Effects of Valley Mixing and Exchange on Excitonsin Carbon Nanotubes with Aharonov-Bohm Flux

Tsuneya ANDO

Department of Physics, Tokyo Institute of Technology2–12–1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

Because of the presence of two valleys and the electron spin, the exciton states have adegeneracy of 16 in the lowest order k·p approximation. Due to a weak short-range part of theCoulomb interaction, they are split into several levels, leaving only a single optically-active brightlevel and making all others optically-inactive or dark. Their ordering and energy splitting areshown to be determined by two parameters characterizing the short-range interaction, sensitiveto detailed form of the potential and wave function. The presence of a magnetic flux throughthe cross section, accessible experimentally, modifies these energy levels strongly and causes theappearance of two bright excitons.

Keywords: carbon nanotube, Aharonov-Bohm effect, exciton, effective-mass theory, exchange effect,intervalley mixing

§1. Introduction

A single-wall carbon nanotube (CN) consists of arolled two-dimensional (2D) graphite sheet and its elec-tronic states change critically from metallic to semicon-ducting depending on its tubular circumferential vec-tor. Because of the one-dimensional (1D) structure withcylindrical form, combined with unique electronic struc-tures of the 2D graphite, the Coulomb interaction playsan important role in the band structure and the excitoneffect is extremely important in optical spectra.1−3) Thepurpose of this paper is to study effects of exchangeinteraction and mixing between different valleys on theexciton and to determine the energy levels of bright(optically active) and dark (optically inactive) excitonsin the absence and presence of a magnetic flux.

Experimentally, the 1D electronic structure of CNwas observed by scanning tunneling spectroscopy4,5) andthe resonant Raman scattering.6) Optical absorptionspectra reflect the band structure directly,7,8) though ex-citonic effects are of vital importance.9,10) Quite recently,optical absorption and photoluminescence of individualCN were observed.11,12) Splitting of the absorption andemission peaks due to an Aharonov-Bohm (AB) effectassociated with magnetic flux passing through the crosssection was observed also.13−17)

Quasi-particle spectra of CN were calculated usinga first-principles GW method,18) and calculations wereperformed also for optical absorption spectra with the in-clusion of excitonic final state interactions,19,20) althoughlimited to CN’s with very small diameter. There havebeen some reports on a phenomenological description ofexcitons also.21,22)

Because of the presence of two valleys, associatedwith the K and K’ points at the corner of the first Bril-louin zone, and the electron spin, the exciton states havea degeneracy of 16 in the lowest order k·p approximation.If we take into account a weak short-range part of theCoulomb interaction, they split into several levels due tointer-valley mixing and exchange effect, leaving only asingle bright level and making all others dark.23−26) The

presence of an AB flux accessible experimentally modifiesthese energy levels strongly.

In this paper, we calculate the energy levels ofexcitons taking into account these effects extending thek·p scheme in the absence and presence of AB flux.It is shown that these effects are characterized by twoparameters specifying a short-range part of the Coulombinteraction. In the absence of flux the ordering of asinglet bright (optically active) exciton and many otherdark (optically inactive) excitons depend sensitively onthe parameters. The presence of an AB flux causes theappearance of two bright excitons.

The paper is organized as follows: In §2 the k·pscheme is reviewed very briefly and a short-range partof the Coulomb interaction is introduced. In §3 ef-fects of the short-range interaction on exciton levels arediscussed. In §4 some examples of explicit results arepresented. A summary and conclusion are given in§5. An effective Hamiltonian is derived for intra- andinter-valley processes in Appendix A and B, respectively.

§2. Effective-Mass Approximation

2.1 Energy Bands and Wave Functions

Figure 1 shows the lattice structure of 2D graphite,the first Brillouin zone, and the coordinate system inCN. In a graphite sheet the conduction and valencebands consisting of π states cross at K and K’ points ofthe Brillouin zone, where the Fermi level is located.27)

Electronic states of the π-bands near a K point aredescribed by the k·p equation:28,29)

γ(σxkx+σyky)F (r) = εF (r), (2.1)

where γ is a band parameter, σx and σy are the Paulispin matrices, and k=(kx, ky) is a wave-vector operator.

The structure of CN is specified by a chiral vectorL corresponding to the circumference as shown in Fig.1. It is written as L = naa + nbb in terms of twointegers na and nb, where a and b are the primitivetranslation vectors. In the following we shall choosethe x axis in the circumference direction and the y axis

Submitted to Journal of Physical Society of Japan

Page 2 T. Ando

in the axis direction, i.e., L = (L, 0), where L is thecircumference. The angle η between L and the horizontalaxis is called the chiral angle. Electronic states ofCN with a sufficiently large diameter are obtained byimposing the boundary conditions in the circumferencedirection:

F (r+L) = F (r) exp[2πi

(ϕ− ν

3

)], (2.2)

with ϕ = φ/φ0, where φ is AB flux passing through thecross section of CN and φ0 = ch/e is the magnetic fluxquantum. Further, ν is an integer determined uniquelyfor given L as ν =0 or ±1 through na+nb =3M+ν withinteger M .

The energy bands are specified by s = ±1, integern corresponding to the discrete wave vector along thecircumference direction, and the wave vector k in theaxis direction, where s = −1 denotes the valence bandand +1 the conduction band. The wave function for aband associated with the K point with wave vector K iswritten as

F Ks,n,k(x, y) =

1√AL

exp[iκνφ(n)x+iky]F Ks,n,k, (2.3)

where A is the length of CN,

κνϕ(n) =2π

L

(n+ϕ− ν

3

), (2.4)

and

F Ks,n,k =

1√2

(bνϕ(n, k)

s

), (2.5)

with

bνϕ(n, k) =κνϕ(n)−ik√κνϕ(n)2+k2

. (2.6)

The corresponding energy is given by

εKs,n,k = s εK

nk, (2.7)

with

εKnk = γ

√κνϕ(n)2+k2. (2.8)

The equation for the K’ point with wave vector K ′

is obtained by replacing ky by −ky and the boundaryconditions ν by −ν. Correspondingly, we have κ′

νϕ(n)=κ−νϕ(n) and b′νϕ(n, k)=b−νϕ(n, k)∗. The band structureobtained above is illustrated in Fig. 2 for ν =1. The bandstructure for ν = −1 can be obtained by exchanging Kand K’ points.

