Manifestations of electron interactions in photogalvanic effect in chiral nanotubes
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Manifestations of electron interactions in the photogalvanic effect in chiral nanotubes
Raphael Matthews, Oded Agam, Anton Andreev, and Boris Spivak The Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Department of Physics, University of Washington, Seattle, Washington 98195, USA (Dated: October 14, 2018)Carbon nanotubes provide one of the most accessible experimental realizations of one dimensionalelectron systems. In the experimentally relevant regime of low doping the Luttinger liquid formed byelectrons may be approximated by a Wigner crystal. The crystal-like electronic order suggests thatnanotubes exhibit effects similar to the M¨ossbauer effect where the momentum of an emitted photonis absorbed by the whole crystal. We show that the circular photovoltaic effect in chiral nanotubesis of the same nature. We obtain the frequency dependence of the photovoltage and characterize itssingularities in a broad frequency range where the electron correlations are essential. Our predictionsprovide a basis for using the photogalvanic effect as a new experimental probe of electron correlationsin nanotubes.
PACS numbers: 78.67.Ch
I. INTRODUCTION
Carbon nanotubes (CN) can be viewed as long cylin-ders made from a graphene sheet [1–3]. Depending onthe way this sheet is rolled up, the cylinder may havechiral (i.e., helical) structure. Chirality implies the exis-tence of the circular photogalvanic effect (CPGE), wherea circularly polarized electromagnetic wave induces a dccurrent. The magnitude of the photo-induced voltage isfinite even when the momentum of the photon is negligi-ble. This effect was predicted in Refs. [4, 5] and since thenwas investigated both experimentally and theoretically -see, for example, Refs. [6–9] and references therein.Usually the circular photogalvanic effect in non-centrosymmetric media arises due to a spin-orbit inter-action. In carbon nanotubes the spin-orbit interaction isweak and the circular photogalvanic effect arises becausethe electron motion along the tube axis and circumfer-ence is strongly mixed in chiral nanotubes. The singleparticle theory of chirality related optical effects in car-bon nanotubes has been developed in Ref. [10]. This the-ory holds for non-degenerate electrons in semiconductornanotubes or for metallic ones at high enough frequenciesof the electromagnetic field.However, at low temperatures the single particle de-scription of one-dimensional interacting electron sys-tems, such as metallic CN, is invalid. Being a quasi-one-dimensional system, CN exhibits distinctive featureswhich result from the strong correlations of the electronsin the system [11], for instance, the Fermi edge singu-larity which is manifested by a characteristic power-lawsingularity of the tunneling density of states. The pho-tovoltaic effect in this regime was not investigated, andwill be studied here.The paper is organized as follows. In Sec. II we re-view the non-interacting theory of the CPGE in CN. InSec. III we study the problem with electron-electron in-teractions. Next, in Sec. IV, we discuss the limit where ∆ j =−1 ∆ j =+1 K K b a (a) (b)
K KKK k k x x FIG. 1: (a) The honeycomb lattice of graphene and its prim-itive lattice vectors a and b . (b) The Brillouin zone ofgraphene. K and K ′ denote the Dirac points of the energyspectrum. The straight lines cut the zone at the discrete setof allowed transverse momenta. This set is determined bythe way a graphene sheet is rolled into a tube, such that thewave function is invariant with respect to the translation vec-tor L = n a + m b . The example shown here corresponds to( n, m ) = (5 , K and K ′ points of the Brillouin zone are associated with a changeof the angular momentum by ± ¯ h , thus a circularly polarizedlight directed along the tube excites electrons only near oneof the Dirac points. interactions are strong enough so that the electronic cor-relations are governed by Wigner crystal order. We drawparallels between the CPGE and the M¨ossbauer effect,and show that the photovoltage, as a function of the fre-quency of the electromagnetic wave, exhibits a series ofdistinct singularities. Our results are summarized in Sec-tion V. II. SINGLE PARTICLE PICTURE
Let us begin with a brief review of the noninteract-ing theory of the CPGE in CN’s. Fig. 1(a) presentsthe honeycomb lattice structure of a graphene sheet,and its two primitive lattice vectors a = a (0 ,
