Conversion of twisted light to twisted excitons using carbon nanotubes

Conversion of twisted light to twisted excitons using carbon nanotubes

Item Type Article

Authors Zang, Xiaoning;Singh, Nirpendra;Lusk, Mark T.;Schwingenschlögl, Udo

Citation Zang, X., Singh, N., Lusk, M. T., & Schwingenschlögl, U.

(2022). Conversion of twisted light to twisted excitons using carbon nanotubes. Npj Computational Materials, 8(1). https://

doi.org/10.1038/s41524-022-00726-6 Eprint version Publisher's Version/PDF

DOI 10.1038/s41524-022-00726-6

Publisher Springer Science and Business Media LLC Journal npj Computational Materials

Rights Archived with thanks to npj Computational Materials. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/

by/4.0/.

Download date 2023-12-03 19:26:38

Link to Item http://hdl.handle.net/10754/676125

ARTICLE

^OPEN

Conversion of twisted light to twisted excitons using carbon nanotubes

Xiaoning Zang¹, Nirpendra Singh ^1,2, Mark T. Lusk³✉and Udo Schwingenschlögl ¹✉

Carbon nanotubes are explored as a means of coherently converting the orbital angular momentum of light to an excitonic form that is more amenable to quantum information processing. An analytical analysis, based on dynamical conductivity, is used to show that orbital angular momentum is conserved, moduloN, for a carbon nanotube illuminated by radially polarized, twisted light. This result is numerically demonstrated using real-time time-dependent density functional theory which captures the absorption of twisted light and the subsequent transfer of twisted excitons. The results suggest that carbon nanotubes are promising candidates for constructing optoelectronic circuits in which quantum information is more readily processed while manifested in excitonic form.

npj Computational Materials (2022) 8:42 ; https://doi.org/10.1038/s41524-022-00726-6

INTRODUCTION

Centrosymmetric molecules can host twisted excitons carrying orbital angular momentum (OAM)¹, and this can be employed to mediate algebraic operations involving twisted light². A chain of these molecules constitutes a conduit for OAM transfer in the form of twisted exciton wave packets with controllable linear momen- tum^3,4. The interconversion and manipulation of optical and excitonic manifestations of OAM represents a rich field for quantum information processing⁵. While optical methodologies are already well-established for encoding and transmitting data as OAM^6–11, a distinct advantage of excitons is that they can be readily combined and manipulated for quantum information processing¹. A chain of molecules is therefore qualitatively different than simply directing twisted light through optical fibers¹². The experimental realization of twisted excitons, however, is challenging, because it requires rigidly connected molecules with low reorganization energy.

Carbon nanotubes (CNTs) offer a means transferring the angular momentum of light to excitonic form, processing it, and then converting it back into an optical form. This is because they are structurally rigid and have the requisiteC_Nsymmetry¹³. The Bloch wave functions, therefore, can be constructed using the rotational operator with the OAM as eigenvalue. The fact that the electrons carry OAM has been confirmed theoretically¹⁴and experimentally¹⁵. Ultra-long CNTs have been experimentally fabricated^16,17to lengths of as much as half a meter¹⁸. CNTs with carbon purity above 99.98%

can be synthesized very efficiently¹⁹ and 92% yield has been reported for growth of CNTs with specific chirality²⁰. Combined with tunable band gaps, CNTs are promising candidates for optoelec- tronic applications as in, for example, solar cells^21–29. Recently, CNTs are demonstrated to be high-purity and high-efficiency single- photon sources, which makes CNTs promising candidates for on- chip quantum light sources for quantum information³⁰. Although a lot is known about the optoelectronic properties of CNTs³¹ and their exciton dynamics^32–35, the interaction with twisted light is essentially unexplored³⁶.

Here we computationally demonstrate that OAM is conserved when CNTs convert twisted light into twisted excitons. We first

establish the optical selection rules by deriving the dynamical conductivity based on the helical and rotational symmetries. Then real-time time-dependent density functional theory (RT-TDDFT) is employed to directly simulate the transfer of OAM from twisted light to twisted excitons³⁷. RT-TDDFT includes the multi-level and multi- electron effects omitted by the tight-binding model. We show that CNTs have effectively the same structure as a molecular chain except that there is a constant helical misorientation between neighboring sites as illustrated in Fig.1a. CNTs are readily available with a wide range of electronic band structures. They are also rigid and have low reorganization energy so that exciton phonon coupling is minimized, helping to support coherence³⁸. Although the extended states of CNTs make this intuitively reasonable, a reorganization energy of tens of meV has been reported in a recent experiment³⁹. The reorganization energy estimated by the peak width of photolumi- nescence spectra ranges from tens of meV⁴⁰to as low as 1 meV⁴¹. This is comparable to the most rigid porphyrins^42,43. CNTs therefore offer an ideal setting for the realization of twisted exciton circuitry.

The central concept can be elucidated using a tight-binding model with coupling between the pz orbitals of neighboring carbon atoms. The resulting band structure and eigenstates can then be expressed in terms of crystal momentum due to translational symmetry along the principle axis of the CNT⁴⁴. However, the required computational time can be dramatically reduced by exploiting helical and rotational symmetries instead⁴⁵. This approach also is advantageous in that the OAM is related to irreducible representations of SO(2) for continuous space andC_N for discrete space. Each eigenstate can be identified with a helical quantum number,κ, for the twist of subsequent structural units along the axis and a rotational quantum number,q, due to the rotational symmetry of structural strands of which the tube is composed. Phonon effects are disregarded under the reasonable assumption that reorganization energy is so low for the extended states of this system exciton wave packet coherence is maintained during the information processing. This can be elucidated by considering a dimer consisting of thefirst two sites in Fig.1a^38,46, for which the decoherence rateRcan be analytically calculated as

1Applied Physics Program, Physical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia.

2Department of Physics, Khalifa University of Science and Technology, Abu Dhabi 127788, United Arab Emirates.³Department of Physics, Colorado School of Mines, Golden, CO 80401, USA.✉email: mlusk@mines.edu; udo.schwingenschlogl@kaust.edu.sa

www.nature.com/npjcompumats

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences

1234567890():,;

the sum of a pure dephasing rateRdand a relaxation rateRr: R¼R_dþR_r

Rd¼δε²Sð0Þ=ð2b²Þ (1)

R_r4J²SðbÞ=ð2b²Þ: (2)

Here b²=δε²+4J² with δε and J being the onsite energy difference and the coupling between these two sites, respectively.