2.2 Coulomb Interaction

Figure 3 shows elementary processes of electron-electron interaction. Intra-valley processes (a) involvescattering within the same valley and thus a small mo-mentum transfer and inter-valley processes (b) involvemomentum transfer K −K ′. Because both K − K ′

and 2(K−K ′) are different from any reciprocal latticevectors, processes other than shown in the figure are notallowed due to the momentum conservation.

As shown in Appendix A, the Coulomb interactioninvolving scattering within each valley is separated intothe conventional long-range part and a small short-range

part. For the K point, in particular, the matrix elementof the long-range part can be written as

V(K2,K1)(K4,K3) =∫

dr

∫dr′ [F K

2 (r)†F K1 (r)]

× v(r−r′)[F K4 (r′)†F K

3 (r′)], (2.9)

with

v(r−r′) =e2

κD(r−r′), (2.10)

where D(r−r′) is the three-dimensional distance betweenr and r′, κ is a static dielectric constant describing effectsof polarization of electrons of core states, σ bands, πbands away from the K and K’ points, and a surroundingmaterial. The similar results can be written down formatrix elements involving electrons in different valleys.

For the K point, the short-range part not includedin the conventional k·p scheme is written as

V(1)(K2,K1)(K4,K3) = Ω0 w1

∫dr [F K

2 (r)†σzFK1 (r)]

×[F K4 (r)†σzF

K3 (r)],

(2.11)where σz is the z component of the Pauli matrix, Ω0 =(√

3/2)a2 is the area of a unit cell of 2D graphite, andw1 characterizes the interaction strength. A similarexpression can be written down for matrix elementsinvolving electrons in different valleys. In terms of theeffective Coulomb potential V (R−R′) between the πorbitals at R and R′, we have

w1 ≈ 12

∑R

[V (R) − V (R − τ)], (2.12)

where R is the lattice translation vector and τ is thevector connecting nearest-neighbor A and B sites asshown in Fig. 1.

The short-range part of the Coulomb interactioncauses also inter-valley scattering between K and K’points as shown in Appendix B. The inter-valley termis written as

V(2)(K′2,K1)(K4,K′3)

= Ω0w2

∫dr [FK′A

2 (r)∗FKA1 (r)FKA

4 (r)∗FK′A3 (r)

+ FK′B2 (r)∗FKB

1 (r)FKB4 (r)∗FK′B

3 (r)],(2.13)

withw2 ≈

∑R

V (R) ei(K−K′)·R. (2.14)

A similar expression is obtained for V(2)(K2,K′1)(K′4,K3).

When we approximately set V (R) ∝ e2/|R|, forR �=0, we have

w1 ≈ 12[V (0) − c1V (a)],

w2 ≈ V (0) − c2V (a),(2.15)

with dimensionless constants c1 = 2.67 · · · and c2 =1.54 · · ·. The above shows that

w1 <∼12w2. (2.16)

We can also use the Ohno potential,30,31) which

Valley Mixing and Exchange in Carbon Nanotubes Page 3

realistically describes organic polymer systems:

V (R) =U√

(U |R|/e2)2+1, (2.17)

where U ≈ 11.3 eV is the energy cost to place twoelectrons on the same site (R = 0). The resulting w1

and w2 are shown in Fig. 4, which gives w1/w2 ∼ 0.17,w1 ∼ 0.7 eV, and w2 ∼ 4.0 eV for e2/a = 5.85 eVcorresponding to a=2.46 A. The actual value of w1/w2 ishighly likely to be sensitive to model potentials, however.Further, w1 and w2 are reduced by screening, but theamount of the effect is likely to be smaller than thatgiven by the static dielectric constant κ introduced forthe long-range Coulomb interaction.

A simplest nearest-neighbor tight-binding model hasbee used to derive the short-range Coulomb interactionin Appendices A and B. It is worth being pointed outthat the parameterization into w1 and w2 is much moregeneral although their actual values can be different fromthose estimated above. As will be demonstrated below,the relative ordering of exciton levels are quite sensitiveto the ratio w1/w2. Therefore, we should leave w1 andw2 rather as adjustable parameters to be determinedexperimentally.

§3. Exciton

3.1 Long-Range Coulomb Interaction

The exciton energy levels and wave functions aredetermined mainly by the long-range Coulomb interac-tion V . Because short-range interactions w1 and w2

constitute only small corrections, we shall consider themin the lowest-order perturbation theory. The excitonwave function for an electron in the K valley and a holein the K valley, to be called the KK exciton, is given by

|KK〉 =∑

n

∑k

ψKKn (k) cK†

+,n,kcK−,n,k|g〉, (3.1)

where |g〉 is the ground-state wave function correspond-ing to the filled valence bands and empty conductionbands and cK†

s,n,k and cKs,n,k are the creation and annihi-

lation operators for states s, n, k of the valley K. Usingthe velocity operator vy = (γ/h)σy , the matrix elementfor optical transition is calculated as

〈KK|vy|g〉 = +iγ

h

∑n

∑k

κνϕ(n)√κνϕ(n)2+k2

ψKKn (k)∗.

(3.2)The equation of motion for ψKK

n (k) becomes

ε ψKKn (k) =

(εK

nk+εKnk+ΔεKK

nk

)ψKK

n (k)

−∑m

∑q

V KK(+,n,k ; +,m,k+q)(−,m,k+q ;−,n,k) ψKK

m (k+q),

(3.3)with

V KK(+,n,k ; +,m,k+q)(−,m,k+q ;−,n,k)

=2e2

Aκ ε|n−m|(|q|)I|n−m|(L|q|

2π

)K|n−m|

(L|q|2π

)

× (F K†+,n,kF K

+,m,k+q)(FK†−,m,k+qF

K−,n,k),

(3.4)

and

ΔεKKnk = ΣK

+,n,k − ΣK−,n,k, (3.5)

where ε|n−m|(|q|) is the dielectric function describingscreening effects due to virtual interband transitions inthe vicinity of the K and K’ points, and In(t) and Kn(t)are the modified Bessel function of the first and secondkind, respectively. The self-energy is written as32,33)

ΣK±,n,k = ±1

2(ΣK

+,n,k−ΣK−,n,k

), (3.6)

with

ΣKn,±,k = −

∑m

∑q

2e2

Aκ εn−m(q)I|n−m|

(L|q|2π

)