1) and b = a ( − , √ /
2, where a = 2 . n and m which specify the trans-lation vector L = n a + m b that wraps around the cylin-der. The requirement that the wave function be invari-ant with respect to a translation by L , ψ ( x + L ) = ψ ( x ),imposes a quantization condition on the component ofthe wave vector in the transverse direction k ⊥ | L | = 2 πj ,where j is an integer. Fig. 1(b) shows the Brillouin zoneof graphene. The corners of this zone are Dirac pointswhere the spectrum is degenerate and the two inequiva-lent points are denoted by K and K ′ . The straight linescutting through the zone represent the aforementionedquantization condition. Each line, i.e, each value of j ,defines a subband of the nanotube spectrum which cor-responds to a definite angular momentum of the electronaround the cylinder (see, for example, Refs. [10, 12, 13]).A characteristic feature of chiral nanotubes is that theminima of subbands with different values of angular mo-mentum are shifted with respect to each other by a mo-mentum δp , as illustrated in Fig. 2. To be concrete weconsider the situation where the lowest conduction sub-band with angular momentum ¯ h j is partially occupied(say due to doping by an an external gate), and the an-gular momentum of the first unoccupied band of the K valley is ¯ h ( j + 1). Now let us assume that a circularly po-larized electromagnetic wave is radiated in the directionof the tube. Absorption of a photon will be accompaniedby a unit change of the angular momentum, which at lowenough frequencies might take place only in one valley,say K, since the corresponding transition in the opposite(K’) valley is forbidden by conservation of angular mo-mentum; see Fig. 1(b). A finite value of the photovoltageemerges from the asymmetry of velocities and light ab-sorption probabilities for right- and left- moving electronsin the K valley. The single particle absorption threshold ω spc is determined by the electron band structure and isillustrated in Fig. 2. Because of the constant densityof states at the Fermi level, the photo-voltage V ( ω ) as afunction of the frequency ω experiences a jump from zeroto a finite value at the threshold frequency ω = ω spc . III. INTERACTING PARTICLES PICTURE
Generally, the frequency ω c of the many-body absorp-tion threshold is different from the single particle thresh-old ω spc . At sufficiently large values of δp the absorptionthreshold can be significantly lower. It corresponds to anindirect transition in which one electron is transferred toa state near the bottom of the upper subband. The to-tal momentum of the system does not change as a resultof photoabsorption. Therefore a momentum − δp is im-parted to the electrons in the lower (partially occupied)subband. Since the velocity of the excited electron inthe upper band is small, only the electrons in the lowerband contribute to the photocurrent. The photovoltage V ( ω ) necessary to nullify the current is determined by δ K’ K p ω csp pE ∆ jj−1 j+1j j+1j FIG. 2: Subbands of chiral nanotubes demonstrating theasymmetric band structure near each valley. The shaded re-gion in the lower box represents the occupied states withinthe single particle picture. the momentum flux required to compensate for the fluxof momentum to the lower band electrons due to pho-ton absorption. Denoting the photon absorption rate by W ( ω ), we obtain for the photo-induced voltage V ( ω ) = δpne W ( ω ) , (1)where n is the electron density in the lower band, and e is the electron charge. The precise criteria of validityof Eq. (1) depend on the value of r s and will be dis-cussed below. Here r s = U/E kin is the ratio betweenthe characteristic potential, U ∼ e n , and the kinetic, E kin ∼ (¯ hn ) /m ( m being the electron mass) energies ofthe electrons.Another manifestation of the many body effects isthat near the threshold, | ω − ω c | ≪ ω c , the photovolt-age exhibits a power law singularity V ( ω ) ∼ ( ω − ω c ) η characteristic of one dimensional (1D) systems. We willshow that η >