The thermally weighted spectral density is SðωÞ ¼2πcoth½_ω=ð2kBTÞX

i

ωid²_iδðωωiÞ (3) withd_i,ωiandK_Bbeing the strength of electron-phonon coupling, the frequency of the phonon mode, and Boltzmann’s constant, respectively. The reorganization energy is λ¼P

i_ωid²_i P

iλðωiÞ. Since all sites in a given CNT have same onsite energy, we haveR_d=0 andb=2∣J∣=2∣V_ppπ∣whereV_ppπ=−2.77 eV⁴⁷ is the coupling between nearestpzorbitals in CNTs. ThusR=πλ(b)/ℏ is much less thanπλ/ℏ=1/21fs⁻¹and the coherence time 1/Ris much larger than 21 fs. This result is based on the estimate that λ=10 meV. Both the coherence time and the exciton life time, which is estimated to be in the order of 10 ps⁴¹, are much larger than the exciton transfer time in a 4 nm (3,3) CNT (see Fig.7). Then a momentum conservation is associated with the photo-excitation of an electron from a valence state with quantum numbersκand q to a conduction state with quantum numbers κ⁰ and q⁰.

Specifically, q⁰q and κ⁰κ must be equal to the photonic angular momentum (PAM) and helical momentum of the twisted light, respectively.

The OAM conservation between excitons and photons can be studied within the context of band structure. The OAM of the exciton (XAM) is the total OAM of all occupied Bloch states and is also the sum of the OAM of all electrons in conduction bands (EAM) and all holes in valence bands (HAM). The fact that the ground state has zero OAM indicates that the EAM associated with an excited electron is the negative of the total EAM from all other electrons. This means that the electron and hole in the same Bloch state are of opposite OAM. It is shown that OAM is conserved during an optical excitation, i.e., PAM=XAM=EAM+HAM. In the absence of meaningful phonon effects, a twisted exciton wave packet transfers down the CNT with the same OAM as was absorbed from twisted light.

RESULTS

Tight-binding analysis

A CNT can be created by rolling up graphene along a lattice vector

!R

¼n₁!e

1þn₂!e

2 with the resulting structure characterized by the pair of integers (n₁,n₂). Here !e

2 and !e

2 are the primitive lattice vectors (Fig.2). A band gap is present provided⁴⁴

n₁n₂¼3l± 1; l2Z: (4)

Fig. 1 CNTs and OAM. aCNTs have the same structure as a chain of coaxial parallel molecules with helical misorientation. A chain of centrosymmetric molecules exhibits a structural helicity in which each molecule is misoriented relative to its neighbors. Exciton hopping can still occur between nearest-neighbor arms of the same color.bMolecule withC3symmetry.cClockwise 2πphase variation for OAM=1 anddcounterclockwise 4πphase variation for OAM=

−2. Applied to the molecule inb, by taking the phases ofcanddat the positions of the arms, twisted light witheOAM=1 andfOAM

=−2 leads to the same phase difference between neighboring arms modulo 2π.

Ar m

2

3

4

5

e

2

^

A B e

1

^

R

^{^}

H

^{^}

1

1 ⁸

Site

^{16 24}

Fig. 2 Decomposition of the (5,10) CNT into a spiral of 5-arm sites.

The CNT can be viewed as a piece of graphene rolled up along the lattice vector !R ¼5!e

1þ10!e

2 (!e

1 and !e

2 being the primitive lattice vectors). Each arm consists of two inequivalent carbon atoms, denoted asAandB. The 5 arms are highlighted by different colors.

!H

is the primitive lattice vector along the spiral. The black dashed lines indicate different sites.

2

1234567890():,;

Examination of the resulting helical and rotational symmetries⁴⁵ reveals that each CNT can be viewed as anM-site system, withN arms at each site, in which each arm consists of two inequivalent carbon atoms denoted asAand B. For the (5, 10) CNT of Fig. 2, for instance, there are five arms at each site which are created by rotational operations of thefirst arm, and thesefive arms give rise to the entire CNT byM−1 subsequent helical operations. Figure 3 shows the spiral formed by following one arm along the different sites. The primitive lattice vector along the helical direction is given by!H

¼p₁!e

1þp₂!e

2

with p₁≤p₂ being the smallest nonnegative integers fulfilling p₁n₂−p₂n₁=N.

Since our goal is not to generate quantitatively precise band structures, aπ-band approximation is applied to demonstrate the key idea that CNTs can serve as conduits for OAM transport and thus as building blocks for optoelectronic circuits. The symmetry group generated by the helical operation is isomorphic to the one-dimensional translation group, so that Bloch’s theorem can be readily generalized⁴⁷. Moreover, because the helical and rotational operators commute, we obtain the following symmetry-adapted, generalized Bloch sums⁴⁵:

jΦ^α_κ;qi ¼ 1 ffiffiffiffiffiffiffi pMNX^M

m¼1

X^N

n¼1

ε^ðm1Þκ_M ε^ðn1Þq_N jϕ^α_m;ni: (5) HereεM=e^i2π/MandεN=e^i2π/N. Each Bloch sum is given by three parameters: α, κ, and q. While α is Aor B (inequivalent carbon atoms), κ and qare the quantum numbers associated with the helical and rotational operators, respectively, taking on integer values within the domains ð^M₂;^M₂ and ð^N₂;^N₂. M is the site number afterM−1 helical operations andNis the arm number afterN−1 rotational operations. Moreover,jϕ^α_m;niis the orbital of theα-type carbon atom in armnat sitem.

The matrix elements between Bloch sums with differentκandq are zero. Thus, the wave functions are linear combinations of basis states,

jΨκ;qi ¼c^A_κ;qjΦ^A_κ;qi þc^B_κ;qjΦ^B_κ;qi; (6) with constantsc^A_κ;qandc^B_κ;q. Inserting this into the Schrödinger equation gives a 2 × 2 eigenvalue problem with energies,

±∣HAB∣, that are functions of the quantum numbersκandq:

H_AB hΦ^A_κ;qjH^₀jΦ^B_κ;qi

¼V_ppπ P

h1;1;i;jiε^ði1Þκ_M ε^ðj1Þq_N : (7) HereVppπis the coupling strength between nearest neighbors (with a value of −2.77 eV⁴⁷), 〈1, 1;i,j〉 denotes the nearest neighbors of atom A in arm 1 at site 1, and H^₀ is the one- electron Hamiltonian of the CNT. This allows the band structure (which is a function ofκandq) to be immediately constructed.

