title_full |
Electrostatic screening and Friedel oscillations in semiconducting nanotubes |
| title_fullStr |
Electrostatic screening and Friedel oscillations in semiconducting nanotubes |
| title_full_unstemmed |
Electrostatic screening and Friedel oscillations in semiconducting nanotubes |
| title_sort |
electrostatic screening and friedel oscillations in semiconducting nanotubes |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Carbon nanotubes, quantum wires and Luttinger liquid |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/117552 |
| citation_txt |
Electrostatic screening and Friedel oscillations
in semiconducting nanotubes / A.V. Chaplik, L.I. Magarill, R.Z. Vitlina // Физика низких температур. — 2008. — Т. 34, № 10. — С. 1094–1097. — Бібліогр.: 8 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT chaplikav electrostaticscreeningandfriedeloscillationsinsemiconductingnanotubes AT magarillli electrostaticscreeningandfriedeloscillationsinsemiconductingnanotubes AT vitlinarz electrostaticscreeningandfriedeloscillationsinsemiconductingnanotubes |
| first_indexed |
2025-11-28T19:50:00Z |
| last_indexed |
2025-11-28T19:50:00Z |
| _version_ |
1850064940368396288 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10, p. 1094–1097
Electrostatic screening and Friedel oscillations
in semiconducting nanotubes
A.V. Chaplik, L.I. Magarill, and R.Z. Vitlina
Institute of Semiconductor Physics, Novosibirsk 630090, Russia
E-mail: magarill_lev@hotmail.com
Received April 8, 2008
In 3D and 2D electronic systems the singular contribution to the static permittivity ε (Kohn singularity)
is a small correction to the regular part of ε but it results in the leading term in asymptotic behavior of the
screened potential (Friedel oscillations). In the present letter we show that for nanotubes quite different re-
sults are valid: ε becomes infinitely large at the singular point and the Friedel oscillations do not play the
dominant role in the screening at the large distances. Moreover, the zero and highest cylindrical harmonics
of the effective potential are screened by quite different mechanisms.
PACS: 73.63.Fg Nanotubes.
Keywords: static permittivity, Friedel oscillations, nanotubes.
Collective effects in quasi-1D systems are remarkable
by a number of distinguishing features. Probably the most
non-trivial one relates to the strictly 1D system of inter-
acting fermions with the linear single particle dispersion
law (the Luttinger model). However, there exist also dif-
ferent 1D objects not described in terms of the Luttinger
liquid first of all due to necessity to account for the
transversal subbands and transitions between them. The
corresponding example is a nanotube — a hollow cylin-
der with 2D electron gas on its surface. As usually, avail-
ability of the mobile electrons results in renormalization
of the electron–electron interaction, i.e., screening. The
specific form of the screened e–e interaction is deter-
mined by the effective dimension and by the energy spec-
trum of the electron gas. As is known in 3D plasma the
bare Coulomb interaction is replaced (due to screening)
by the Yukawa law plus the Friedel oscillations
cos ( ) /2 3p r rF if the electron gas is degenerate; here pF
is the Fermi momentum, r is the distance from a point
charge, and � = 1. Similar results in the 2D case read: the
regular part of the screened potential at r → ∞ is Qa rB
2 3/
[1], where Q is the initial point charge, aB is the effective
Bohr radius, whereas the oscillating contribution is pro-
portional to cos ( / ) /2 4 2p r rF − π . We see that both in 3D
and in 2D problems the Kohn singularity of ε( )k at
k pF= 2 gives the leading term of the screened potential
for r → ∞; here ε( )k is the static permittivity depending on
the momentum k.
Consider now a nanotube with the semiconductor type
of the single particle energy spectrum:
ε p l
p
m
Bl B
ma
l, ; , , , ,= + ≡ = ± ±
2
2
22
1
2
0 1 2 � (1)
Here p is the electron momentum parallel to the tube axis,
l is the number of the subband, m is the effective mass and
a is the nanotube radius. Thus, the problem in question
differs from the similar one for a planar 2D system by
only quantization of one of the component of momentum:
p l ax = / . The considered system is infinite and uni-
form in z-direction (the nanotube axis) and periodic and
uniform in the azimuthal direction ϕ. Hence, the elect-
ron Green function G depends on only differences
z z− ′ − ′,ϕ ϕ . In the Fourier representation G is diagonal:
G p p l l G p lpp ll( , ; , ) ( , )′ ′ = ′ ′δ δ .
In what follows we will apply the standard diagram
technique to find the e e− interaction renormalized by
screening and we will use the linear theory of screening.