0, for δp ≫ ¯ hnr / s , and η < δp ≪ ¯ hnr / s . Thus interactions suppress the photogal-vanic effect near the threshold in the first case and en-hance it in the second. We shall also discuss additionalsingularities in W ( ω ) which occur above the threshold atintervals of the Debye frequency of the Wigner crystaland are associated with the generation of hard plasmons.In principle, recombination processes create an addi-tional contribution to the electron distribution functionin the lower band which is asymmetric in p and whichaffects the photovoltage V ( ω ). In this case however, thesingularity of V ( ω ) at the absorption threshold remainsintact. Recombination with a large energy transfer usu-ally takes place through a multiphonon emission via deepimpurities which are associated with a large momentumtransfer to individual phonons. Therefore the momen-tum asymmetry of the distribution function in the lowerband associated with such processes is small and can beneglected.The significance of many-body effects for the absorp-tion and the photovoltaic effect can be appreciated evenin the framework of perturbation theory using simplekinematic considerations. Let us assume that δp is largerthan the characteristic momenta of electrons in the lowerband. Consider an absorption process in which one elec-tron is transferred to a state near the bottom of theupper band and N electrons in the lower band receivea recoil momentum of order p ∗ ≈ − δp/N and energy ǫ ∗ ∼ ( p ∗ ) / m . The energy of such a photoexcitationis ¯ hω N ≈ ∆ + ( δp ) / (2 mN ), where ∆ is the energy dif-ference between the bottoms of the bands (Fig. 1). Itdecreases with increasing N and may be significantlylower than the single particle threshold ω spc ≈ ω N =1 . If ǫ ∗ ≫ max[ U, E F ], regardless of the details of the elec-tron wave function in the ground state, the excited elec-trons may be treated as free particles and the absorp-tion rate W ( ω ) can be calculated using the N -th orderof the perturbation theory in a way similar to that ofRefs. [14, 15]. The increase of N with decreasing fre-quency can be considered as a precursor of the Lut-tinger liquid regime. Practically, however, the condition( δp ) / (2 m ) ≫ max[ U, E F ] is difficult to realize in car-bon nanotubes, and therefore there is no parametricallybig interval of frequencies where the perturbation theoryworks. For this reason, below, we focus our attention onthe case where frequency is close to the threshold. IV. THE WIGNER CRYSTAL LIMIT
Consider the experimentally relevant regime of lowelectronic densities n , where the ratio of the interac-tion energy to the kinetic energy of the electrons is large, r s ≫
1. In this case exchange processes associated withthe tunneling of electrons are exponentially suppressedin r s , and will therefore be neglected in what follows. Inthis approximation electrons may be labeled by the sitenumber ν in the Wigner crystal lattice.The wave functions of electrons localized near a givensite are superpositions of Bloch functions either in thevalley K or K ′ . The two possibilities are realized withequal probability. In a broad frequency range electronsexcited by light into the upper subband remain localizedat their original sites of the Wigner crystal. For a cir-cularly polarized light only electrons from valley K canparticipate in the absorption. Thus the wave functions ofexcited electrons are superpositions of Bloch wave func-tions of the upper band in the K valley. Since they arelocalized near the Wigner crystal sites, their average mo-mentum must be δp ; see Fig. 1. Conservation of mo-mentum dictates that the opposite momentum − δp must be transferred into the collective motion of the Wignercrystal, which leads to Eq. (1). A similar transfer ofthe recoil momentum from a given atom to the wholecrystal occurs during the emission of γ rays. One of thestriking manifestations of the transfer of the recoil mo-mentum to the collective motion in three dimensionalcrystals is the M¨ossbauer effect [16] in which no phononsare excited during a γ -ray emission. The probabilityfor such a transition is given by the Debye-Waller fac-tor exp[ − ( δp ) h x i / ¯ h ], where δp is the recoil momen-tum and h x i is the variance of the displacement of thenucleus from its equilibrium position. In a one dimen-sional system the latter diverges logarithmically with thesystem size and the M¨ossbauer effect is impossible. How-ever light absorption with the emission of collective exci-tations (plasmons) remains possible. The probability ofplasmon emission determines the absorption rate abovethe threshold, ω > ω c . Though these plasmons carry theexcess energy, their average momentum is zero. Thuseach photon absorption provides a momentum δp whichis transferred to the Wigner crystal of the electrons inthe occupied subband. Since the average velocity of theexcited electron in the upper band is zero, we again