The eigenstates ofH^₀are the Bloch wave functionsjs;q;κi,

H^0js;q;κi ¼εs;q;κjs;q;κi; (8)

wheres= +/−refers to conduction/valence states.

Turning to the interaction of CNTs with twisted light, an ideal Bessel mode can be written as^48,49

Eðρ;ϕ;z;tÞ ¼J_lðkρ;ρÞe^ilϕe^ikzze^iωt; (9) whereJ_ldenotes thel-th order Bessel function of thefirst kind, (ρ,ϕ,z) the polar coordinates,k_zandk_ρthe wave numbers, and ω the frequency. For cylindrically polarized twisted light^50,51 with polarization singularity on the main axis of the CNT, the rotational center of the SO(2) symmetry (twisted light) coincides with that of theC_Nsymmetry (CNT). Thus, the combined system (twisted light and CNT) and, particularly, the Hamiltonian of the light-matter interaction possess rotational symmetry, i.e., the OAM may be conserved. We adopt radially polarized twisted light, i.e.,

!E

¼!e

ρE₀sinðlϕþk_zzωtÞ; (10)

with!e

ρbeing the radial unit vector andE₀the magnitude of the electricfield on the surface of the CNT.

Note that e^ilϕwithϕ∈(0, 2π] are the irreducible representations of the rotation group of continuous two-dimensional space, SO(2), wherelcan be any integer. When twisted light is applied to a CNT the rotation group becomesC_Nwith the defining representation

e^iðn1Þ2πN . Since the group is abelian, each element (n=1,⋯,N) forms

an equivalence class and the number of equivalence classes equals the number of irreducible representations (which must be orthogonal to each other). The dimensionsdr of the irreducible representations must satisfy P_N

r¼1d²_r ¼N, implying d_r=1. The identity and defining representation are irreducible representa- tions. As P_N

n1e^{iðqq0Þn1N2π}¼δq;q⁰, the irreducible representations have the form e^iqn1N2π, with q to be determined. The character orthogonality requiresP

qe^{iðnn0ÞNq2π}¼δn;n⁰, which is equivalent to choosingqas integers in the intervalð^N₂;^N₂. Therefore,CNhas the irreducible representations e^iqðn1Þ2πN ,n=1,⋯,Nandqis an integer in the interval ð^N₂;^N₂. Thus, l is reduced to q modulo N. For example,l=1 and−2 are equivalent in the case ofC3symmetry, as both are reduced toq=1, see Fig.1b–f. This implies

!E

¼!e

ρE₀sin qðn1Þ2π

N þκ0ðm1Þ2π

M ωt

; (11)

whereκ0=Mlθ/2πwithθ¼2πð!R !H

Þ=j!R

j², see Fig.3. Note that k_zin Eq. (10) is set to zero based on the dipole approximation, i.e., the wave length of the twisted light is assumed to be much larger than the size of the CNT. The helical momentumκ0is nonzero, as the phase of the twisted light is zero along the z-direction, see Fig. 3 Geometry.Spiral of arm 1 (red color) with angleθbetween sites 1 and 2. The phase of the twisted light is zero along thez-direction (black line).

X. Zang et al.

3

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2022) 42

Fig.3. Figure4a shows the values ofθfor CNTs withCNsymmetry (N> 2). ForN≤2 only excitons with XAM=0 can be conducted, which is not of our interest. According to Fig. 4b, θ decreases whenn2increases for both the zigzag and armchair CNTs.

The Hamiltonian of the light-matter interaction is H^int¼ e r! !e

ρE0sinðψωtÞ

¼ D₁E₀sinðωtÞ D₂E₀cosðωtÞ; (12) where we defineψ¼^qðn1Þ2π_N þ^κ0ðm1Þ2π_M as the angle between!e

x

and!e

ρ,D1er0cosψ, andD2er0sinψ(r0being the radius of the CNT). The Cartesian coordinate system is chosen such thatD₁ andD2refer to thexandy-directions, respectively. The electron density propagates according to the quantum Liouville equation

∂^ρðtÞ

∂t ¼_{i_}¹½HðtÞ;^ ρðtÞ, where^ HðtÞ ¼^ H^₀þH^_int. Considering H^_int as perturbation, separating ρðtÞ^ into equilibrium and perturbation parts,^ρðtÞ ¼^ρ₀þΔ^ρðtÞ, and omitting the second order term, we obtain

∂Δ^ρðtÞ

∂t ¼_{i_}¹½H^₀;Δ^ρðtÞ

þE₀sinðωtÞ½D1;^ρ0 E₀cosðωtÞ½D2;^ρ0g: (13) This equation is solved by

Δ^ρðtÞ ¼ _{i_}¹R_t

1½D^₁ðt⁰tÞ;^ρ0E0sinðωt⁰Þdt⁰ _{i_}¹R_t

1½D^₂ðt⁰tÞ;^ρ0E0cosðωt⁰Þdt⁰; (14) whereD^iðtÞ ¼^U^y₀ðtÞDiU^0ðtÞwithU^0ðtÞ ¼e^{_iH^0t}. The current then is given by

!j

ðtÞ ¼ _{i_}¹R_t

1Tr ½D^₁ðt⁰tÞ;^ρ0^

!j ð0Þ

E₀sinðωt⁰Þdt⁰ _{i_}¹Rt

1Tr ½D^₂ðt⁰tÞ;^ρ₀^

!jð0Þ

E₀cosðωt⁰Þdt⁰; (15) with

^j

!ðtÞ ¼^j₁ðtÞ!e

xþ^j₂ðtÞ!e

yþ^j₃ðtÞ!e

z

¼_πr^e2 0M

_i^U^y₀ðtÞ½H^₀;!^r U^₀ðtÞ: (16) From Eq. (15), we have

j_iðtÞ ¼ R₁

1Θðtt⁰Þαi;1ðtt⁰ÞE0sinðωt⁰Þdt⁰ þ R₁

1Θðtt⁰Þαi;2ðtt⁰ÞE0cosðωt⁰Þdt⁰; (17)

whereΘ(t) is the step function and αi;i⁰ðtÞ _{i_}¹Tr½D^_i⁰ðtÞ;ρ^0^j_ið0Þ

¼ _{i_}¹Tr^ρ₀½D^_i0ð0Þ;^j_iðtÞ

: (18)