× K|n−m|(L|q|

2π

)g0

[εK

m(k+q)]

× (F K†±,n,kF K

−,m,k+q)(FK†−,m,k+qF

K±,n,k),

(3.7)

where g0(ε) is a cutoff function, defined by

g0(ε) =εαc

c

|ε|αc +εαcc

. (3.8)

The symmetrization of the self-energy given in eq. (3.6)is necessary to satisfy the symmetry of the energy bandsaround ε=0 in the presence of interaction.32,33)

The strength of the Coulomb interaction is speci-fied by the dimensionless parameter (e2/κL)(2πγ/L)−1,which is independent of the circumference L and es-timated as (e2/κL)(2πγ/L)−1 ≈ 0.35/κ for γ = 6.46eV·A, which corresponds to γ =

√3a|γ0|/2 with nearest-

neighbor transfer integral γ0 = 3 eV and a = 2.46 A.In bulk graphite we have κ≈ 2.4.34) In CN, this simpleconstant screening is valid only approximately becauseof the cylindrical form with hollow vacuum inside andsurrounding material outside. Therefore, κ should betreated as a parameter in the following. As long as κ isindependent of L, the interaction parameter is indepen-dent of the CN diameter and therefore the band gap andalso the exciton binding energy are inversely proportionalto the circumference length L or the diameter d=L/π.

The cutoff function (3.8) contains two parametersαc and εc and should be chosen in such a way that onlythe contributions from states in the vicinity of the Fermilevel, where the k·p scheme is valid, should be included.The self-energy shift depends on the cutoff energy loga-rithmically, but is essentially independent of the parame-ter αc as long as the cutoff function decays smoothly butrapidly enough.32,33) The appropriate value of εc is thehalf of the π band width ∼ 3γ0, where −γ0 denotes thehopping integral between nearest-neighbor carbon atomsand related to γ through γ = (

√3/2)γ0a. This shows

that the self-energy shift has an extra weak logarithmicdependence on the diameter ∝ ln(L/a) in addition to theuniversal L−1 scaling mentioned above.

We can obtain similar expressions for the K’K’exciton consisting of an electron and a hole in the K’valley, KK’ for an electron in the K valley and a hole inthe K’ valley, and K’K for an electron in the K’ valleyand a hole in the K valley. The velocity operator for

Page 4 T. Ando

the K’ point is given by vy =−(γ/h)σy, and therefore itsmatrix element becomes

〈K ′K ′|vy |g〉 = −iγ

h

∑n

∑k

κ′νϕ(n)√

κ′νϕ(n)2+k2

ψK′K′n (k)∗,

(3.9)with ψK′K′

n (k) being the wave function of the K’K’exciton. Obviously, matrix elements for KK’ and K’Kexcitons vanish identically. In the absence of a magneticflux (ϕ=0), we have the symmetry relations ψK′K′

n (k)=ψKK−n (k) and κ′

νϕ(n)=−κνϕ(−n). Therefore, the opticalmatrix elements are exactly the same between the K andK’ points. In the presence of a magnetic flux (ϕ �=0), thissymmetry is destroyed.

If we consider the contribution of the lowest con-duction band and the higher valence band, both wavefunction and binding energy for the KK’ and K’K exci-tons are essentially same as those of the KK and K’K’excitons in the absence of a magnetic flux. When interband mixing is included, they become different, however.Consider the KK’ exciton in CN with ν =1, for example.When an electron is scattered from the lowest conductionband n=0 to the first excited conduction band n=+1 atthe K point, a hole at the K’ point in the highest valenceband is scattered into the valence band n=1 which lieslower than the second valence band n=−1. As a result,effects of interband scattering are smaller and the KK’and K’K excitons have an energy slightly higher thanthat of the KK and K’K’ excitons. This difference canbe comparable to energy shifts due to short-range partof the Coulomb interaction as shown below.

3.2 Short-Range Coulomb Interaction

Including spins we can write the exciton wave func-tions for an electron in the valley v and spin σ and ahole in the valley v′ and spin σ′ as |(v, σ)(v′, σ′)〉 withv, v′ = K or K′ and σ, σ′ =↑ or ↓. There are 16 statesin total. They are first classified into singlet and tripletdepending on the total spin. The singlet states consistof four states

1|v, v′〉 =1√2[|(v, ↑)(v′, ↑)〉+|(v, ↓)(v′, ↓)〉], (3.10)

and the triplet states consist of twelve states

3|v, v′, +1〉 = |(v, ↑)(v′, ↓)〉,3|v, v′, 0〉 =

1√2

[|(v, ↑)(v′, ↑)〉 − |(v, ↓)(v′, ↓)〉],3|v, v′, +1〉 = |(v, ↓)(v′, ↑)〉.

(3.11)Note that the spin of a hole is denoted by that of thecorresponding electron in the valence band in the aboveand therefore is opposite to notations used frequently.

They are split further due to the inter-valley short-range interaction. In the absence of a magnetic flux,the singlet states split into the bonding and antibondingof 1|KK〉 and 1|K ′K ′〉, and two degenerate 1|KK ′〉and 1|K ′K〉. The triplet states split in the same wayalthough each state has a degeneracy of three due to thedirection of the total spin of the electron and hole.

Figure 5 shows the matrix elements of the short-range part of the Coulomb interaction contributing tosinglet excitons 1|KK〉 and 1|K ′K ′〉. The effectiveHamiltonian becomes

1VKK,K′K′ =Ω0

L2

( 1|KK〉 1|K ′K ′〉−w1 p(KK)(KK)+2w1 qKK −w2 r(KK)(K′K′)+2w1qKK′

−w2 r(K′K′)(KK)+2w1qK′K −w1 p(K′K′)(K′K′)+2w1 qK′K′

), (3.12)

with

p(v1v2)(v3v4) = Tr σzG†v1v2

σzGv3v4 , (3.13)

qv1v2 = TrσzG†v1v1

Tr σzGv2v2 , (3.14)

r(v1v2)(v3v4) = Tr Gd†v1v2

Gdv3v4

, (3.15)

where we have defined the (2×2) matrix

Gvv′ =L

A

∑n,k

F v+,n,kψvv′

n (k)F v′†−,n,k, (3.16)

and Gdvv′ consists of the diagonal elements of Gvv′ .