arriveat the expression for V ( ω ) given by Eq. (1).Near the absorption threshold the probability of inter-band transitions exhibits a power law singularity, whichcan be described phenomenologically in terms of a motionof a mobile impurity in a Luttinger liquid [17–23]. Thestrength of electron-electron interactions in the Wignercrystal regime enables us to determine the light absorp-tion probability not only near the absorption thresholdbut in a much wider frequency range.Although many of our conclusions have general charac-ter, we will consider the case ( δp ) / m ≪ ne where theWigner crystal may be viewed as a weakly anharmonicchain [24]. We model our system by the Hamiltonian H = H + H ep , where the unperturbed Hamiltonian is H = X ν ∆ + ( p ν − δp ) m p ν m ! + X µ>ν V ( x µ − x ν ) . (2)Here p ν and x ν denote the momentum and position of the ν -th electron, ∆ is the energy shift between the bands,and V ( x ) is the interaction potential. For simplicity theelectron mass is assumed to be the same in both sub-bands. The matrix in Eq. (2) acts on the subband index.In the rotating wave approximation the interaction withthe electromagnetic field is described by H ep = D X ν ( σ + ν ae − iωt + h.c. ) . (3)Here a is the photon annihilation operator, σ + ν is theraising operator of the ν -th electron from the lower tothe upper subband, and D is the dipole matrix elementof the transition. The electron positions and momenta ina Wigner crystal regime can be expressed in terms of theplasmon annihilation and creation operators, b q and b † q , x ν = ν/n + X q s ¯ h mN ω q ( b q + b †− q ) e iqν , (4a) p ν = − i X q r ¯ hmω q N ( b q − b †− q ) e iqν , (4b)where n and N are the electron density and the numberof sites, respectively, in the Wigner crystal, and ω q isthe frequency of the a phonon with (dimensionless) wavenumber q which stratifies the dispersion relation: ω q = 2 m ∞ X ν =1 d V ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) x = ν/n [1 − cos( qν )] . (5)Denoting by c and c † the annihilation and creation ofphotoexcitations, respectively, on site ν = 0, one canwrite the Hamiltonian in the form H = X q ¯ hω q b † q + iδp c † c p mN ¯ hω q ! b q − iδp c † c p mN ¯ hω q ! + ∆ c † c, (6)where we have subtracted the zero point energy of theplasmons. The absorption of photons happens indepen-dently at different lattice sites; thus, within the dipoleapproximation, it is given by the Kubo formula, W ( ω ) = N D ¯ h Re Z ∞ dte iωt h [ c ( t ) , c † (0)] i , (7)where h· · · i denotes thermal averaging. To decouple theinteraction between the plasmons and the excited particlewe perform a unitary transformation of the operatorsˆ U = exp iδp c † c X q b q + b †− q p mN ¯ hω q ! . (8)In the transformed basis the Hamiltonian takes the formˆ U † H ˆ U = X q ¯ hω a b † q b q + ¯ hω c c † c, (9)while the creation and annihilation operators areˆ U † c ˆ U = ce iδpx / ¯ h , and ˆ U † c † ˆ U = ce − iδpx / ¯ h . (10)Thus in the limit of zero temperature Eq. (7) reduces to W ( ω ) = N D ¯ h Z ∞−∞ dt exp (cid:18) i ( ω − ∆ / ¯ h ) t − ( δp ) F ( t )2 m ¯ hω D (cid:19) , (11)where ω D = ω π is the Debye frequency, and F ( t ) = Z π dq π ω D ω q (1 − e − iω q t ) . (12) Ω - Ω c Ω D Ñ Ω D W H Ω L D N FIG. 3: The absorption spectrum obtained by numerical in-tegration of (11) for the case of short range interactions anda momentum shift δp , for which η = − . The singular behavior of the absorption rate is deter-mined by the long time asymptotic behavior of F ( t ). Inthis limit the integral over q may be evaluated by thestationary phase approximation. For a typical disper-sion, the stationary phase is at q = π and the dispersion,near this point, may be approximated as ω q ≃ ω D + 12 ω ′′ π ( q − π ) , (13)with ω ′′ π = d ω q /dq | q = π . But there are also contribu-tions from the end points of the integration interval, at q = 0 and q = 2 π . Near these points, the dispersion maybe linearized. In particular, near q = 0, ω q ≃ ω ′ q, (14)where ω ′ = dω q /dq | q =0 , and similarly near q = 2 π , ω q ≃ ω ′ (2 π − q ). Evaluating the integral (12) using theseapproximations we obtain F ( t ) ∼ ω D πω ′ ln( iω ′ t ) + γ − exp( − iω D t ) p πiω ′′ π t , (15)where γ is a constant of order unity which depends onthe precise form of ω q . The first term of this asymp-totic formula accounts for the soft plasmons generated bythe excitation, while the last term is associated with theexcitation of hard plasmons with energy near the plas-mon Debye