We are aware that in strictly 1D system with linear sin-
gle-particle dispersion law the Luttinger liquid model is
valid. The linear screening theory relates to only long-
wave length limit. This theory breaks down for the mo-
mentum of the order of pF just when the Friedel oscilla-
tions become essential [2–4]*.
� A.V. Chaplik, L.I. Magarill, and R.Z. Vitlina, 2008
* There is a misprint in the Eq.(8) of Ref. 2. Asymptotic behavior of the screened potentialUeff must be1 2/| | ln | / |x x d rather than
1/| | ln | / |x x d , otherwise U x dxeff∫ ( ) diverges while the Fourier component U qeff ( ) remains finite at q → 0.
However we actually deal with 2D object and we use
the parabolic dispersion law for electrons. For such a situ-
ation the Luttinger model is not applicable and we do not
have any other instrument except RPA to investigate the
problem. After that it is not surprising that our results for
the Friedel contribution to the screened potential differ
qualitatively from the ones obtained in [2–4] and also [5].
The main difference is dependence of our effective
(screened) potential not only on z (coordinate along the
tube axis) but also on ϕ — azimuthal separation between
two electrons on the surface of a hollow cylinder. We dis-
covered that average potential (zero cylindrical harmo-
nic) and all the other harmonics are screened qualitatively
different.
We guess that in the present situation — there is an ex-
actly solvable model but for only strictly 1D system with
linear dispersion law and an approximate method for gen-
eral case — it is worth to find general results within RPA
and to see what experiments will show. This will be done
in what follows.
The Gell-Mann–Brueckner [6] chain of electron loops
determining the Fourier component of the screened e–e
interaction V k n( , ) comes to the geometrical progression:
V k n
V k n
V k n k n
( , )
( , )
( , ) ( ; , )
,
( )
( )
=
+
0
01 Π ω
(2)
where V ( )0 is the bare Coulomb interaction:
V k n e
dzd
z a
ikz in
( )( , ) ~
sin ( / )
0 2
2 2 2
0
2
4 2
=
+
=
− −
−∞
∞
∫∫ e ϕπ
ϕ
ϕ
= 4 2π~ (| | ) (| | ) .e I k a K k an n
(3)
Here ~ /e e2 2= χ, χ is the background dielectric constant,
I Kn n, are the modified Bessel functions of the 1th and
the 3rd type. The polarization operator (electron loop) has
a form
Π( ; , )
, ,
, ,
ω
π ε ε
k n dp
f f
l
p k l n p l
p l p k l
=
−
−
−∞
∞
= −∞
∞
− −
− −∫∑1
2 2
n i− −ω δ
( )δ = + 0 ,
(4)
where f fp l p l, ,( )≡ ε are the Fermi occupation numbers.
We are interesting here in the static screening and we
put from now on ω = 0. By transforming V k n( , ) from
Eq. (2) back to z-space we obtain the expansion of the
screened interaction in cylindrical harmonics. As
V k n( )( , )0 and Π( ; , )0 k n are even functions of n we come
to the series
V z
dk
ikz
V k V k n
n
( , )
( )
exp ( )
[ ( , ) ( , ) co
ϕ
π
= ×
× +
−∞
∞
=
∞
∫
∑
2
0 2
2
1
s ( )] .nϕ (5)
The slowest decreasing term at | |z → ∞ is given by the
zero harmonic and reads
V z
dk
kz
e I ka K ka
e k I
0
0
2
0 0
2
0 0
4
1 4
( ) cos ( )
~ ( ) ( )
~ ( )
=
+
∞
∫ π
π
π Π ( ) ( )
,
ka K ka0
(6)
where
Π Π0 2
0 0
2
2
( ) ( ; , ) ln ,k k
m
k
k p
k p
L
L
l
l
≡ =
+
−
⏐
⏐
⏐ ⏐
⏐
⏐
−
∑π
(7)
L is the number of the highest occupied subband at zero
temperature, p m E Bll F= −2 2( ) is the Fermi momentum
of the lth subband. Eq. (7) demonstrates the qualitative
difference of the quasi-1D problem and the 2D and 3D
problems. In 2D and 3D systems the Kohn singularity at
k pF= 2 gives small corrections to the regular part of the
polarization operator: ξ ξln | | in 3D [7] and | |ξ in 2D
case, where ξ = − <<( ) /k p pF F2 2 1. But in 1D situation
Π0( )k becomes infinitely large at k pl= 2 and the re-
normalized interaction vanishes at these points.
To calculateV z0( ) in the regime | |z a>> we write in the
Eq. (6) cos ( ) Re (exp ( ))kz ikz= , and turn the path of inte-
gration to the upper imaginary semiaxis. At large | |z we
have two contributions to V z0( ): V0 from the point k = 0,
where K 0 has logarithmic singularity but Π0 remains fi-
nite, and a number of the Kohn contributions from the
points k pl= 2 resulting in the Friedel oscillations
~
( )V z0 .