Figure 6 shows the matrix elements of the short-range part of the Coulomb interaction contributing to theenergy shift of the singlet excitons 1|KK ′〉 and 1|K ′K〉.The resulting energy shifts are given by

1VKK′ =Ω0

L2

(2w2 r(KK′)(KK′)− w1 p(KK′)(KK′)

),

1VK′K =Ω0

L2

(2w2 r(K′K)(K′K)−w1 p(K′K)(K′K)

).

(3.17)Figure 7 shows the matrix elements contributing to

the triplet excitons 3|KK〉 and 3|K ′K ′〉. The effective

Hamiltonian becomes

3VKK,K′K′ =Ω0

L2

( 3|KK〉 3|K ′K ′〉−w1 p(KK)(KK) −w2 r(KK)(K′K′)−w2 r(K′K′)(KK) −w1 p(K′K′)(K′K′)

).

(3.18)Similarly, the matrix elements shown in Fig. 8 give theenergy shift for triplet excitons 3|KK ′〉 and 3|K ′K〉

3VKK′ = −Ω0

L2w1 p(KK′)(KK′),

3VK′K = −Ω0

L2w1 p(K′K)(K′K).

(3.19)

Among 16 states, eigenstates of 1VKK,K′K′ can beoptically allowed. In the absence of a magnetic flux, wehave

p ≡ p(KK)(KK) = p(K′K′)(K′K′),

r ≡ r(KK)(K′K′) = r(K′K′)(KK),

q ≡ qKK = qK′K′

q′ ≡ qKK′ = qK′K ,

(3.20)

because of the symmetry mentioned above. Therefore,

Valley Mixing and Exchange in Carbon Nanotubes Page 5

the eigenstates are given by

1|KK-K ′K ′(±)〉 =1√2

(1|KK〉 ± 1|K ′K ′〉). (3.21)

Only state 1|KK-K ′K ′(+)〉 is optically allowed becauseof the identical optical matrix elements between the Kand K’ points. When w1/w2 < r/2q, optically allowed1|KK-K ′K ′(+)〉 is lower than 1|KK-K ′K ′(−)〉 in en-ergy, but this ordering is reversed when w1/w2 >r/2q.

Without explicit calculations, we can make veryrough estimation of the matrix elements p, q, and r inthe absence of a magnetic flux. We shall replace F K

s,n,k

etc. by those with n=0 and k=0, i.e.,

F K±,0,k =

1√2

(1±1

), F K′

±,0,k =1√2

(1∓1

). (3.22)

Then, we have

GKK ∼ 12

(1 −11 −1

)[LψKK(0)]2,

GK′K′ ∼ 12

(1 1−1 −1

)[LψK′K′(0)]2,

GKK′ ∼ 12

(1 11 1

)[LψKK′(0)]2,

GK′K ∼ 12

(1 −11 −1

)[LψK′K(0)]2,

(3.23)

where the effective amplitude of the exciton wave func-tion is defined by

ψvv′(0) =1√AL

∑n,k

ψvv′n (k), (3.24)

satisfying

ψKK(0) = ψK′K′(0) ≈ ψKK′(0) = ψK′K′(0) ≈ ψ(0).(3.25)

Therefore, we have

p ∼ 0, q ≈ q′ ∼ [Lψ(0)]2, r ∼ 12[Lψ(0)]2. (3.26)

The effective Hamiltonian and energy shift areroughly given by

1VKK,K′K′ ∼ Ω0ψ(0)2(

2w1 − 12w2+2w1

− 12w2+2w1 2w1

), (3.27)

1VKK′ = 1VK′K ∼ Ω0ψ(0)2w2, (3.28)

3VKK,K′K′ ∼ Ω0ψ(0)2(

0 − 12w2

− 12w2 0

), (3.29)

3VKK′ = 3VK′K ∼ 0. (3.30)

The critical value corresponding to the change in theordering of optically active and inactive states is givenby w1/w2∼1/4.

As was shown previously,1,2) the exciton is almostone-dimensional and its wave function is extended overthe circumference almost equally. Further, the bindingenergy is scaled by e2/κL∝L−1 and therefore the extentof the wave function in the axis direction is proportionalto L. As a result ψ(0) is scaled by L−1 and Lψ(0)

is independent of the circumference or the diameter.This shows that the energy splitting and the shift dueto the short-range part of the Coulomb interaction isproportional to (Ω0/L2)w2 and decreases rapidly withthe increase of the circumference in proportion to (a/L)2

in contrast to a/L dependence of the band gap andexciton energy themselves. They can play an appreciablerole only in very thin CN’s. The dependence ∝ (a/L)2

is quite in contrast to the suggestion ∝ a/L suggestedrecently.25)

§4. Numerical Results and DiscussionIn the following some examples of numerical results

are shown for the cutoff energy εc(2πγ/L)−1 =10 corre-sponding to typical CN’s with diameter ∼1.4 nm and theeffective interaction strength (e2/κL)(2πγ/L)−1 = 0.1and 0.2. The results for other values of the parametersare qualitatively same.

Figure 9 shows the fundamental (n = 0) band gapsεG and ε0

G in the presence and absence of interaction,respectively, and the lowest exciton energy-level εex as afunction of the AB flux. First, we notice that althoughthe band gap is enhanced considerably by interaction,the exciton energy level remains almost independent ofthe interaction strength due to the strong cancellation ofinteraction effects on the band gap and exciton bindingenergy.2) The energies of the KK and K’K’ excitonsdecrease and increase, respectively, with the flux due tothe Aharonov-Bohm effect. The KK’ and K’K excitonshave an energy slightly larger than the KK and K’K’excitons in the absence of the flux as discussed in §3.1although the difference is unrecognizable in this scale.Their energies remain essentially independent of the flux.

Figure 10 shows the parameters p, q, and r de-scribing effects of the short-range part of the Coulombinteraction as a function of the flux. As discussed inthe previous section, we have p q and r ∼ (1/2)q,showing that the rough approximation turns out to besurprisingly good.