frequency ω D . Substituting this expressionin (11) and expanding the hard plasmons contributionin a power series we find that the absorption exhibitssingularities at frequencies ω j = ∆ / ¯ h + jω D , which areassociated with the generation of j hard plasmons. Thusfor | ω − ω j | ≪ ω D we get W ( ω ) ∼ s j ( ω − ω j ) | ω − ω j | η j , (16)where η j = δp πm ¯ hω ′ + j − , (17)and s j ( ω ) = α j + β j θ ( ω ) is a step function between twovalues which depend on j .The existence of higher frequency singularities, j > r s → ∞ . At finite values of r s thesesingularities are smeared by anharmonic interactions be-tween plasmons. The power of the higher frequency sin-gularities, η j , increases with j . Therefore only the firstfew of them are significant. An example of this behaviorin the case where ω q = ω D | sin( q/ | and the value of δp is chosen such that η = − . j = 0) is not broadened by anharmonic interactions. Itsexponent is η ≈ δp n √ r s − η j dueto the sudden change of the effective interaction strengthbetween the excited electron and the rest of electrons,which leads to a contribution to η associated with theorthogonality catastrophe.Let us turn now to the case r s ≪ ω spc and usually δp ≪ p F where p F is the Fermi momentum of the occupied band.The photoexcitation induces a direct transition whichdoes not change the electron momentum. Therefore theM¨ossbauer mechanism described above does not apply.However, if the energy relaxation time of an excited elec-tron in the upper subband is shorter than the momen-tum relaxation time of the electrons in the bottom band,then each photoexcitation results in an effective momen-tum transfer of − p F to the electrons in the lower band.Thus the photovoltage is given by Eq. (1) in which δp should be replaced by p F . The absorption rate has apower law singularity exponent at the absorption thresh-old, W ( ω ) ∼ ( ω − ω spc ) η with the exponent η ∼ r s . Theabove threshold singularities shown in Fig. 3 are expectedto smear out. V. SUMMARY
In this paper we have shown that the inclusion of in-teractions into the CPGE problem in CN changes thefrequency dependence of the photovoltage V ( ω ) dramat-ically. Instead of a jump at the threshold frequency, theinteraction produce a singularity with a power exponentwhich depends on the relation between the momentumshift between subbands and the strength of the interac- tion between neighboring electrons, similar to the classi-cal M¨ossbauer effect. We also show that V ( ω ) exhibitsadditional singularities at higher frequencies. These re-sults provide a new experimental probe by which the na-ture of electronic correlations in carbon nanotubes canbe examined.We thank E. L. Ivchenko, T. Giamarchi, A. Kamenev,and K. Matveev for useful discussions. This research hasbeen supported by the United States-Israel BinationalScience Foundation (BSF) grant No. 2008278 and by theU.S. DOE grant DE-FG02-07ER46452. [1] S. Ijima, Nature (London), 354, 56, (1991).[2] C. Schnenberger and L. Forro, Physics World Vol 13, No6, 37-41 (2000)[3] V. N. Popov, Mater. Sci. Eng. R. ,61 (2004).[4] E. L. Ivchenko and G. E. Pikus, Pis’ma Zh. Exsp. Teor.Phys. , 640 (1978) [JETP Lett. , 604 (1978)].[5] V.I. Belinicher, Phys. Lett. A 66, 213 (1978).[6] B. I. Sturman and V. M. Fridkin, The Photovoltaic andPhotorefractive effects in Noncentrosymmetric Materials ,Gordon and Breach Science Publishers, 1992.[7] E. L. Ivchenko and G. E. Pikus,
Superlattices and OtherHeterostructures: Symmetry and Optical Phenomena ,Springer Series in Solid State Sciences, vol. 110, Springer-Verlag, 1995; second edition 1997; Ch. 10.[8] S. D. Ganichev et al., Appl. Phys. Lett. , 3146 (2000).[9] S. D. Ganichev et al., Physica E , 52 (2001).[10] E. L. Ivchenko and B. Spivak, Phys. Rev. B , 155404(2002).[11] T. Giamarchi, Quantum Physics in One Dimension .Clarendon Press, Oxford (2004).[12] S. Tasaki, K. Maekawa, and T. Yamabe, Phys. Rev. B , 9301 (1998).[13] H. Ajiki and T. Ando, J. Phys. Soc. Japan , 1255(1993).[14] M. B. Voloshin, Nuc. Phys. B , 233 (1992).[15] L. S. Brown, and C. Zhai, Phys. Rev. D 47 , 5526, (1993).[16] H.J. Lipkin,
Quantum Mechanics: New Approaches toSelected Topics , (Dover, 2007).[17] T. Ogawa, A. Furusaki, and N. Nagaosa Phys. Rev. Lett. , 3638 (1992).[18] L. Balents, Phys. Rev. B , 4429 (2000).[19] M. B. Zvonarev, V. V. Cheianov and T. Giamarchi, Phys.Rev. Lett. , 240404 (2007); ibid. , 116804 (2011).[24] K. A. Matveev, A. V. Andreev, and M. Pustilnik, Phys.Rev. Lett.105