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The periodic orbit magnetic ordering transition on mesoscopic tubes
Solid State Communications,
Pergamon
003%1098(95)00353-3 THE PERIODIC ORBIT MAGNETIC
ORDERING
Shi-Liang Zhu
TRANSITION
Vol. 95, No. 11, pp. 765-769, 1995 Elsevier Science Ltd Printed in Great Britain 0038-1098/95 $9.50+.00
ON MESOSCGPIC
TUBES
and Yi-Chang Zhou
department oI Physics, Zhouahang University, Guangzhou, China 5 10275 (Accepted 21 April 1995 by Z. Gun)
In this paper, two simple models for considering the magnetic responses of the recently synthesized carbon nanotubes (CN) are designed: one is a single tube and the other consists of concentric tubes in bulk material with some distribution of radii. We discuss the case of fixed chemical potential. We predict that while the radius of the single tube alters, the diamagnetic response and paramagnetic response of the tube will occur alternately, and that the magnetic response of CN in bulk materials is always diamagnetic, which is in agreement with experiment. Keywords: A. fullerenes, D. mechanical properties, D. phase transitions.
1.
would alternately occur. In contrast of this, in the case discussing by Ref.[5], there is only a critical radius a~. According to our knowledge, theoretical analyses of CN are limited to discussiug the properties of a single tube. It should be noticed, however that the measurement magnetization of a single tube is not easy due to the microscopic size of each tube. therefore, a certain alternative model that estimates the average intrinsic magnetization of tubes in bulk material is used in this paper. The average magnetization of the tubes in bulk material that the diameters sat@ the average distribution and Gaussian distribution am investigated. Our results are coincident with the experiment[7] that a CN bulk material is always diamagnetic though the intrinsic magnetization of a single tube may be diamagnetic or paramagnetic depending on their diameter. This paper is organized as follows. In sec.2 we present two simple physical models: the model A is a single tube in a magnetic flux, and the model B is many concentric tubes in a bulk graphite. In sec.3 we discuss the periodic magnetic ordering transition on mesoscopic tube and the average magnetization of CN in bulk material. We conclude with a few remarks in sec.4.
Introduction
Physics at meaoacoPic tubes have attracted many interest since Graphite treedIe called carbon nanotubes (CN) were recently discovere~l]. A CN is a few concentric tubes conakting of carbon atom hexagons arranged in a helical fashion about the axis. Such novel structura character&s of the tubes have indeed been
attracting much interest to its physical properties. Both experimental and theoretical investigations have led to many important discoveries[2-61. Ref.[S] predicted a striking magnetic ordering transition of electrons on mesoscopic tube in a magnetic field as the tube radius is varied. The magnetic property of the tube associated with the A-B effect has alao been theoretically analyzed in Ref.[6]. In this paper, the magnetic response of single tube and many concentric tubes in bulk mate&l are studied by using independent electron model and assuming the chemical potential to be fixed while the rack of tubes ia altered. A CN nanotubes is a new type of carbon fibre consisting of coaxial cylinders of graphite sheet. Although the number of concentric tubes so far reported is at least two, many theories [2-51 consider a CN as a single tube, because interlayer coupling is known to be quite weak in bulk graphite. In this paper, we report that the magnetic response is very difkent from the previous predicted properties considering a single tube, if we assume the chemical potential (Fermi energy Ef at zero temperature) to be fixed among the tubes in bulk graphite even if their radius are difference. Our results show that a similar magnetic ordering transition of a nanoscale tube will occur while the radius of tube is altered, which has been predicted by Ref.151 in the condition of fixed electron density but the chemical potential being altered with the radius of tube. In our discussing case, however, there are a series of critical radius a~ of a mesoscopic tube, i.e. when the radius of tube were altered continuously, the diamagnetic response and the paramagnetic response
2. Model and Approach In this section the magnetization of a single tube and the average magnetization of a CN in bulk material are obtained at zero temperature within the free-electron model. Model A : Single Tube in Magnetic Fhrx We consider a 2D degenerate electron gas on a cylindrical surface (Fig.l(a)). The Hamiltonian for free electrons confined on the surface of the cylinder is given by 8= (P+eA:c)* where 765
we
choose
the
2m symmetric
gauge
field
766
PERIODIC ORDERING TRANSITION ON MESOSCOPIC TUBES such
that the applied external
> field g points in the Z directionThe Schrodinger equation at #itnlri~al coordinats is:
magnet&
and the flux quantum where the flux 0 = na2B & = k/e, the wave function is Y($,Z) = @(e&Z), and the boundary conditions are: Q(2n+e)=0(8) (3) i Z( 0) = Z(b) = 0 where a, b are the radius and height of the cylinder, respectively. In this paper, we restrict our a&yses about the case of long cylinder i.e. b&+1. Using of the boundary conditions (3), the single electron energy spectrum and wave function for an electron on a cylindrical surface can be obtained:
Vol. 95, No. 11
number 1 but depends on the subband index n, i.e. the electrons in the same subband index n but dit%rent number I contribute equal quantity of q-tun magnetization. Therefore the total magnetic moment of a tube at zero temperature is given by
In the numerical calculation we solve for the lowest and highest occupied snbband number in the presence of an externally applied magnetic field through the equation[5]. We solve for tbe number of occupied subbands Nocc (in Fig.2) under a tixed Fermi energy in the presence of a magnetic field through the inquality En0 - Ef -< 0 (because of Ed -> En@, i.e. -EfsO The Nocc
of n which satisfies the is the maximum
inequality (8). Model B: Concentric Tubes in Bulk Material (4)
where 4, = n&b,
n= 0, 21, +2 .......