Figure 11 shows the eigenvalues of the effectiveHamiltonian 1VKK,K′K′ , · · · describing short-range Coul-omb interaction as a function of w1/w2 in the absenceof the flux. Triplet excitons have a tendency to lielower in energy than the singlet excitons. The singletexciton consisting of the bonding combination of the Kand K’ points denoted by ‘Singlet KK-K’K’(+)’ in thefigure is optically active and all other levels are inactive.The optical active exciton is frequently called the brightexciton and inactive excitons are called dark excitons.The figure shows that the energy of the bright excitondepends on w1, while dark excitons remain independentof w1. The relative ordering of the singlet KK-K’K’(+)and KK-K’K’(−) excitons is reversed around w1/w2 ≈0.25. For the Ohno potential with U = 11.3 eV givingw1/w2 < 0.2, the bright exciton has the lowest energyamong all singlet excitons. The singlet KK-K’K’(−)state is always degenerate with triplet KK-K’K’(−).

It should be noticed that the actual energy levels ofKK’ and K’K excitons relative to those of KK and K’K’excitons are obtained by adding the contribution aris-ing from the small difference of interband contributionsdiscussed in §3.1 to those shown in Fig. 11. In fact,

Page 6 T. Ando

the momentum conservation along the circumferencedirection makes the KK’ and K’K exciton slightly higherin energy than the KK and K’K’ excitons. This differenceis smaller than the splitting due to the short-rangeCoulomb interaction for thin CN’s as shown in Fig. 12,but becomes comparable for thicker tubes.

Figure 12 shows some examples of the energy levelsof the singlet and triplet excitons as a function of the fluxfor the parameter w2/γ0 =1.5 in CN’s with circumferenceL/a = 20. In the absence of a flux, for w1/w2 = 0.4shown in Fig. 12 (a) the highest-energy exciton is bright,while for w1/w2 = 0.2 shown in Fig. 12 (b) the brightexciton has the lowest energy among singlet excitons.Mixing between the K and K’ points diminishes rapidlywith the increase of the flux and excitons for the Kand K’ points become essentially independent of eachother when the Aharonov-Bohm splitting between themexceeds the splitting of their bonding and antibondingcombinations in the absence of the flux.

Figure 13 shows the optical intensity of the singletKK-K’K’(+) and KK-K’K’(−) excitons as a function ofthe flux. In the absence of the flux, only the bondingor antibonding combination of KK and K’K’ excitonsare bright depending on the ratio w1/w2. With theincrease of the flux, however, both combinations becomeoptically active and change into independent KK andK’K’ excitons for sufficiently large flux. The opticalintensity of the KK exciton becomes larger than that ofthe K’K’ exciton because of the decrease in the energydue to AB flux.2)

Figure 14 shows some examples of the dynamicalconductivity describing the absorption spectra. A phe-nomenological broadening Γ(2πγ/L)−1 = 0.01 has beenintroduced in this example. In the absence of the flux, asingle absorption peak appears only at the energy of thebonding or antibonding combination of KK and K’K’excitons depending on the ratio w1/w2 as mentionedabove. With the increase of the flux, clear double peaksappear corresponding to the KK and K’K’ exciton, splitby the AB flux.

The splitting of the exciton peaks due to theAharonov-Bohm effect was observed recently.13−17) Animportant characteristic feature of the results is that thesplitting does not seem to become observable until themagnetic flux reaches a certain critical value and thenstarts to increase with the flux. This is consistent withthe present result that two peaks appear only when theAharonov-Bohm splitting exceeds the coupling energy ofthe KK and K’K’ exciton due to the short-range part ofthe Coulomb interaction.

It has been shown that the triplet excitons tend tohave energies lower than the singlet excitons. Further,the ordering of the singlet excitons turns out to be quitesensitive to the parameters w1 and w2 characterizing thestrength of the short-range Coulomb interaction. It isvery difficult to give a reliable value of these parameterseven by a first-principles calculation of energy bandsand excitons because they are sensitive to details of thepotential and the wave functions. This ordering playsimportant roles in determining the quantum efficiencyof photoluminescence in thin CN’s with relatively large

splitting. With the increase of the thickness, the split-ting becomes rapidly smaller and such problems becomeirrelevant.

First-principles calculations of exciton energy levelswere reported recently for a very thin zigzag nanotubewith L/a = 10 in the absence of AB flux.26) The re-sults show that the bright bonding state of singlet KKand K’K’ excitons lies much higher than the dark anti-bonding states, but lies slightly lower than the singletKK’ and K’K excitons. This roughly corresponds tothe present result for w1/w2 ∼ 3.5 shown in Fig. 11.Further, the highest triplet exciton corresponding to thebonding of KK and K’K’ lies very close to the lowestsinglet exciton in excellent agreement with the presentresult that they are completely degenerate. Quantita-tively, however, the amount of the splitting of the tripletexcitons is almost the same as that of the singlet excitons,which is in contrast to the larger splitting shown in Fig.11. The origin of such disagreement is not known.

§5. ConclusionThe effective k·p Hamiltonian describing effects of

the short-range part of the Coulomb interaction onexcitons has been derived based on a nearest-neighbortight-binding model. It describes the splitting of excitonsinto singlet and triplet due to exchange and into differentintervalley combinations such as coupled KK and K’K’,KK’, and K’K, where, for example, KK’ stands for anexciton with an electron in the K valley and a hole in theK’ valley. Only the bonding combination of KK and K’K’singlet excitons is optically active (bright) and others areall inactive (dark) in the absence of an Aharonov-Bohmmagnetic flux passing through the cross section of thenanotube. With the increase of the flux, the KK andK’K’ excitons split and both become bright.

The short-range Coulomb interaction is character-ized by two parameters w1 and w2 and the resultingenergy splitting and shift of the exciton energy levels areproportional to a2ψ(0)2, where ψ(0) is the exciton am-plitude for the vanishing electron-hole distance. Becauseψ(0) ∝ L−1, they decrease in proportion to (a/L)2 andbecome unimportant for thick nanotubes. The relativeordering of the bright and dark excitons is sensitive tow1/w2. This ratio satisfies w1/w2 <∼ 1/2, but its exactvalue depends sensitively on the form of the effectivepotential and the π-band wave functions. A detailed andcareful experimental study of the Aharonov-Bohm effecton the exciton may be used to determine this relativeordering.

AcknowledgmentsThis work was supported in part by a 21st Century

COE Program at Tokyo Tech “Nanometer-Scale Quan-tum Physics” and by Grant-in-Aid for Scientific Researchfrom Ministry of Education, Culture, Sports, Scienceand Technology Japan. Numerical calculations were per-formed in part using the facilities of the SupercomputerCenter, Institute for Solid State Physics, University ofTokyo.