Taking the Fourier transform of Eq. (15), we obtain

~j_iðωÞ ¼ ~σi;1ðωÞ~EsinðωÞ þ~σi;2ðωÞ~EcosðωÞ (19) with

σ~i;i⁰ðωÞ ¼ ^πr_{_ω}^20MR1

0 Tr^ρ0½^j_i0ð0Þ;^j_iðtÞ

e^{iðωþiηÞt}dt

~E_sinðωÞ ¼ R1

1E₀sinðωtÞe^{iðωþiηÞt}dt

~E_cosðωÞ ¼ R₁

1E₀cosðωtÞe^{iðωþiηÞt}dt;

(20)

whereηis the phenomenological rate of relaxation and we have used^j_ið0Þ ¼_πr¹2

0M

∂D^iðtÞ

∂t

t¼0. The expression for~αi;i⁰ðωÞ is the Kubo formula for the dynamical conductivity. Optical selection rules can be established by transformation to the basis fjs;q;κig. The position operator is written as^52,53

^r

! ¼r₀cosψ!e

xþr₀sinψ!e

yþ i sinβ∂

∂κþsinβΩ^

!e

z; (21)

whereβis the angle between the rotational and helical directions, and

Ω^¼ X

s;s⁰;q;κ

i Z

u_s0;q;κð!Þr ∂us;q;κð!Þr

∂κ d!rjs⁰;q;κihs;q;κj (22) withu_s;q;κð!Þr being the Bloch function. We have

^ρ0¼2X

s;q;κ

n_Fðεs;q;κÞjs;q;κihs;q;κj (23)

withnFbeing the Fermi distribution. Substitution of Eqs. (21)–(23) into Eq. (20) yields the nonzero dynamical conductivity

~

α1;1ðωÞ ¼~α2;2ðωÞ

¼ i2e² πr²₀M_²ω

P

s;s⁰;q⁰;κ

n_Fðεs;q⁰;κÞ n_Fðεs⁰;q⁰q;κκ0Þ

´jhs;q⁰;κj½H^₀;r₀e^iψjs⁰;q⁰q;κκ0ij² jεs⁰;q⁰q;κκ0εs;q⁰;κj þ_ωþi_η ;

(24)

where we have used Eq. (16) and the rotating wave approximation.

Therefore, ~j₁ðωÞ ¼ ~σ1;1ðωÞ~E_sinðωÞ and ~j₂ðωÞ ¼~σ2;2ðωÞ~E_cosðωÞ.

Equation (24) implies that transitions are allowed between bands

(a) (b)

Fig. 4 Evaluation ofθ.Values ofθfor (a) (n₁,n2) CNTs with rotational symmetryCN(N> 2) and (b) zigzag (red) and armchair (green) CNTs.

4

with the OAM difference equal to the PAM, since in the numerator the OAM difference between the two states (q) is the OAM of the twisted light. In other words, the OAM is conserved moduloN. We show in Fig.5a the band structure of the (5, 10) CNT with the OAM of the bands indicated. The transitions of an electron from the state j;2;0i to different conduction states are illustrated in Fig.5b, showing thatκ0depends on the PAM.

Former studies of the optical selection rules byk⋅ptheory⁵⁴or, equivalently, by linear expansion of the Hamiltonian at the Dirac points of graphene⁵²did not consider the change of the symmetry group from SO(2) toC_N with the consequence that the OAM of both the twisted light and electron could take any integer value for all CNTs. This leads to wrong conclusions. For example, in the case ofC₃symmetry, electrons transfer from the valence band withq= 1 under illumination with l=2 twisted light to the conduction band withq=3 according tok⋅ptheory, while they transfer to the conduction band withq=0 according to Eq. (24).

Equation (5) has exactly the same form as the tight-binding model for anM-site chain ofN-arm cofacial molecules⁴. Therefore P_N

n¼1ε^ðn1Þq_N ϕ^α_m;nE

can be considered as a twisted electronic state with OAM=qat sitem. The two carbon atoms in one arm have the same phase terms,ε^ðm1Þκ_M andε^ðn1Þq_N , so that each CNT can be considered as a spiral ofN-arm sites. In order for the exciton to carry nonzero OAM, we require N≥3, withN being the greatest common divisor ofn1 andn2. This condition must be fulfilled in addition to the requirement that the CNT is semiconducting, see Eq. (4). From Eq. (7), one can expect that each band is associated with a specific OAM. This is demonstrated in Fig. 5, where the band structure of the (5,10) CNT is plotted with different colors encoding the OAM. The electrons and holes fulfill EAM=−HAM for a specific band.

OAM conservation between the photon of twisted light absorbed by the CNT and the twisted exciton (XAM=EAM+ HAM) is confirmed by our analysis. For example, when an electron in the valence band with OAM=−1 absorbs twisted light with

PAM= +2 it is excited into a conduction band with OAM= +1.

Thus we have an electron with EAM= +1 and a hole with HAM= +1, such that the exciton carries XAM=EAM+HAM=PAM. In general, a twisted exciton with XAM=q+nN(q2 ½^N1₂ ;^N1₂ ) for arbitraryn2Zis equivalent to the twisted exciton with XAM=q.

This is analogous to the lack of uniqueness in the definition of crystal momentum, i.e.,ℏk=ℏ(k+2πn/a) wherea is the lattice spacing. As a consequence, a twisted exciton with XAM=qcan be generated by twisted light with PAM=q+nN. WhenNis even, excited states induced by twisted light with∣q+nN∣=N/2 have no specific circularity and therefore are not considered⁵⁵.

It is worth emphasizing the difference between the important role of the PAM here in contrast to the typically neglected role of the photonic linear momentum in standard band structure theory.

In the latter case the linear momentum of the electrons in the valence band is so large that their change in momentum due to photon absorption can be safely ignored at room temperature.

Within the CNT setting, though, the PAM is on the same order of magnitude as the EAM of the valence band electrons and the OAM conservation is nontrivial.

RT-TDDFT analysis

RT-TDDFT considers multi-level and multi-electron effects omitted in the tight-binding paradigm⁵⁶. And it is a computationally cheaper way to study excitonic effects comparing with the Bethe- Salpeter equation. The evolution of the electronic states during the interaction with twisted light can be explicitly tracked within a Kohn–Sham (KS) formulation of the electronic structure, see the Methods section for details. Exciton transport with conserved OAM is demonstrated for a 4 nm long (3, 3) CNT, see Fig.6a, with hydrogenated ends, because a (5, 10) CNT with sufficient length exceeds the available computational resources. Note that the length of 4 nm is much larger than the diameter (~exciton size).