are
energy
subband indices, and 1 =1,2 ,I... are axial energy quantum numbers. In the case of long cylinder (b/a >> l),Each fixed index n and d&rent I form a subband. The ek&onic numbersunderthe Fermi energy in subband index n (we have are two times of the maximum of I considered two degeneration in each n) . So we can obtain the electronic numbers Nn in a fixed subband
fromenergy spectrum(4Xi.e. where
k,(#l&)=
N, =int(~k,((l#,)),
2ma3Eflh’-(n+(/#o)2,
and
int(x) is the integer part of x. Thus the electron density o in the system is obtained:
According to our knowledge, although the number of concentric tubes so fz reported is at least two, theoretical analyses of CN are limited to discussing the properties of a single tube. It should be noticed, however, that the me asurement of the in&sic magnetization of an siugle tube is not easy due to the microscopic six&of tube. Therefore, a certain altemative mode1 each (F&i@)) is designed and the average intrinsic magnetization of tubes in bulk material is calculated with this model. F&l(b) is a model of a CN nanotubes in a bulk graphite, which in&de many concenttk tubes. We assume interlayer coupling to be quite weak in bulk graphite just as previous theories have assumed[2-51 except that the chemical potential of electron is fixed in the tubes of d&rent diameters even if each tube is in different CN but all CN’s needles are in a bulk
where Np and Nf are the
lowest and highest occupied subband indices, respectively.~ Although the electron densityodependsontheradiusoftubejitstillhasa limitedvalueata-p~ An elecuxui in the eigenstate w has orbital magnetic moment
I
(a)
(6) where pLs is the Bohr magneton. It is noticed that the elecnon magnetization does not depend on the quantum
(b)
1
\\
Y X
. I
Fig. 1. The physical models considered. (a) a cylinder carrying magnetic flux (; (b) a simple model of CN multicylinder structure.
Vol. 95, No. 11
PERIODIC ORDERING TRANSITION ON MESOSCOPIC TUBES
767
graphite.This is the main difkmnce between our model and the model in Ref[4,5]. (see Fig.3 in Ref.[4], the Fermi energy Ef varies with tubule radius in the absence of magnetic field. for example, 15.285 ev at a = 0.35nm, 15.257 ev at a = 0.45mu). We think that the chemical potential should be equal among those tubule surfaces, because the electrons can tunnel from one tubule surface to another tubule surf-. Because each carbon atom contributes a x -electron and the chemical bond has an average length 0.142 mu. Ref.[4] has e&mated the electron density on the tubule to be 00 = 63.14 run-2 . Therefore, we choose the chemical potential E/=lS.OOev in the absence of the magnetic field in our nwnerical calculations, and the Fetmi energy varies with magnetic field at the fixed radius. CN tubes were obtained with inner and outer diameter of 3-7 mu and 1S-20 nm respectively, and up to 1pminlength.htourpaper,wechooseb=lpm,and discuss the properties at two different types of the tubule radius distriiutiou: One is Gaussion distribution:
-cd0 . . . . j . . . * 1 5
10
.
.
*
c
1’ 15
.
’
’
1
-10 20
6)
44 =--_p(_~) & we define average magnetktion average over the radius a:
G
= p” _&~,~L+,&~+
c
by Gaussian
2L &,K&do+,~
(10)
and we choose a~ = 60.0, A = 20.0 in our future numfficat calculations. we have considered the Gaussian distribution is symmetry according to %, and the radius of tube should not be a negative number. Another is average distribution: l/w arlu
f(a) = 0
aia,
or u>az
where w = a, - a, and the average magnetization M&,=, is defined as: (12) we choose 4 =2.0 mu., and + = 10.0 nm, w = 80.0, we think that the average magnetization (10) and (12) can be coincided with the magnetic response of CN in bulk material. 3.