Appendix A: Intra-Valley InteractionConsider a tight-binding model in which only a sin-

gle resonance integral between nearest-neighbor carbon

Valley Mixing and Exchange in Carbon Nanotubes Page 7

atoms is included and others are completely neglectedtogether with overlapping integrals. In terms of theenvelope functions F K and F K′

, the wave functions atK and K’ points are written as

ψK(q) = Ω1/20

∑RA

eiK·RAFKA(RA)φ(q−RA)

+ Ω1/20

∑RB

(−ω)eiηeiK·RBFKB(RB)φ(q−RB),

ψK′(q) = Ω1/2

0

∑RA

eiηeiK′·RAFK′A(RA)φ(q−RA)

+ Ω1/20

∑RB

eiK′·RBFK′B(RB)φ(q−RB),

(A1)where η is the chiral angle, Ω0 = (

√3/2)a2, ω =

exp(2πi/3), q is the electron coordinate in three dimen-sion, φ(q) is the wave function of a π orbital at the origin,and RA and RB denote A and B sites, respectively.

First, we consider an intra-valley process shownby the top-left diagram of Fig. 3 (a). Neglecting theoverlapping of orbitals localized at difference sites, wehave

V(K2,K1)(K4,K3) = Ω20

∑α,β=A,B

∑Rα,R′

β

V (Rα−R′β)

× FKα2 (Rα)∗FKα

1 (Rα)FKβ4 (R′

β)∗FKβ3 (R′

β), (A2)

where V (R − R′) is the effective Coulomb potentialbetween an electron localized at the π orbital at R andR′, defined by

V (R−R′) =∫

d3q

∫d3q′ vC(q−q′)|φ(q−R)|2|φ(q′−R′)|2,

(A3)where vC(q−q′) represents the Coulomb energy betweencharges at q and q′.

Introduce a smoothing function g(r) which variessmoothly in the range |r|<∼ a and decays rapidly andvanishes for |r|�a.3) It should satisfy the conditions

∑RA

g(r−RA) =∑RB

g(r−RB) = 1, (A4)

and ∫g(r−RA)dr =

∫g(r−RB)dr = Ω0. (A5)

This function g(r−R) can be treated as Ω−10 δ(r−R)

when multiplied by slowly-varying functions such as wavefunction F .

We multiply the inside of the summation in theright-hand side of eq. (A2) by

Ω−10

∫dr g(r−Rα)Ω−1

0

∫dr′ g(r′−R′

β),

which is unity identically. Then, using the slowly-varyingnature of the wave functions, we can rewrite eq. (A2) as

V(K2,K1)(K4,K3) =∑

α,β=A,B

∫dr

∫dr′vαβ(r−r′)

× FKα2 (r)∗FKα

1 (r)FKβ4 (r′)∗FKβ

3 (r′), (A6)

with

vαβ(r − r′) =∑Rα

∑R′

β

g(r−Rα)g(r′−R′β)V (Rα−R′

β).

(A7)Obviously, we have

vAA(r−r′) = vBB(r−r′),vAB(r−r′) = vBA(r−r′).

(A8)

Define the symmetric and antisymmetric combina-tions of the potential as

v0(r−r′) =12[vAA(r−r′)+vAB(r−r′)],

v1(r−r′) =12[vAA(r−r′)−vAB(r−r′)].

(A9)

Then, the Coulomb matrix elements are written as

V(K2,K1)(K4,K3) = V(0)(K2,K1)(K4,K3) + V

(1)(K2,K1)(K4,K3),

(A10)with

V(0)(K2,K1)(K4,K3) =

∫dr

∫dr′ v0(r−r′)

× [F K2 (r)†F K

1 (r)][F K4 (r′)†F K

3 (r′)], (A11)

and

V(1)(K2,K1)(K4,K3) =

∫dr

∫dr′ v1(r−r′)

× [F K2 (r)†σzF

K1 (r)][F K

4 (r′)†σzFK3 (r′)], (A12)

where σz is the z component of the Pauli spin matrix.The term V (0) represents the long-range Coulomb inter-action included already in the lowest-order k·p theory.There can be a small correction if we use the actualform of v0(r−r′) in stead of v(r−r′) = e2/κD(r−r′)where D(r − r′) is the distance between r and r′ inthree dimension. However, this correction causes a slightshift in the whole exciton energies and therefore doesnot contribute to mixing between K and K’ points andexchange effects. Therefore, we shall confine ourselves tothe term V (1) in the following.

Because main contributions to v1(r−r′) come fromshort-range interactions, we can approximately set

v1(r−r′) = Ω0 w1 δ(r−r′), (A13)

and

V(1)(K2,K1)(K4,K3) = Ω0 w1

∫dr [F K

2 (r)†σzFK1 (r)]

×[F K4 (r)†σzF

K3 (r)],

(A14)where

w1 ≈ Ω−10

∫v1(r)dr. (A15)

This gives eq. (2.14). Because w1 is dominated byshort-range interaction, it can be estimated safely byusing lattice sites of the 2D graphite without inclusion

Page 8 T. Ando

of finite curvature of CN.

Appendix B: Inter-Valley Interaction

The process shown in the upper part of Fig. 3 (b) isgiven by

V(2)(K′2,K1)(K4,K′3)

= Ω20

∑RA,R′

A

FK′A2 (RA)∗FKA

1 (RA)FKA4 (R′

A)∗FK′A3 (R′

A)

× ei(K−K′)·(RA−R′A)V (RA−R′

A)

+ Ω20

∑RA,RB

FK′A2 (RA)∗FKA

1 (RA)FKB4 (RB)∗FK′B

3 (RB)

× (−ω−1)e−2iηei(K−K′)·(RA−RB)V (RA−RB)

+ Ω20

∑RA,RB

FK′B2 (RB)∗FKB

1 (RB)FKA4 (RA)∗FK′A

3 (RA)

× (−ω)e+2iηei(K−K′)·(RB−RA)V (RB−RA)

+ Ω20

∑RB ,R′

B

FK′B2 (RB)∗FKB

1 (RB)FKB4 (R′

B)∗FK′B3 (R′

B)

× ei(K−K′)·(RB−R′B)V (RB−R′

B). (B1)

In terms of the smoothing function g(r), this is rewrittenas

V(2)(K′2,K1)(K4,K′3)

=∫

dr

∫dr′FK′A

2 (r)∗FKA1 (r)FKA

4 (r′)∗FK′A3 (r′)

× v′AA(r−r′)

+∫

dr

∫dr′FK′A

2 (r)∗FKA1 (r)FKB

4 (r′)∗FK′B3 (r)

× (−ω−1)e−2iηv′AB(r−r′)

+∫

dr

∫dr′FK′B

2 (r)∗FKB1 (r)FKA

4 (r′)∗FK′A3 (r′)

× (−ω)e+2iηv′BA(r−r′)

+∫

dr

∫dr′FK′B

2 (r)∗FKB1 (r)FKB

4 (r′)∗FK′B3 (r′)

× v′BB(r−r′), (B2)

with

v′αβ(r−r′) =∑

Rα,R′β

g(r−Rα)g(r′−R′β)V (Rα−R′

β)

× ei(K−K′)·(Rα−Rβ),(B3)

Obviously, we have

v′AA(r−r′) = v′BB(r−r′),v′AB(r−r′) = v′BA(r−r′)∗.