The fact that the (3, 3) CNT is metallic does not prohibit demonstration of the core idea since it still has rotational and

M (a)

(+, ) (+, ) (+, ) (+, ) (+, )

( , )

(b)

0 1

2 3 4

5 6

7 8

9

10

M

Fig. 5 Momentum conservation. aBand structure of the (5,10) CNT. The bands are associated with OAM=−2 (red),−1 (green), 0 (black),+1 (blue), or+2 (orange).bAbsorption of helical moment by an electron in statej;2;0iwhen illuminated by twisted light with different PAM (numbers at arrows).

X. Zang et al.

5

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2022) 42

helical symmetries and has a small gap between highest occupied and lowest unoccupied orbitals for afinite length tube. A short laser pulse is applied to excite an exciton wave packet and a long laser pulse is applied to verify the OAM conservation, see Fig.6c.

The short laser pulse is localized in the time domain and delocalized in the frequency domain, see Fig.6d, i.e., states in a broad range of energy are excited. In contrast, the long laser pulse is localized around 2 eV, which is less than the energy difference between the occupied and unoccupied KS orbitals with OAM=

±1, see Fig.6b.

Solution of the Bethe-Salpeter equation shows that strongly bound excitons dominate the excitations of the (3, 3) carbon nanotube⁵⁷. The absorption spectra without and with electron- hole interaction both feature a single prominent peak, which is found below the unbound electron-hole excitations in the case with electron-hole interaction due to the large exciton binding energy of ~100 meV. These results agree with the experimental observations of ref.⁵⁸. The excitons could dissociated in an applied electric field⁵⁹ and in the internal electricfield present at a p-n junction²⁶. In our case, however, there are no such fields. The excitons could also dissociate spontaneously, which is believed to be the reason for the observed photocurrent in carbon nanotubes despite the large exciton binding energy⁵⁹. The spontaneous dissociation is accompanied by phonon emission, as direct transition has been shown to be very weak⁶⁰. Since we do not consider phonons in our simulations, we can conclude that the excitations are excitons during and after the illumination.

To demonstrate the transport of an exciton wave packet along the CNT, the laser pulse is applied only to thefirst site of the CNT.

The physics would not change when it was applied to more sites simultaneously, but this would require a longer CNT to observe

the exciton wave packet. With phonon coupling disregarded, the laser pulse results in a coherent superposition of the ground and excited states. The electron and hole populations at each site are calculated through integration of the attachment and detachment densities⁶¹over a cylinder with the site in the center and with the length being the projection of !H

on the axis of the CNT. The exciton population is defined as the average of the electron and hole populations. Figure7demonstrates that the electrons, holes, and excitons propagate along the CNT. Due to reflection at the other end,finally an equal distribution is established.

To verify the OAM conservation, we first calculate the exciton populations when the system is initially in the ground state. The idea is that small excitations with different OAMs do not influence each other. Note that the long laser pulse in Fig.6c is employed in the following. Let us denote the excited states with the same OAM (moduloN) byjΨðOAMÞiand the corresponding populations byp (OAM). If there is OAM conservation, then the excited state is

ΨðqÞ

j i and its population isp(q) for twisted light with PAM=q.

The total exciton population is ∑qp(q) when light pulses with differentqare applied. This is consistent with the observationp4~ p1+p3 and p5~p1+p2+p3 in Fig. 8. When two pulses of the same twisted light are applied we have 4 times larger population, which is consistent with the observation p6~ 4p1 in Fig. 8.

However, we yet cannot conclude that there is OAM conservation.

Ifp1in Fig.8corresponds to an excited stateajΨð1Þi þbjΨð1Þi, thenp3must correspond to the excited statebjΨð1Þi þajΨð1Þi and we havep1=p3=a²+b². Thus, the excited state induced by the two twisted light pulses isðaþbÞjΨð1Þi þ ðaþbÞjΨð1Þiand its population is p4=2∣a+b∣². Assuminga;b2R, we finda²~ 2400b²at t=25 fs, which means that twisted light with a specific PAM induces an exciton with pure OAM.

(a)

(c) (d)

(b)

Fig. 6 The (3, 3) CNT. aDecomposition of the (3, 3) CNT into a spiral of 3-arm sites.bEnergies of the KS orbitals. The 402^ndis the highest occupied KS orbital. Orbitals with OAM=0 and ±1 are shown in black and cyan, respectively.cLaser pulses withE0=0.5 V/Å,ℏω=2 eV, and envelope functionexpððt0:3fsÞ²=2ð0:03fsÞ²Þ(red) orexpððt10fsÞ²=2ð3fsÞ²Þ(green). The gray curve is0:5 sinðωtÞ.dNormalized Fourier transforms of the curves inc.

6

Second, we calculate the exciton populations when the system is initially in an excited state with OAM=1 and population 0.05 (p7and p₈in Fig.8). Specifically, one electron is initially excited from a linear combination of the 389^thand 390^thKS orbitals to the 403^rdKS orbital.

Since we know that twisted light with PAM=± 1 induces an exciton with pure OAM=± 1, we will have either p₇=0.05+p₁ or p₈= 0.05+p3. Figure9showsp7~ 0.05+p1, which means that the initial excited state has opposite OAM as compared to the excited state withp₁. In other words, the twisted light with PAM=−1 induces an exciton with the same OAM, i.e., the OAM is conserved. Note that the initial excited state has an energy of 2.3 eV, which is not resonant with the frequency of the laser pulse. Thus, it contributes little to the excited state induced by the laser pulse andp₈is similar to 0.05+p₃. Due to the OAM conservation, the population for each XAM can be determined, see the Methods section for details. Figure 10a shows XAM for twisted light with PAM=−1, 0, and 1. We can conclude that twisted light with PAM=0 leads to XAM¼0 and twisted light with PAM=± 1 leads to opposite XAMs. Using PAM

=−1 as an example, we have XAM¼ pð1Þ þ2pð2Þ. Together with the normalization conditionp(−1)+p(2)=1 this leads to the exciton populations shown in Fig.10b. The populationp(2) is due

to excitations from occupied to unoccupied orbitals with OAM=

± 1, which have energies of at least 3 eV, see Fig.6b, i.e., far from resonance with the frequency of the laser pulse. Therefore,p(2) can be neglected and the OAM can be determined in this case.