Magnetization
In Fig.2, the magnetization Mtotal(B,a) and the number of occupied subbands N~c against tubule radius are shown when the Fermi energy are fixed. The main results are following: (1) The magnetization is a periodic oscillation function of the tubule radius a at a fixed magnetic field The period depends on the Fermi energy F/. Each oscillatory period begins at the radius where the number of the occupied subbands Nocc increase an iutegral. (2) The magnitude of total magnetic moment is &Ened by the Fermi energy. (3) The total magnetization
and the Fig.2. The total magnetktion Iw,, (-) muuber of occupied subbands No0 (-cc) of a single tube as a function of tubule radius at the Fermi energys (a) Ef =15.0 ev ; (b) Ef =l.O ev . in the magnetic field B=l.O Tesla.
tiom diamagneticinto paramaguetic at the radius where the number of occupied subbands NW increase a integral This is a critical radius a~ which expresses the magnetization translates from diamaguetic response into pammagnetic response. (4) In the region of fixed number of occupied subbands NOW: , the magnetization decreases with the radius increasing. and there is a another critical radius a~ of a mesoscopic tube in the areas, namely the tube is diamagnetic when its radius is larger than a~, but becomes paramagnetic when the radius is smaller than a~. There are a series of critical radius ZQof a mesoscopic tube, i.e. when the radius of tube were altered continuously, the diamagnetic response and the paramagnetic response would alternately occur. So we call it as a periodic maguetic ordering tram&ion. translate sharply
768
PERIODIC ORDERING TRANSITION ON MESOSCOPIC TUBES
we can understand the origin of this periodic transition from energy subband structure of the tube shown schematically in Fii3. In the absence of a magnetic field, every level of a subband with an index n (except n=O) has a twofold degeneracy kn, which correspond to two different projections of the angular momentum along the z axis, Applying a magnetic field along the axial direction lifts this degeneracy( dashed curves) and leads to the separation of diamagentic and pammagnetic behavior. Electrons occupy+ the subbands with positive or negative indices have a diamagnetic or pammagnetic character, respectively. [51 The calculations express that the magnetization mainly depends on the occupied index near Fermi enemy, namely, the tube is pammagnetic when the highest occupied subband index is negative, whereas the tube is diamagnetic when the highest occuptied subband index is positive(see Fig.3). Therefore, when the radius is varied the number of occupied subbands increase an integer, electrons fimtly occupied the negative index because the energy of the electron with negative index is lower than the enemy of electrons with positive index, so the tuble is pammagnetic. Then electrons occupy the positive index when electrons increase due to the radius is varied, thus the tube becomes diamagnetic. There is a critical point in every number of occupied subbands Nocc because the process is the same when the number of occupied subbands increase bv increasing the radius of the tube at the fixed Fermi energy. The calculation shows the magnetization is pammagnetic when the Fermi enera locates at the shaded areas, whereas becomes diamagnetic when the Fermi energy locates at the white areas in Fig.3 As the tubule radius is increased the number of occupied subbands increase (for a fixed Fermi level ) because the energy gap between the subbands decrease, thus the Er can alternately locates at the white and shaded area when the radius of tube is altered. Therefore&e periodic orbital magnetic ordering transition originates from the Fermi energy located periodictly between the white and shaded areas when the tubule radius increases.
The origin of the periodic variations of magnetic order is the variation of subband numbem under Fermi energy while the radius of tube is altering in the condition of fixed Fermi energy. This point is just like the case of de Hass-van-Alphen oscillations (dHvA), in which when the magnetic field increases, the number of subbands decrease. In our problem, the variation of radius of the tube play the role of the magnetic field in dHvA oscillations. In this aspect, we can &ll the phenomenon that the magnetization oscillates with the radius of tube as a new type dHvA oscillations. W7e can estimate the oscillatory period using the following approximate approach. In the absence of the magnetic field, the number of occupied subbands Nocc approach the satisfying the inequality maximum integral n of
2maZEI I# - fir t 0. Thus n_ = int(aJ0726EI)
areas, but becomes diamagnetic when Ef is in the white areas.