(B4)

Considering only short-range contributions, we can ap-proximately set

v′AA(r−r′) = v′BB(r−r′) = Ω0 w2 δ(r−r′),v′AB(r−r′) = v′BA(r−r′)∗ = Ω0 w′

2 δ(r−r′).(B5)

The magnitude of w2 and w′2 may be estimated as

w2 ≈∑R

V (R)ei(K−K′)·R,

w′2 ≈

∑R

V (R+�τ)ei(K−K′)·(R+�τ).(B6)

With the use of the rotational symmetry of the lattice,

we can show w′2 =0. Thus, we have

V(2)(K′2,K1)(K4,K′3)

= Ω0w2

∫dr [FK′A

2 (r)∗FKA1 (r)FKA

4 (r)∗FK′A3 (r)

+ FK′B2 (r)∗FKB

1 (r)FKB4 (r)∗FK′B

3 (r)].(B7)

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A197.

Figure Captions

Fig. 1 (a) Lattice structure of a two-dimensionalgraphite sheet. The coordinates (x′, y′) are fixedon the graphite sheet and (x, y) are chosen in such away that x is along the circumference of a nanotubeand y is along the axis. η is the chiral angle. (b) Thefirst Brillouin zone and K and K’ points. (c) Thecoordinates for a nanotube. The Aharonov-Bohmflux φ is applied in the axis direction.

Fig. 2 (a) A schematic illustration of energy bands of asemiconducting nanotube with ν =1. The quantumnumber n for the motion along the circumferenceis different between the K and K’ points. (b) Aschematic illustration of the shift of the band edgesas a function of the magnetic flux φ. The shift isopposite between the K and K’ points, leading toAharonov-Bohm splitting of absorption peaks.

Fig. 3 Elementary processes corresponding to electron-electron interaction. (a) Intra-valley processes con-sisting of the dominant long-range part V and asmall short-range part V (1). The latter is charac-terized by parameter w1. (b) Inter-valley processesV (2) due to short-range part of the Coulomb inter-action, characterized by w2.

Fig. 4 Parameters w1 and w2 characterizing the short-range part of the Coulomb interaction, calculatedfor the Ohno potential for varying on-site potentialU . For U = 11.3 eV and e2/a = 5.85 eV, w1/w2 ∼0.17, w1∼0.7 eV, and w2∼4.0 eV.

Fig. 5 Interaction processes corresponding to singlet

KK and K’K’ excitons. Upper lines with the left-ward arrow denote an electron in a conduction bandand lower lines with the rightward arrow denote thatin a valence band. Intra-valley scattering due to w1

is described by (a) and its exchange counter partby (b) and (c). Inter-valley scattering due to w2 isdescribed by (d). σ and σ′ denote the spin direction↑ and ↓.

Fig. 6 Interaction processes corresponding to a singletK’K exciton (a) and a singlet KK’ exciton (b).

Fig. 7 Interaction processes corresponding to tripletKK and K’K’ excitons. (a) Intra-valley processesdue to w1 and (b) inter-valley processes due to w2.−σ denote the spin direction opposite to σ.

Fig. 8 Interaction processes corresponding to a tripletKK’ exciton (a) and a triplet K’K exciton (b).

Fig. 9 Calculated excitation energies for the funda-mental band gap of a semiconducting nanotube andcutoff energy εc(2πγ/L)−1 = 10 as a function ofmagnetic flux φ. (a) (e2/κL)(2πγ/L)−1 = 0.1 and(b) 0.2. The band gap in the presence and absenceof interaction is denoted by εG and ε0

G, respectively,and the exciton energy is denoted by εex.

Fig. 10 Parameters p, q, and r corresponding tothe matrix elements of the short-range part of theCoulomb interaction as a function of the flux. (a)(e2/κL)(2πγ/L)−1=0.1 and (b) 0.2.

Fig. 11 The eigenvalues of the effective Hamiltonian1VKK,K′K′ , 1VKK′ , 1VK′K , 3VKK,K′K′ , 3VKK′ , and3VK′K describing the short-range Coulomb interac-tion as a function of w1/w2 in the absence of theflux. (a) (e2/κL)(2πγ/L)−1=0.1 and (b) 0.2.

Fig. 12 Calculated energy levels of the singlet andtriplet excitons as a function of the flux for theparameter w2/γ0 = 1.5 and L/a = 20. (a) (e2/κL)(2πγ/L)−1=0.1 and (b) 0.2.

Fig. 13 Calculated optical intensity of the singlet KK-K’K’(+) and KK-K’K’(−) excitons as a function ofthe flux. L/a = 20. (e2/κL)(2πγ/L)−1 = 0.2 (a)w1/w2 =0.4 and (b) 0.2. Thin dotted lines representthe results for w1 =w2 =0.

Fig. 14 Calculated dynamical conductivity describ-ing the absorption spectra. A phenomenologicalbroadening Γ(2πγ/L)−1 =0.01 has been introduced.The flux φ is varied with interval Δφ/φ0 = 0.001up to φmax/φ0 = 0.02. The curves are shifted inthe vertical direction. The solid lines represent theconductivity and the dotted lines represent con-tributions of two singlet KK-K’K’ excitons. Thepeak positions in the presence and absence of theshort-range interaction are shown also. L/a = 20.(e2/κL)(2πγ/L)−1 = 0.2. (a) w1/w2 = 0.4 and(b) 0.2. Thin dotted lines represent the results forw1 =w2 =0.