We next demonstrate conservation of the spin of circularly polarized light, which can be decomposed into radially and azimuthally polarized twisted lights^1,62,

1ffiffi

p2ð!e

x± i!e

yÞ

¼ ¹ffiffi

p2½ðcosϕ!e

r ∓sinϕ!e

ϕÞ± iðsinϕ!e

r± cosϕ!e

ϕÞ

¼ ¹ffiffi

p2e^{± iϕ}ð!e

r± i!e

ϕÞ;

(25)

where!e

x,!e

yand!e

r,!e

ϕ are the unit vectors of Cartesian and polar coordinate systems, respectively, and e^±iϕrepresents OAM

=± 1. Right-hand circularly polarized light with spin=−1 is applied along the axis of the CNT. According to Fig.11,p10~ 4p1

andp11~p3+p9confirm the equivalence of right-hand circularly polarized light with spin=−1 and twisted light with PAM=−1 in the sense that they induce an exciton with the same XAM. Thus, the CNT can convert circularly polarized light into twisted light.

(a)

t = 2 fs t = 4 fs

t = 6 fs t = 8 fs

(b)

Fig. 7 Exciton wave packet. aElectron and hole populations at different sites as functions of time.bExciton population at different sites at specific times. The results refer to the short laser pulse of Fig.6c.

X. Zang et al.

7

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2022) 42

DISCUSSION

We have shown that the orbital angular momentum is conserved during the excitation and subsequent transfer of twisted excitons

in CNTs. These have extremely low reorganization energy in comparison to molecular chains implying the possibility of strong band transport and long coherence time for twisted exciton wave packets. CNTs therefore hold promise for exploiting the interplay of photonic and excitonic angular momentum for quantum information processing. The present work initially employs tight- binding analysis based on the helical and rotational symmetries to derive the dynamical conductivity. The obtained optical selection rules show OAM conservation and non-vertical transitions between bands for OAM≠0. The advantage of using the helical and rotational symmetries is that theCNrotation group of CNTs is considered automatically. Traditional derivations of the optical selection rules, by employingk⋅ptheory or equivalently by linear expansion of the Hamiltonian around the Dirac points of graphene, are actually based on the rotation group SO(2), which leads to wrong conclusions on the optical excitations of CNTs illuminated by twisted light.

We employ the RT-TDDFT approach, which includes multi-level and multi-electron effects, being computationally cheaper than the Bethe-Salpeter equation, for the study of excitons. The results confirm that the OAM is conserved in the presence of electron- hole interaction, between photon and exciton as well as during exciton transfer. Former studies have shown that OAM cannot be exchanged between twisted light and the electrons of isolated atoms by dipole transition^63–67. However, this does not apply to a ring of atoms withCNsymmetry¹, as each atom can be excited by dipole transition and pick up the specific phase of the twisted light at its position, giving rise to a collective mode. The reverse process, generation of twisted light via the dipole transition of molecules in a ring, has been demonstrated in ref.⁵⁵.

3

Fig. 9 Exciton population differences.p7−(0.05+p1) (green) and p8−(0.05+p3) (blue), compare Fig.8. Black dashed lines indicate the average in the respective region.

XAM=1

XAM=-2 (a)

(b)

XAM=-1

XAM=2 PAM=1

PAM=-1 PAM=0

Fig. 10 OAM conservation. a XAMobtained for twisted light with PAM=−1 (green), 0 (black), and 1 (blue).bCorresponding exciton populations.

p

₂

p

₄

p

₅

p

₆

p

₇

4p

₁

p

₁+ +

p

₂

p

₃

p

₁+

p

₃

p

₁₌

p

₃

p

₈

Fig. 8 Twisted light to twisted exciton.Exciton populations in a 4 nm long (3, 3) CNT illuminated by a twisted light pulse with PAM=

−1 (p1,p7), 0 (p2), or 1 (p3,p8), twisted light pulses with PAM=−1 and 1 of the same amplitude (p4), twisted light pulses with PAM=

−1, 0, and 1 of the same amplitude (p5), and two twisted light pulses with PAM=−1 of the same amplitude (p6). The CNT is initially in the ground state for p1 to p6 and in a 0.95:0.05 superposition of the ground state and excited states forp7andp8.

8

METHODS

Time-dependent Kohn–Sham formulation

Denoting the external potential (nuclei and lightfield) asν^ext, the Hartree potential asνHa, and the exchange-correlation potential asνxc, we have

i__∂t^∂jψiðr;tÞi ¼ h _2m^_2∇²þνextðr;tÞ þνHa½ρðr;tÞ þνxc½ρðr;tÞi

ψiðr;tÞ

j i (26)

ρðr;tÞ ¼2X^N

i¼1jψiðr;tÞihψiðr;tÞj: (27)

Eq. (27) is the spin-reduced density operator of 2Nelectrons. RT-TDDFT simulations are carried out using the Octopus package⁶⁸ with Troullier- Martins pseudopotentials and the Perdew–Burke–Ernzerhof exchange- correlation potential within the generalized gradient approximation. A time step of 0.66 × 10⁻¹⁸s is employed together with an atomic sphere radius of 3Åand a grid size of 0.15Å.

Average OAM of the exciton

In principle, the total OAM can be calculated as the sum of the OAMs of all states weighted by their populations, which is, however, computationally impossible. The total OAM can also be calculated by summation of the OAMs over all occupied time-propagated KS orbitals³⁷. Because of theCN

symmetry, the KS orbitals with OAM=0 are nondegenerate and the KS orbitals with OAM≠0 are twofold degenerate. As we can always construct KS orbitals ψ^qjðr;0ÞE

with OAM=qfrom ψjðr;0Þ

by the method of ref.¹,

the OAM of thei-th time-propagated KS orbital can be readily calculated as OAM_i¼X^M

j¼1

jhψ^qjðr;0Þjψiðr;tÞij²q: (28) Thus,PN

i¼1OAMiis the total OAM of the time-propagated multi-electron state, which is a linear combination of the ground and excited states. Since the ground state has OAM=0, the average OAM of the exciton is

XAM¼X^N

i¼1

OAMi=X^M=2

j¼1

n²_j; (29)

with the total number P_M=2

j¼1n²_j of excitons derived from the positive eigenvalues {nj} of (ρ(r,t)−ρ(r, 0))/2⁶¹. The overline indicates that this is an average quantity.

Eq. (29) does not recognize the equivalence of OAMs moduloN³⁷. For example, in the case of the (3, 3) CNT we have

XAM¼ X¹

q;q⁰¼1

p^q_q0ðqq⁰Þ; (30)

wherep^q_q0is the population of the state with one electron excited from the KS orbital with OAM¼q⁰to that with OAM=q(the total population of the excited states being normalized to 1). When twisted light with PAM=1 is applied, we have XAM¼p¹₀ð1Þ þp¹₁ ð2Þ, which is not equal to the PAM whenp¹₁ ≠0. While Eq. (29) thus cannot be used to determine the total OAM of the excitons, it helps to determine the population of each exciton with distinct OAM.