( free
electron int(x) is the integer part of x), so the oscillatory period Ta = an-an-l = 1I Jq
(we have ignored the
difference between x with int(x)). Therefore, the oscillatory period is approxmation 0.051 run, 0.2nm at L?, = 15.0 ev and E,. = 1.0 ev, respectively. This estimation qualitatively agree with Fig.2. This qualitative physical picture interprets the oscillatory periods mainly depend on the Fermi energy and the electron effective mass iu the free electron model. The average magnetization of concentric tubes in a bulk material against magnetic field are shown in Fig.4, in which F&t(a) is the case of tubes radius with the Gaussian distribution, whereas Fig.4(b) is the case of tubes radius with the average distribution. The main results are: (1) The average responses to magnetic field are always diamagnetc. It expresses the diamagnetic is dominant although the magnetization of a single tube may be diamagnetic or paramagnetic depends on the tubule radius. We can estimate the energy splitting of the +n states due to a magnetic field along the axial direction &d compare it with the orbital energy level difference En+l- En. At a 1 tesla magnetic field and a = 5.0 nm, the splitting due to the magnetic field is about one-twentieth of the orbital energy level difference. The tube is paramagnetic only when the Fermi energy locates at the splitting areas due to applied magnetic field, and becomes diamagnetic when the Fermi energy locates at the splitting areas of the orbital energy Itye difference (see Fig.3). Thus we can estimate the diamagnetic is dominant character in the average magnetization of a CN in bulk material. (2)The average magnetimtion closely relates with the Fermi energy and the tubule radius distribution. M&)
Fig.3. Schematic subband electronic spectrum showing the splitting of subbands (dashed curves). the tube is paramaguetic when the Fermi energy .!Zr is in the shaded
Vol. 95, No. 11
with the average distribution oscillates with the
with Gaussian apphed magnetic field, but M&, . * distnbuuon does not. The MPB,afin average distribution oscillates with the magnetic field, because the 0fasingletubeisperiodicinIp magneu=tion ~*L%~(B,~) with &, period, and the periodic results were counteracted at almost all the tubule radius except for the boundary radius such as a = 2.0 mu and a = 10.0 nm. therefore : The period is approximate 6.5 tesla at Fk4(b) because
Vol. 95, No. 11
PERIODIC ORDERING TRANSITION ON MESOSCOPIC TUBES
769
6.5 tesla correspond to one qrantum +a at a = 10.0 nm (E = &/xa2). The oscillatory period is independent on Er. The average magnetization Mx
with the Gaussian
distribution is not oscillate with the applied magnetic field because the periods all have beeu cut beause the radius of Gaussian ~~b~on has no bo~~ radius. 4.
20 3
0
10
20 3
30
40
50
(Tesla)
30
40
50
(Tesla)
as a function of Fig.4. The average rn~e~on extemai mqaetic field at Ef=15.0 ev; in the conditions of tubule radiuses satisfy (a) Gausskn distribution (b) and average axon _
Conch&on
In this paper, we have considered two simple physical models (model A and B), which directly relate with the muhkylinder nanotubes in a recently synthesized magnetic fieeld,to investigate the ma8netiic respome. The ma8uetic effect of orbital motion is only considered and the Pauli parama8netic effect is ne&mted, because Ref.[S] has estimated that the Zeeman’s enemy splittin is about one-tenth of the split&g due to the orbital motion of electron at rtqnetic field of 1 Tesla. Our results are following: (l)We have predicted that a periodic retie ordaing Nixon of electrons on mesoscopic tubes will occur, if the chemical potential is fixed among the tubule surfaces, i.e. when the radhts of tube were altered contnmously, the diamagnetic response and the optic respouse would step occur. In some aspect, we can caIl the phenomenou that the magnetization oscillates with the radius of tubes as a new type dHvA os&lIatious. Cafe it with Ref.[S], the periodic ma8netic order& transition of electrou will not occur (only one critical radius exist) in the case of tied electron density in Ref[5] because the nqnetic transition will be restrained by the altering of the chemical potent&. (2) The average maguetizations, when the tube radius are Gauss&n or average ~~buti~ are always retie, this result is coincide with the experiment [7]. Finally, we note that the above results are directly applicable to coaxial graphite microtubles. In particular, experiments on all CNk needles in a bulk material are parallel. A~~owIe~~~t - This work is supported in part by the Foundation of ~~g~~ Undo Abranced Research Center, and is supported by the establishment Foundation of Key Descipliue provided by Hi& Education Bureau of Crag ~o~e~~
References [l] S.Iijima,Nature 56,354 (1991). f2] J.W.~~e,B.I.~p and C.T.Wbite, Phys.Rev. Lett.68. 631f1991). [3] NHamada,S.Sawada and A.Oshiyama,Phys.Rev.Lett. 68,1579(1992). [4]L.W~~F.S.Da~~,A.S~~~~.R.B~op,Phys. Rev.B46,7175 (1992).
[5]P.S.Davids,L.Wan& A.Saxena, andA.R.Bishop, Phys.Rev.B48,17545(1993). [6] H.Ajii and T.Audo, J.Phys.Soc.Jap.62,247~1~3). [7] K.Tanaka, T.Sato, T.Yamabe, KOkahara, KUchida, M.Yumura, H,Niino, S.Oh&ma, Y.Kur@ K.Yase, F.lkar.al@ C!hem.Phys.Lett. 223, 65(1994).