Page 10 T. Ando

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Fig. 7 Fig. 8

Page 12 T. Ando

0.00 0.05 0.100.0

0.5

1.0

1.5

Flux (units of φ0)

Exc

itatio

n E

nerg

y (u

nits

of 2

πγ/L

)

εG0

εG

εex

ν = 1(e2/κL)(2πγ/L)-1=0.1εc(2πγ/L)-1= 10

(a)

0.00 0.05 0.100.0

0.5

1.0

1.5

Flux (units of φ0)

Exc

itatio

n E

nerg

y (u

nits

of 2

πγ/L

)

εG0

εG

εex

ν = 1(e2/κL)(2πγ/L)-1=0.2εc(2πγ/L)-1= 10

KK K’K’ KK’, K’K

(b)

Fig. 9 (a) Fig. 9 (b)

0.00 0.05 0.100.0

0.5

1.0

1.5

Flux (units of φ0)

Par

amet

ers:

p, q

, and

r

ν = 1(e2/κL)(2πγ/L)-1=0.1εc(2πγ/L)-1= 10

p(KK)(KK) p(K’K’)(K’K’)

p(KK’)(KK’), p(K’K)(K’K)

q(KK)

q(K’K’)q(K’K’)

r(KK)(KK)r(K’K’)(K’K’)

r(KK)(K’K’), r(K’K’)(KK)

r(KK’)(KK’), r(K’K)(K’K)

(a)

0.00 0.05 0.100.0

0.5

1.0

1.5

2.0

Flux (units of φ0)

Par

amet

ers:

p, q

, and

r

ν = 1(e2/κL)(2πγ/L)-1=0.2εc(2πγ/L)-1= 10

p(KK)(KK)p(K’K’)(K’K’)

p(KK’)(KK’), p(K’K)(K’K)

q(KK)

q(K’K’)q(K’K’)

r(KK)(KK)r(K’K’)(K’K’)

r(KK)(K’K’), r(K’K’)(KK)

r(KK’)(KK’), r(K’K)(K’K)

(b)

Fig. 10 (a) Fig. 10 (b)

Valley Mixing and Exchange in Carbon Nanotubes Page 13

0.0 0.1 0.2 0.3 0.4 0.5-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

w1/w2

Ene

rgy

(uni

ts o

f w2Ω

0/L2

)

Triplet KK’ & K’K

Triplet KK-K’K’(+)

Singlet KK’ & K’K

Singlet KK-K’K’(-)

Triplet KK-K’K’(-)

Singlet KK-K’K’(+)

ν = 1(e2/κL)(2πγ/L)-1=0.1εc(2πγ/L)-1= 10

(a)

0.0 0.1 0.2 0.3 0.4 0.5-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

w1/w2

Ene

rgy

(uni

ts o

f w2Ω

0/L2

)

Triplet KK’ & K’K

Triplet KK-K’K’(+)

Singlet KK’ & K’K

Singlet KK-K’K’(-)

Triplet KK-K’K’(-)

Singlet KK-K’K’(+)ν = 1(e2/κL)(2πγ/L)-1=0.2εc(2πγ/L)-1= 10

(b)

Fig. 11 (a) Fig. 11 (b)

0.00 0.01 0.02

-0.05

0.00

0.05

0.10

Flux (units of φ0)

Ene

rgy

Shi

ft (u

nits

of 2

πγ/L

)

KK-K’K’(-)

K

KK-K’K’(+)

K’

KK’ & K’K

KK-K’K’(+)

KK-K’K’(-)

KK’ & K’K

L/a= 20.0 (ν =1)(e2/κL)(2πγ/L)-1=0.2εc(2πγ/L)-1=10w2/γ0=1.5w1/w2=0.4

Singlet Triplet Without KK’ Mixing and Exchange

(a)

0.00 0.01 0.02

-0.05

0.00

0.05

0.10

Flux (units of φ0)

Ene

rgy

Shi

ft (u

nits

of 2

πγ/L

)

KK-K’K’(+)

K

KK-K’K’(-)

K’

KK’ & K’K

KK-K’K’(+)

KK-K’K’(-)

KK’ & K’K

L/a= 20.0 (ν =1)(e2/κL)(2πγ/L)-1=0.2εc(2πγ/L)-1=10w2/γ0=1.5w1/w2=0.2

Singlet Triplet Without KK’ Mixing and Exchange

(b)

Fig. 12 (a) Fig. 12 (b)

Page 14 T. Ando

0.00 0.05 0.100.0

0.5

1.0

1.5

Flux (units of φ0)

Opt

ical

Inte

nsity

K

K’KK-K’K’(-)

KK-K’K’(+)

L/a= 20.0 (ν =1)(e2/κL)(2πγ/L)-1=0.2εc(2πγ/L)-1=10w2/γ0=1.5w1/w2=0.4

Without KK’ Mixing and Exchange

(a)

0.00 0.05 0.100.0

0.5

1.0

1.5

Flux (units of φ0)

Opt

ical

Inte

nsity

K

K’

KK-K’K’(+)

KK-K’K’(-)

L/a= 20.0 (ν =1)(e2/κL)(2πγ/L)-1=0.2εc(2πγ/L)-1=10w2/γ0=1.5w1/w2=0.2

Without KK’ Mixing and Exchange

(b)

Fig. 13 (a) Fig. 13 (b)

-0.05 0.00 0.05 0.100

50

100

150

Energy Shift (units of 2πγ/L)

Dyn

amic

al C

ondu

ctiv

ity (

units

of e

2 /h)

L/a=20.0 (ν=1)(e2/κL)(2πγ/L)-1=0.2w2/γ0=1.5, w1/w2=0.4 εc(2πγ/L)-1=10

φmax/φ0=0.02Δφ/φ0=0.001Γ(2πγ/L)-1=0.01

(a)

-0.05 0.00 0.05 0.100

50

100

150

Energy Shift (units of 2πγ/L)

Dyn

amic

al C

ondu

ctiv

ity (

units

of e

2 /h)

L/a=20.0 (ν=1)(e2/κL)(2πγ/L)-1=0.2w2/γ0=1.5, w1/w2=0.2 εc(2πγ/L)-1=10

φmax/φ0=0.02Δφ/φ0=0.001Γ(2πγ/L)-1=0.01

(b)

Fig. 14 (a) Fig. 14 (b)