For the nonoscillating part V z0( ) we get the expansion in
inverse powers of the value Λ ≡ −ln ( | |/ )2 z a C (C is the
Euler constant)
V z
e
z
ma ma CB B
0
2
0
2
0
4
1
4
2
( )
~ /
...=
⎛
⎝
⎜
⎞
⎠
⎟ −
+
+⎛
⎝⎜
⎞
⎠⎟πκ
πκ
Λ Λ
, (8)
where
κ π0 0 0
1
20 1 2 1≡ → = +
=
∑Π ( ) [ / ( / )] /k m p p
l
L
l ;
πκ 0 is the sum of the partial density of states in occupied
subbands. Thus, the Coulomb interaction in nanotubes is
screened rather weakly, V z e z z a0
2 2( ) ~ / ( ln ( / )).
To find the oscillating part
~
( )V z0 we consider a small
vicinity of the point k pl= 2 and note that the factor
~e I K2
0 0 in the integrand of Eq. (6) cancels. Thus, we see
yet another peculiarity of the 1D problem: Friedel oscilla-
tions of the screened potential do not depend on the
charge! (see below Eqs. (9), (18)). This is due to infinitely
large magnitude of Π0( )k at k pl= 2 . Then one can intro-
duce the parameter ξ = −( ) /k p pl l2 2 and take the inte-
gral over ξ from −∞ till ∞. By the same shift of the integra-
tion path as described above we arrive at the series in
powers of 1 4/ ln ( )p zl :
Electrostatic screening and Friedel oscillations in semiconducting nanotubes
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1095
~
( )
cos ( )
ln ( ) ln (
V z
p
m
p z
z p z
Cl
l L
l L
l
l
0
2
2
2 2
4
1
2
4
=
= − −
= −
=
∑ π
p zl )
.+
⎡
⎣
⎢
⎤
⎦
⎥� (9)
We see that the amplitude of the Friedel oscillations in
nanotubes decreases with increasing distance not slower
than the regular part V z0( ) does. Moreover, the ratio
~
/V V0 0 at any z in the order of magnitude can be estimated
as 1/ ( )p aF B , and for «metallic» limit of the dense elec-
tron gas ( p aF B >> 1) the Friedel oscillations become ne-
gligibly small.
Qualitatively different results are obtained for non-
zero harmonics. The factor I ka K kan n( ) ( ) for n ≠ 0 tends
to the constant 1 2/ n for k → 0 andΠ( , )k n at k = 0 remains
finite either. Thus, all non-zero harmonics of the Cou-
lomb interaction undergo the screening of the dielectric
type, that is, their dependence on the distance z coincides
with that of the bare Coulomb law. Indeed, from Eq. (3)
we have for the nth harmonic of the bare interaction
V z
e
a
Q
z
a
V z a
n n
n
( )
/
( )
( )
~
,
( )
(
0
2
1 2
2
2
0
1
2
= +
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
>>
−π
�
Γ n
n
e
z
a
z
n+ ⎛
⎝⎜
⎞
⎠⎟
1 2 2 2
/ )
!
~
,
π
(10)
where Qν is the spherical function of the second type. The
regular part of the screened interaction stems again from
the region of small k. Eq. (4) gives:
Im ( ( , )| ,
( ( , )) ln
(
Π
Π
k n
k n
m
k
k a n kp
k
L
L
=
−
=
=
+ +∑
0
2
2 2 2
0
2
2
Re
π
l
l
a n l
k a n kp a n l
2 2 2 2
2 2 2 2 2 2 2
4
2 4
)
( )
−
+ − −
⏐
⏐
⏐
⏐
⏐
⏐
(11)
and there are two possibi l i t ies for the quanti ty
κ n n≡ Π( , )0 . If the subband occupation for T = 0 termi-
nates at l L= ± and n L2 24> , then
κ
π
n
l
l L
L
ma p
n l
=
−= −
∑4
4
2
2 2 2
. (12)
Otherwise, for n L2 24≤ and n even the nth harmonic of
the potential couples the degenerate states l n0 2= / and
l n0 2= − / . In this case we calculate the limit of the uncer-
tainty in Eq. (11) at n l= 2 0, k → 0 and find
Πn l
l
lm
p
p a
n
= = +
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟2 2
2
20
0
02 1 2
π
. (13)
After that the terms with l l= ± 0 in the sum of Eq. (12)
should be replaced byΠn l= 2 0
from Eq. (13). Thus, for the
nth harmonic of the screened potential at large | |z we ob-
tain the contribution from the point k = 0 :
V z
n
n
e
z
a
z ma n
n
n
n
B
( )
( / )
!
~
=
+ ⎛
⎝⎜
⎞
⎠⎟
+
⎛
⎝
⎜
⎞
⎠
⎟
−Γ 1 2
1
2 2 2
π
πκ
. (14)
The role of the effective dielectric constant for the nth
harmonic is played by the quantity
ε
πκ
n
n
Bma n
= +
⎛
⎝
⎜
⎞
⎠
⎟1
2
(15)
that tends to 1 with increasing n. The dielectric type of
screening of harmonics with n ≥ 1 can be understood in
terms of the classical electrostatics (suggested by M.V.
Entin). Each term of the series in Eq. (5) corresponds to
the one of the expansion in multipoles: n = 1 gives di-
pole–dipole contribution, n = 2 — quadrupole–quadru-
pole and so on. At very large | |z the fields of all multipoles
on the nanotube surface are practically normal to the axis,
hence, they cause only the displacements of electrons in
the azimuthal direction ϕ. That is why the z-dependence
of the e–e interaction does not change; the system simply
is polarized in accord with the dielectric mechanism when
only the bound charges are available.
As to the oscillating part
~
( )V zn it is determined by ze-
ros of the argument of logarithm in the Eq. (11). First of
all one can see that such contributions exist not for all val-
ues of n. For a fixed concentration of electrons p a LF ≤
whereas the singularity in the right-hand-side of Eq. (11)
occurs only if p a n lF
2 2 2> −( ) with | |l L< . Thus, the
Friedel oscillations of the nth harmonic exist for n L≤ 2
but for n L> 2 we have only nonoscillating contribution
V zn ( ) given by Eq. (14). The calculation of the oscillating
part
~
( )V zn for n L≤ 2 is totally similar to the derivation of
Eq. (9). The singular points k c are given by the relation
k p p
n l
a
c l F= ± − −⎛
⎝⎜
⎞
⎠⎟
2
2
(16)
and the polarization operator at k k c→ takes the form
Π ≈
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ −
=
+
−
m
k
q
k k
q
k p k p a nl
nl k
c
l
c
l
c l c l
c
2
2
2
2
π
ln ,
( )
( pl )
.
(17)
The result reads
~
( )
cos ( )
ln ( )
V z
mz
k k z
zq
n
c c
ll
= − ∑2 2
2
π
. (18)
Thus, only in the case n L≤ 2 the Friedel oscillations in
nanotubes give a noticeable contribution at large dis-
tances that is oscillating part of the nth harmonic of the
1096 Fizika Nizkikh Temperatur, 2008, v. 34, No. 10
A.V. Chaplik, L.I. Magarill, and R.Z. Vitlina
screened potential decreases slower with increasing z as
compared with the regular partV zn ( ). It is reasonable also
that the Friedel oscillations look like superposition of the
monochromatic waves with various periods 2π / k c (de-
pending both on n and l) rather than the single wave
cos ( )2 p rF as it is in 2D and 3D systems.
The problem of the screened Coulomb potential in
nanotubes was considered in [8]. However, the authors
have given only numerical results for zeroth harmonic
V z0( ) and did not discuss the Friedel oscillations.
To conclude, we have analyzed the e–e interaction in
nanotubes in the framework of the linear screening theo-
ry. Qualitative difference between quasi-1D systems and
3D and 2D systems is established. The Friedel oscilla-
tions in nanotubes do not determine the asymptotic be-
haviour of the screened potential at large distances in con-
trast with 3D and 2D systems.
This work has been supported by the RFBR, by the RF
President grant for scientific schools, as well as by the
Programs of the Russian Academy of Sciences.
1. T. Ando, A. Fowler, and F. Stern, Rev. Mod. Phys. 54, 437
(1982).
2. R. Egger and H. Grabert, Phys. Rev. Lett. 79, 3463 (1997).
3. Q. Yuan, H. Chen, Y. Zhang, and Y. Chen, Phys. Rev. B58,
1084 (1998).
4. R. Egger and H. Grabert, Phys. Rev. Lett. 75, 3505 (1995).
5. H.J. Schulz, Phys. Rev. Lett. 71, 1864 (1993).
6. M. Gell-Mann and K. Brueckner, Phys. Rev. 106, 364 (1957).
7. E.M. Lifshitz and L.P. Pitaevskii, Fizicheskaya Kinetika,
Nauka, Moskva (1979), v. 10, § 40.
8. M.F. Lin and D.S. Chuu, Phys. Rev. B56, 4996 (1997).
Electrostatic screening and Friedel oscillations in semiconducting nanotubes
Fizika Nizkikh Temperatur, 2008, v. 34, No. 10 1097
|