Pergamon
003%1098(95)00353-3 THE PERIODIC ORBIT MAGNETIC
ORDERING
Shi-Liang Zhu
TRANSITION
Vol. 95, No. 11, pp. 765-769, 1995 Elsevier Science Ltd Printed in Great Britain 0038-1098/95 $9.50+.00
ON MESOSCGPIC
TUBES
and Yi-Chang Zhou
department oI Physics, Zhouahang University, Guangzhou, China 5 10275 (Accepted 21 April 1995 by Z. Gun)
In this paper, two simple models for considering the magnetic responses of the recently synthesized carbon nanotubes (CN) are designed: one is a single tube and the other consists of concentric tubes in bulk material with some distribution of radii. We discuss the case of fixed chemical potential. We predict that while the radius of the single tube alters, the diamagnetic response and paramagnetic response of the tube will occur alternately, and that the magnetic response of CN in bulk materials is always diamagnetic, which is in agreement with experiment. Keywords: A. fullerenes, D. mechanical properties, D. phase transitions.
1.
would alternately occur. In contrast of this, in the case discussing by Ref.[5], there is only a critical radius a~. According to our knowledge, theoretical analyses of CN are limited to discussiug the properties of a single tube. It should be noticed, however that the measurement magnetization of a single tube is not easy due to the microscopic size of each tube. therefore, a certain alternative model that estimates the average intrinsic magnetization of tubes in bulk material is used in this paper. The average magnetization of the tubes in bulk material that the diameters sat@ the average distribution and Gaussian distribution am investigated. Our results are coincident with the experiment[7] that a CN bulk material is always diamagnetic though the intrinsic magnetization of a single tube may be diamagnetic or paramagnetic depending on their diameter. This paper is organized as follows. In sec.2 we present two simple physical models: the model A is a single tube in a magnetic flux, and the model B is many concentric tubes in a bulk graphite. In sec.3 we discuss the periodic magnetic ordering transition on mesoscopic tube and the average magnetization of CN in bulk material. We conclude with a few remarks in sec.4.
Introduction
Physics at meaoacoPic tubes have attracted many interest since Graphite treedIe called carbon nanotubes (CN) were recently discovere~l]. A CN is a few concentric tubes conakting of carbon atom hexagons arranged in a helical fashion about the axis. Such novel structura character&s of the tubes have indeed been
attracting much interest to its physical properties. Both experimental and theoretical investigations have led to many important discoveries[2-61. Ref.[S] predicted a striking magnetic ordering transition of electrons on mesoscopic tube in a magnetic field as the tube radius is varied. The magnetic property of the tube associated with the A-B effect has alao been theoretically analyzed in Ref.[6]. In this paper, the magnetic response of single tube and many concentric tubes in bulk mate&l are studied by using independent electron model and assuming the chemical potential to be fixed while the rack of tubes ia altered. A CN nanotubes is a new type of carbon fibre consisting of coaxial cylinders of graphite sheet. Although the number of concentric tubes so far reported is at least two, many theories [2-51 consider a CN as a single tube, because interlayer coupling is known to be quite weak in bulk graphite. In this paper, we report that the magnetic response is very difkent from the previous predicted properties considering a single tube, if we assume the chemical potential (Fermi energy Ef at zero temperature) to be fixed among the tubes in bulk graphite even if their radius are difference. Our results show that a similar magnetic ordering transition of a nanoscale tube will occur while the radius of tube is altered, which has been predicted by Ref.151 in the condition of fixed electron density but the chemical potential being altered with the radius of tube. In our discussing case, however, there are a series of critical radius a~ of a mesoscopic tube, i.e. when the radius of tube were altered continuously, the diamagnetic response and the paramagnetic response
2. Model and Approach In this section the magnetization of a single tube and the average magnetization of a CN in bulk material are obtained at zero temperature within the free-electron model. Model A : Single Tube in Magnetic Fhrx We consider a 2D degenerate electron gas on a cylindrical surface (Fig.l(a)). The Hamiltonian for free electrons confined on the surface of the cylinder is given by 8= (P+eA:c)* where 765
we
choose
the
2m symmetric
gauge
field
766
PERIODIC ORDERING TRANSITION ON MESOSCOPIC TUBES such
that the applied external
> field g points in the Z directionThe Schrodinger equation at #itnlri~al coordinats is:
magnet&
and the flux quantum where the flux 0 = na2B & = k/e, the wave function is Y($,Z) = @(e&Z), and the boundary conditions are: Q(2n+e)=0(8) (3) i Z( 0) = Z(b) = 0 where a, b are the radius and height of the cylinder, respectively. In this paper, we restrict our a&yses about the case of long cylinder i.e. b&+1. Using of the boundary conditions (3), the single electron energy spectrum and wave function for an electron on a cylindrical surface can be obtained:
Vol. 95, No. 11
number 1 but depends on the subband index n, i.e. the electrons in the same subband index n but dit%rent number I contribute equal quantity of q-tun magnetization. Therefore the total magnetic moment of a tube at zero temperature is given by
In the numerical calculation we solve for the lowest and highest occupied snbband number in the presence of an externally applied magnetic field through the equation[5]. We solve for tbe number of occupied subbands Nocc (in Fig.2) under a tixed Fermi energy in the presence of a magnetic field through the inquality En0 - Ef -< 0 (because of Ed -> En@, i.e. -EfsO The Nocc
of n which satisfies the is the maximum
inequality (8). Model B: Concentric Tubes in Bulk Material (4)
where 4, = n&b,
n= 0, 21, +2 .......