DATA AVAILABILITY

All data generated or analyzed during this study are included in the article.

Received: 2 October 2021; Accepted: 13 February 2022;

REFERENCES

1. Zang, X. & Lusk, M. T. Twisted molecular excitons as mediators for changing the angular momentum of light.Phys. Rev. A96, 013819 (2017).

2. Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes.

Phys. Rev. A45, 8185–8189 (1992).

3. Zang, X., Montangero, S., Carr, L. D. & Lusk, M. T. Engineering and manipulating exciton wave packets.Phys. Rev. B95, 195423 (2017).

4. Zang, X. & Lusk, M. T. Angular momentum transport with twisted exciton wave packets.Phys. Rev. B96, 155104 (2017).

5. Unuchek, D. et al. Room-temperature electrical control of excitonflux in a van der Waals heterostructure.Nature560, 340–344 (2018).

6. Marrucci, L., Manzo, C. & Paparo, D. Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media.Phys. Rev. Lett.96, 163905 (2006).

7. Nagali, E. et al. Quantum information transfer from spin to orbital angular momentum of photons.Phys. Rev. Lett.103, 013601 (2009).

8. Persuy, D. et al. Four-wave mixing in quantum wells using femtosecond pulses with Laguerre-Gauss modes.Phys. Rev. B92, 115312 (2015).

9. Fickler, R., Campbell, G., Buchler, B., Lam, P. K. & Zeilinger, A. Quantum entan- glement of angular momentum states with quantum numbers up to 10,010.Proc.

Natl Acad. Sci.113, 13642–13647 (2016).

10. Krenn, M. et al. Twisted light transmission over 143 km.Proc. Natl Acad. Sci.113, 13648–13653 (2016).

11. Shigematsu, K., Yamane, K., Morita, R. & Toda, Y. Coherent dynamics of exciton orbital angular momentum transferred by optical vortex pulses.Phys. Rev. B93, 045205 (2016).

12. Brunet, C. & Rusch, L. A. Opticalfibers for the transmission of orbital angular momentum modes.Opt. Fiber Tech.35C, 2 (2017).

13. Robertson, D. H., Brenner, D. W. & Mintmire, J. W. Energetics of nanoscale gra- phitic tubules.Phys. Rev. B45, 12592–12595 (1992).

14. Ajiki, H. & Ando, T. Electronic states of carbon nanotubes.J. Phys. Soc. Jpn.62, 1255–1266 (1993).

15. Minot, E. D., Yaish, Y., Sazonova, V. & McEuen, P. L. Determination of electron orbital magnetic moments in carbon nanotubes.Nature428, 536–539 (2004).

16. Zheng, L. X. et al. Ultralong single-wall carbon nanotubes.Nat. Mater.3, 673–676 (2004).

4p

₁

p

₃+

p

₉

p

₁₀

p

₁₁

p

₉

p

₁

= p

₃

Fig. 11 Circularly polarized light to twisted exciton. Exciton populations in a 4 nm long (3, 3) CNT illuminated by right-hand circularly polarized light (p9), right-hand circularly polarized light with spin=−1 and twisted light with PAM=−1 of the same amplitude (p10), and right-hand circularly polarized light and twisted light with PAM=1 of the same amplitude (p11). The CNT is initially in the ground state. Thep1=p3curve is reproduced from Fig.8.

X. Zang et al.

9

Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences npj Computational Materials (2022) 42

17. Wang, X. et al. Fabrication of ultralong and electrically uniform single-walled carbon nanotubes on clean substrates.Nano Lett.9, 3137–3141 (2009).

18. Zhang, R. et al. Growth of half-meter long carbon nanotubes based on Schulz- Flory distribution.ACS Nano7, 6156–6161 (2013).

19. Hata, K. et al. Water-assisted highly efficient synthesis of impurity-free single- walled carbon nanotubes.Science306, 1362–1364 (2004).

20. Yang, F. et al. Chirality-specific growth of single-walled carbon nanotubes on solid alloy catalysts.Nature510, 522–524 (2014).

21. Freitag, M., Martin, Y., Misewich, J. A., Martel, R. & Avouris, P. Photoconductivity of single carbon nanotubes.Nano Lett.3, 1067–1071 (2003).

22. Stewart, D. A. & Léonard, F. Photocurrents in nanotube junction.Phys. Rev. Lett.

93, 107401 (2004).

23. Lee, J. U. Photovoltaic effect in ideal carbon nanotube diodes.Appl. Phys. Lett.87, 073101 (2005).

24. Itkis, M. E., Borondics, F., Yu, A. & Haddon, R. C. Bolometric infrared photoresponse of suspended single-walled carbon nanotubefilms.Science312, 413–416 (2006).

25. Jia, Y. et al. Nanotube-silicon heterojunction solar cells.Adv. Mater.20, 4594–4598 (2008).

26. Gabor, N. M., Zhong, Z., Bosnick, K., Park, J. & McEuen, P. L. Extremely efficient multiple electron-hole pair generation in carbon nanotube photodiodes.Science 325, 1367–1371 (2009).

27. Li, Z. et al. Light-harvesting using high density p-type single wall carbon nano- tube/n-type silicon heterojunctions.ACS Nano3, 1407–1414 (2009).

28. Wadhwa, P., Liu, B., McCarthy, M. A., Wu, Z. & Rinzler, A. G. Electronic junction control in a nanotube-semiconductor Schottky junction solar cell.Nano Lett.10, 5001–5005 (2010).

29. Yang, L. et al. Efficient photovoltage multiplication in carbon nanotubes.Nat.

Photon.5, 672–676 (2011).

30. Kawabe, R. et al. Pure and efficient single-photon sources by shortening and functionalizing air-suspended carbon nanotubes. ACS Appl. Nano Mater. 3, 682–690 (2020).

31. Scarselli, M., Castrucci, P. & Crescenzi, M. D. Electronic and optoelectronic nano- devices based on carbon nanotubes.J. Phys. Condens. Matter.24, 313202 (2012).

32. Ostojic, G. N. et al. Interband recombination dynamics in resonantly excited single-walled carbon nanotubes.Phys. Rev. Lett.92, 117402 (2004).

33. Hagen, A. et al. Exponential decay lifetimes of excitons in individual single-walled carbon nanotubes.Phys. Rev. Lett.95, 197401 (2005).