are
energy
subband indices, and 1 =1,2 ,I... are axial energy quantum numbers. In the case of long cylinder (b/a >> l),Each fixed index n and d&rent I form a subband. The ek&onic numbersunderthe Fermi energy in subband index n (we have are two times of the maximum of I considered two degeneration in each n) . So we can obtain the electronic numbers Nn in a fixed subband
fromenergy spectrum(4Xi.e. where
k,(#l&)=
N, =int(~k,((l#,)),
2ma3Eflh’-(n+(/#o)2,
and
int(x) is the integer part of x. Thus the electron density o in the system is obtained:
According to our knowledge, although the number of concentric tubes so fz reported is at least two, theoretical analyses of CN are limited to discussing the properties of a single tube. It should be noticed, however, that the me asurement of the in&sic magnetization of an siugle tube is not easy due to the microscopic six&of tube. Therefore, a certain altemative mode1 each (F&i@)) is designed and the average intrinsic magnetization of tubes in bulk material is calculated with this model. F&l(b) is a model of a CN nanotubes in a bulk graphite, which in&de many concenttk tubes. We assume interlayer coupling to be quite weak in bulk graphite just as previous theories have assumed[2-51 except that the chemical potential of electron is fixed in the tubes of d&rent diameters even if each tube is in different CN but all CN’s needles are in a bulk
where Np and Nf are the
lowest and highest occupied subband indices, respectively.~ Although the electron densityodependsontheradiusoftubejitstillhasa limitedvalueata-p~ An elecuxui in the eigenstate w has orbital magnetic moment
I
(a)
(6) where pLs is the Bohr magneton. It is noticed that the elecnon magnetization does not depend on the quantum
(b)
1
\\
Y X
. I
Fig. 1. The physical models considered. (a) a cylinder carrying magnetic flux (; (b) a simple model of CN multicylinder structure.
Vol. 95, No. 11
PERIODIC ORDERING TRANSITION ON MESOSCOPIC TUBES
767
graphite.This is the main difkmnce between our model and the model in Ref[4,5]. (see Fig.3 in Ref.[4], the Fermi energy Ef varies with tubule radius in the absence of magnetic field. for example, 15.285 ev at a = 0.35nm, 15.257 ev at a = 0.45mu). We think that the chemical potential should be equal among those tubule surfaces, because the electrons can tunnel from one tubule surface to another tubule surf-. Because each carbon atom contributes a x -electron and the chemical bond has an average length 0.142 mu. Ref.[4] has e&mated the electron density on the tubule to be 00 = 63.14 run-2 . Therefore, we choose the chemical potential E/=lS.OOev in the absence of the magnetic field in our nwnerical calculations, and the Fetmi energy varies with magnetic field at the fixed radius. CN tubes were obtained with inner and outer diameter of 3-7 mu and 1S-20 nm respectively, and up to 1pminlength.htourpaper,wechooseb=lpm,and discuss the properties at two different types of the tubule radius distriiutiou: One is Gaussion distribution:
-cd0 . . . . j . . . * 1 5
10
.
.
*
c
1’ 15
.
’
’
1
-10 20
6)
44 =--_p(_~) & we define average magnetktion average over the radius a:
G
= p” _&~,~L+,&~+
c
by Gaussian
2L &,K&do+,~
(10)
and we choose a~ = 60.0, A = 20.0 in our future numfficat calculations. we have considered the Gaussian distribution is symmetry according to %, and the radius of tube should not be a negative number. Another is average distribution: l/w arlu
f(a) = 0
aia,
or u>az
where w = a, - a, and the average magnetization M&,=, is defined as: (12) we choose 4 =2.0 mu., and + = 10.0 nm, w = 80.0, we think that the average magnetization (10) and (12) can be coincided with the magnetic response of CN in bulk material. 3.
Magnetization
In Fig.2, the magnetization Mtotal(B,a) and the number of occupied subbands N~c against tubule radius are shown when the Fermi energy are fixed. The main results are following: (1) The magnetization is a periodic oscillation function of the tubule radius a at a fixed magnetic field The period depends on the Fermi energy F/. Each oscillatory period begins at the radius where the number of the occupied subbands Nocc increase an iutegral. (2) The magnitude of total magnetic moment is &Ened by the Fermi energy. (3) The total magnetization
and the Fig.2. The total magnetktion Iw,, (-) muuber of occupied subbands No0 (-cc) of a single tube as a function of tubule radius at the Fermi energys (a) Ef =15.0 ev ; (b) Ef =l.O ev . in the magnetic field B=l.O Tesla.
tiom diamagneticinto paramaguetic at the radius where the number of occupied subbands NW increase a integral This is a critical radius a~ which expresses the magnetization translates from diamaguetic response into pammagnetic response. (4) In the region of fixed number of occupied subbands NOW: , the magnetization decreases with the radius increasing. and there is a another critical radius a~ of a mesoscopic tube in the areas, namely the tube is diamagnetic when its radius is larger than a~, but becomes paramagnetic when the radius is smaller than a~. There are a series of critical radius ZQof a mesoscopic tube, i.e. when the radius of tube were altered continuously, the diamagnetic response and the paramagnetic response would alternately occur. So we call it as a periodic maguetic ordering tram&ion. translate sharply
768
PERIODIC ORDERING TRANSITION ON MESOSCOPIC TUBES
we can understand the origin of this periodic transition from energy subband structure of the tube shown schematically in Fii3. In the absence of a magnetic field, every level of a subband with an index n (except n=O) has a twofold degeneracy kn, which correspond to two different projections of the angular momentum along the z axis, Applying a magnetic field along the axial direction lifts this degeneracy( dashed curves) and leads to the separation of diamagentic and pammagnetic behavior. Electrons occupy+ the subbands with positive or negative indices have a diamagnetic or pammagnetic character, respectively. [51 The calculations express that the magnetization mainly depends on the occupied index near Fermi enemy, namely, the tube is pammagnetic when the highest occupied subband index is negative, whereas the tube is diamagnetic when the highest occuptied subband index is positive(see Fig.3). Therefore, when the radius is varied the number of occupied subbands increase an integer, electrons fimtly occupied the negative index because the energy of the electron with negative index is lower than the enemy of electrons with positive index, so the tuble is pammagnetic. Then electrons occupy the positive index when electrons increase due to the radius is varied, thus the tube becomes diamagnetic. There is a critical point in every number of occupied subbands Nocc because the process is the same when the number of occupied subbands increase bv increasing the radius of the tube at the fixed Fermi energy. The calculation shows the magnetization is pammagnetic when the Fermi enera locates at the shaded areas, whereas becomes diamagnetic when the Fermi energy locates at the white areas in Fig.3 As the tubule radius is increased the number of occupied subbands increase (for a fixed Fermi level ) because the energy gap between the subbands decrease, thus the Er can alternately locates at the white and shaded area when the radius of tube is altered. Therefore&e periodic orbital magnetic ordering transition originates from the Fermi energy located periodictly between the white and shaded areas when the tubule radius increases.
The origin of the periodic variations of magnetic order is the variation of subband numbem under Fermi energy while the radius of tube is altering in the condition of fixed Fermi energy. This point is just like the case of de Hass-van-Alphen oscillations (dHvA), in which when the magnetic field increases, the number of subbands decrease. In our problem, the variation of radius of the tube play the role of the magnetic field in dHvA oscillations. In this aspect, we can &ll the phenomenon that the magnetization oscillates with the radius of tube as a new type dHvA oscillations. W7e can estimate the oscillatory period using the following approximate approach. In the absence of the magnetic field, the number of occupied subbands Nocc approach the satisfying the inequality maximum integral n of
2maZEI I# - fir t 0. Thus n_ = int(aJ0726EI)
areas, but becomes diamagnetic when Ef is in the white areas.
( free
electron int(x) is the integer part of x), so the oscillatory period Ta = an-an-l = 1I Jq
(we have ignored the
difference between x with int(x)). Therefore, the oscillatory period is approxmation 0.051 run, 0.2nm at L?, = 15.0 ev and E,. = 1.0 ev, respectively. This estimation qualitatively agree with Fig.2. This qualitative physical picture interprets the oscillatory periods mainly depend on the Fermi energy and the electron effective mass iu the free electron model. The average magnetization of concentric tubes in a bulk material against magnetic field are shown in Fig.4, in which F&t(a) is the case of tubes radius with the Gaussian distribution, whereas Fig.4(b) is the case of tubes radius with the average distribution. The main results are: (1) The average responses to magnetic field are always diamagnetc. It expresses the diamagnetic is dominant although the magnetization of a single tube may be diamagnetic or paramagnetic depends on the tubule radius. We can estimate the energy splitting of the +n states due to a magnetic field along the axial direction &d compare it with the orbital energy level difference En+l- En. At a 1 tesla magnetic field and a = 5.0 nm, the splitting due to the magnetic field is about one-twentieth of the orbital energy level difference. The tube is paramagnetic only when the Fermi energy locates at the splitting areas due to applied magnetic field, and becomes diamagnetic when the Fermi energy locates at the splitting areas of the orbital energy Itye difference (see Fig.3). Thus we can estimate the diamagnetic is dominant character in the average magnetization of a CN in bulk material. (2)The average magnetimtion closely relates with the Fermi energy and the tubule radius distribution. M&)
Fig.3. Schematic subband electronic spectrum showing the splitting of subbands (dashed curves). the tube is paramaguetic when the Fermi energy .!Zr is in the shaded
Vol. 95, No. 11
with the average distribution oscillates with the
with Gaussian apphed magnetic field, but M&, . * distnbuuon does not. The MPB,afin average distribution oscillates with the magnetic field, because the 0fasingletubeisperiodicinIp magneu=tion ~*L%~(B,~) with &, period, and the periodic results were counteracted at almost all the tubule radius except for the boundary radius such as a = 2.0 mu and a = 10.0 nm. therefore : The period is approximate 6.5 tesla at Fk4(b) because
Vol. 95, No. 11
PERIODIC ORDERING TRANSITION ON MESOSCOPIC TUBES
769
6.5 tesla correspond to one qrantum +a at a = 10.0 nm (E = &/xa2). The oscillatory period is independent on Er. The average magnetization Mx
with the Gaussian
distribution is not oscillate with the applied magnetic field because the periods all have beeu cut beause the radius of Gaussian ~~b~on has no bo~~ radius. 4.
20 3
0
10
20 3
30
40
50
(Tesla)
30
40
50
(Tesla)
as a function of Fig.4. The average rn~e~on extemai mqaetic field at Ef=15.0 ev; in the conditions of tubule radiuses satisfy (a) Gausskn distribution (b) and average axon _
Conch&on
In this paper, we have considered two simple physical models (model A and B), which directly relate with the muhkylinder nanotubes in a recently synthesized magnetic fieeld,to investigate the ma8netiic respome. The ma8uetic effect of orbital motion is only considered and the Pauli parama8netic effect is ne&mted, because Ref.[S] has estimated that the Zeeman’s enemy splittin is about one-tenth of the split&g due to the orbital motion of electron at rtqnetic field of 1 Tesla. Our results are following: (l)We have predicted that a periodic retie ordaing Nixon of electrons on mesoscopic tubes will occur, if the chemical potential is fixed among the tubule surfaces, i.e. when the radhts of tube were altered contnmously, the diamagnetic response and the optic respouse would step occur. In some aspect, we can caIl the phenomenou that the magnetization oscillates with the radius of tubes as a new type dHvA os&lIatious. Cafe it with Ref.[S], the periodic ma8netic order& transition of electrou will not occur (only one critical radius exist) in the case of tied electron density in Ref[5] because the nqnetic transition will be restrained by the altering of the chemical potent&. (2) The average maguetizations, when the tube radius are Gauss&n or average ~~buti~ are always retie, this result is coincide with the experiment [7]. Finally, we note that the above results are directly applicable to coaxial graphite microtubles. In particular, experiments on all CNk needles in a bulk material are parallel. A~~owIe~~~t - This work is supported in part by the Foundation of ~~g~~ Undo Abranced Research Center, and is supported by the establishment Foundation of Key Descipliue provided by Hi& Education Bureau of Crag ~o~e~~
References [l] S.Iijima,Nature 56,354 (1991). f2] J.W.~~e,B.I.~p and C.T.Wbite, Phys.Rev. Lett.68. 631f1991). [3] NHamada,S.Sawada and A.Oshiyama,Phys.Rev.Lett. 68,1579(1992). [4]L.W~~F.S.Da~~,A.S~~~~.R.B~op,Phys. Rev.B46,7175 (1992).
[5]P.S.Davids,L.Wan& A.Saxena, andA.R.Bishop, Phys.Rev.B48,17545(1993). [6] H.Ajii and T.Audo, J.Phys.Soc.Jap.62,247~1~3). [7] K.Tanaka, T.Sato, T.Yamabe, KOkahara, KUchida, M.Yumura, H,Niino, S.Oh&ma, Y.Kur@ K.Yase, F.lkar.al@ C!hem.Phys.Lett. 223, 65(1994).