34. Spataru, C. D., Ismail-Beigi, S., Capaz, R. B. & Louie, S. G. Theory and ab initio calculation of radiative lifetime of excitons in semiconducting carbon nanotubes.

Phys. Rev. Lett.95, 247402 (2005).

35. Lüer, L. et al. Size and mobility of excitons in (6,5) carbon nanotubes.Nat. Phys.5, 54–58 (2008).

36. Xiao, F. et al. Control the Raman response of individual carbon nanotubes by orbital angular momentum of light.Opt. Lett.42, 2491–2494 (2017).

37. Zang, X. & Schwingenschlögl, U. First-principles methodology for determining the angular momentum of excitons.Phys. Rev. B100, 045120 (2019).

38. Zang, X. & Lusk, M. T. Designing small silicon quantum dots with low reorgani- zation energy.Phys. Rev. B92, 035426 (2015).

39. Ihly, R. et al. Tuning the driving force for exciton dissociation in single-walled carbon nanotube heterojunctions.Nat. Chem.8, 603–609 (2016).

40. Lefebvre, J. & Finnie, P. Polarized photoluminescence excitation spectroscopy of single-walled carbon nanotubes.Phys. Rev. Lett.98, 167406 (2007).

41. Wang, X. et al. Quantum dot-like excitonic behavior in individual single walled- carbon nanotubes.Sci. Rep.6, 37167 (2016).

42. Mercer, I. P., Gould, I. R. & Klug, D. R. A quantum mechanical/molecular mechanical approach to relaxation dynamics: Calculation of the optical proper- ties of solvated bacteriochlorophyll-a.J. Phys. Chem. B103, 7720–7727 (1999).

43. Ormond, A. B. & Freeman, H. S. Effects of substituents on the photophysical properties of symmetrical porphyrins.Dyes Pigm.96, 440–448 (2013).

44. Charlier, J.-C., Blase, X. & Roche, S. Electronic and transport properties of nano- tubes.Rev. Mod. Phys.79, 677–732 (2007).

45. White, C. T., Robertson, D. H. & Mintmire, J. W. Helical and rotational symmetries of nanoscale graphitic tubules.Phys. Rev. B47, 5485–5488 (1993).

46. Kreisbeck, C. & Kramer, T. Long-lived electronic coherence in dissipative exciton dynamics of light-harvesting complexes.J. Phys. Chem. Lett.3, 2828–2833 (2012).

47. Mintmire, J. & White, C. Electronic and structural properties of carbon nanotubes.

Carbon33, 893–902 (1995).

48. Durnin, J. Exact solutions for nondiffracting beams. I. The scalar theory.J. Opt. Soc.

Am. A4, 651–654 (1987).

49. Andrews, D. & Babiker, M.The Angular Momentum of Light(Cambridge University Press, 2012).

50. Bomzon, Z., Biener, G., Kleiner, V. & Hasman, E. Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings.Opt. Lett.

27, 285–287 (2002).

51. Dorn, R., Qubis, S. & Leuchs, G. Sharper focus for a radially polarized light beam.

Phys. Rev. Lett.91, 233901 (2003).

52. Goupalov, S. V. Optical transitions in carbon nanotubes.Phys. Rev. B72, 195403 (2005).

53. Zang, X., Singh, N. & Schwingenschlögl, U. Valley-dependent current generation in nanotubes by twisted light.Phys. Rev. B98, 085401 (2018).

54. Ajiki, H. Exciton states and optical properties of carbon nanotubes.J. Phys.:

Condens. Matter24, 483001 (2012).

55. Williams, M. D., Coles, M. M., Saadi, K., Bradshaw, D. S. & Andrews, D. L. Optical vortex generation from molecular chromophore arrays.Phys. Rev. Lett. 111, 153603 (2013).

56. Runge, E. & Gross, E. K. U. Density-functional theory for time-dependent systems.

Phys. Rev. Lett.52, 997–1000 (1984).

57. Spataru, C. D., Ismail-Beigi, S., Benedict, L. X. & Louie, S. G. Excitonic effects and optical spectra of single-walled carbon nanotubes.Phys. Rev. Lett.92, 077402 (2004).

58. Li, Z. M. et al. Polarized absorption spectra of single-walled 4Åcarbon nanotubes aligned in channels of an AlPO4-5 single crystal.Phys. Rev. Lett.87, 127401 (2001).

59. Kumamoto, Y. et al. Spontaneous exciton dissociation in carbon nanotubes.Phys.

Rev. Lett.112, 117401 (2014).

60. Hertel, T. et al. Intersubband decay of 1-D exciton resonances in carbon nano- tubes.Nano Lett.8, 87–91 (2008).

61. Head-Gordon, M., Grana, A. M., Maurice, D. & White, C. A. Analysis of electronic transitions as the difference of electron attachment and detachment densities.J.

Phys. Chem.99, 14261–14270 (1995).

62. Zhan, Q. Properties of circularly polarized vortex beams.Opt. Lett.31, 867–869 (2006).

63. Babiker, M., Bennett, C. R., Andrews, D. L. & Dávila Romero, L. C. Orbital angular momentum exchange in the interaction of twisted light with molecules.Phys.

Rev. Lett.89, 143601 (2002).

64. Schmiegelow, C. T. et al. Transfer of optical orbital angular momentum to a bound electron.Nat. Commun.7, 12998 (2016).

65. Babiker, M., Andrews, D. L. & Lembessis, V. E. Atoms in complex twisted light.J.

Opt.21, 013001 (2018).

66. Forbes, K. A. & Andrews, D. L. Optical orbital angular momentum: Twisted light and chirality.Opt. Lett.43, 435–438 (2018).

67. Ji, Z. et al. Photocurrent detection of the orbital angular momentum of light.

Science368, 763–767 (2020).

68. Andrade, X. et al. Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems.Phys. Chem.

Chem. Phys.17, 31371–31396 (2015).

ACKNOWLEDGEMENTS

The research reported in this publication was supported by funding from King Abdullah University of Science and Technology (KAUST). For computer time, this research used the resources of the Supercomputing Laboratory at KAUST.

AUTHOR CONTRIBUTIONS

X.Z. executed the calculations. All authors contributed to the analysis of the data and the writing of the manuscript.

COMPETING INTERESTS

The authors declare no competing interests.

ADDITIONAL INFORMATION

Correspondenceand requests for materials should be addressed to Mark T. Lusk or Udo Schwingenschlögl.

Reprints and permission information is available at http://www.nature.com/

reprints

Publisher’s noteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

10

Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative