Electron-phonon interactions and Resonance Raman scattering in onedimensional systems: application to carbon nanotubes José Menéndez Giovanni Bussi Elisa Molinari Acknowledgements: M. Canonico, C. Poweleit, J. B. Page, G. B. Adams Supported by the National Science Foundation
Infrared absorption and Raman scattering 0 50 100 150 200 250 300 Frequency (cm -1 ) -300-200 -100 0 100 200 300 Raman shift (cm -1 )
Energy units for optical spectroscopy hw = hck = 2phc l k wave vector; 1/l = wavenumber 1/l µ energy 1 ev = 8067.5 cm -1 300 cm -1 = 37.2 mev
Ï Ô Ì Ô Ó Conservation rules for a Raman process hw = hw + h L S W phonon hk = hk + hq L S phonon BACKSCATTERING W (cm -1 ) 300 Raman K S q phonon L L L K L [ wl - ws]( q) = w( KL) - w( q -KL) = ck -c( q -K ) = 2cK -cq 4p 500 nm Brillouin p 0.5 nm q phonon
Phonons in crystals Raman-active
What determines Raman intensities? R È S = 1 R Í R Í M «() t 4pcR 2 2 R Í R Î E ( t) = E coswt Ê ˆ M t P e iwt P ei t a() = 1 Á - * w ag + ag E Á 2Â Ë Ê ˆ P P w, u, u= c( f) d P g ab ab g g Á ab Ë P = P ( w, 0) + ab ab  d d f  f f f f g
Why a frequency shift? M () t = P(, tt ) E( t ) dt Ú M( w) = P( w, w ) E( w ) dw Ú If P(t,t ) = P(t-t ) fi P(w,w ) = P(w) d(w-w ) fim(w) = P(w) E(w) If a phonon is present: Ptt (, ) Pt ( + Tt, + T) Ê PÊw, w ˆ P( n) w d w w n = ( ) n T p Á - -2 Ë Â Á Ë fi w = w -nw phonon ˆ In QM this is just energy conservation!
The carbon nanotube family (n,n) Armchair (n,0) Zigzag (n,m) Chiral C = na + ma h 1 2
The electronic structure of graphene V ppp V ppp = 3.1 ev
Nanotubes: folding the graphene band structure Ê Á Á K Em( k) = E k 2 Á + mk, ( m = 0,..., N -1); g2dá 1 K Á Ë 2 ˆ (11,8) (15,0) (10,10)
Nanotubes: electronic density of states (11,8) (15,0) (10,10)
Vibrational modes in nanotubes
The radial breathing mode in carbon nanotubes w = 1170 cm -1 1003 R. (armchair) J. Kürti et al. Phys. Rev. B 58, R8869 (1998) w = 1170 cm -1 R 0. 982 (zigzag)
Why 1/r dependence? Let us consider an atom of mass M, The radial force is F r = 2F t cos q = F t a C-C /r 0. But F t = K s Da C-C. On the other hand, Da C-C = 2u r cos q = u r a C-C /r 0. Hence F r K a a = Ê s Ë Á ˆ Ê Á r Ë r Therefore C-C C-C 0 0 ˆ ur = K u eff r u r u r F t a C-C u r q K = = M K M Ê a Á Ë r eff s C-C w RBM 0 ˆ r 0 r 0 r0
Raman intensities in carbon nanotubes A.M. Rao et al., Science 275, 187 (1997)
Resonance Raman excitation profiles hw L 100 150 200 250 300 Raman shift (cm -1 ) 100 150 200 250 300 Raman shift (cm -1 ) 100 150 200 250 300 Raman shift (cm -1 ) 100 150 200 250 300 Raman shift (cm -1 ) 100 150 200 250 300 Raman shift (cm -1 ) Intensity (arb. units) 1.65 1.70 1.75 1.80 1.85 1.90 1.95 Laser Photon Energy (ev)
REP for 200 cm -1 RBM E 11 = 1.915 ev Intensity h = 60 mev 1.70 1.80 1.90 2.00 2.10 2.20 Photon energy (ev) M. Canonico et al., Phys. Rev. B 65, 201402 (2002)
Resonance Raman scattering Light Phonon Light 0 j i 0 Light j j phononi i Light I µ Â 0 0 Ê ˆ Ê ij Áhw -hw -E hw -E Á Ë Á Ë L phonon j L i ˆ 2
Raman cross section: quantum theory I The scattering cross section is defined as ds dw = ( radiated power into ) dw incident power per unit area The incident power per unit area is CMP ( ) IP N c Lhw L = n L Other RP IP (R.1) (R.2) where N L = number of incident photons per unit volume and c/n L is the speed of light. The radiated power is RP dw = number of incident photons hw S ( dw ) dt dw = ( NV L ) hw S ( dw ), dt (R.3)
Raman cross section: quantum theory II where dw/dt (dw) is the transition probability per unit time from a state with a photon (w L, k L,l L ) to a state with a photon with polarization l S and a wave vector k S within the solid angle dw plus a phonon of frequency W ph and wave vector q. For an arbitrary scattering wave vector k S the transition probability is given, according to Fermi s golden rule, by dw dt CMP Other d hw hw h 2p 2 W fi l S, l L L S  = ( ) - - h k S ( W ) ph (R.4) where W fi is the matrix element of the transition operator. Combining R.4 with R.1-3, we obtain ds wl nv L = Ê dw Ë Á ˆ ÊË ˆ w c S 2 p  Wfi S L d wl w ( l, l ) 2 - S - h k ( dw ) S ( h h hw ) ph (R.5)
Raman cross section: quantum theory III The sum over k S can be transformed into an integral: k S 3 Â Ú 2 S 3 kdk S S 3 ( dw ) 2p 2p h c Ú CMP Inserting this into R.5, we finally obtain V V n = = Ê w d ( ) ( ) Ë ˆ ( hw ) Other 2 S S (R.6) ds nl dw = Ê Ë Á w L ˆ w Sn ÊË 4 c S ˆ V Wfi ws S w ÊË 2 ˆ 2 L L p (, l ;, l ) (R.7) 2 h The key quantity that contains resonance effects is the transition matrix element W fi
The matrix element W fi I We will use perturbation theory to compute W fi. We assume that the unperturbed hamiltonian H 0 can be written as H 0 = H el + H R + H L, with: A one-electron (band structure) hamiltonian of the form A free radiation term (photons) of the form A harmonic vibrational hamiltonian (phonons): + Hel = E nsc nsc ns CMP k, n,s Other 1 ( 2 ) + HR = a a + H Â k k k Âhw k l k l k l k, l ( 1 ) 2 + = b b + L ÂhW q m q m q m q, m (R.8) (R.9) (R.10)
The matrix element W fi II Therefore the quantum states of the system are characterized by three sets of occupation numbers: (R.11) We will be able to simplify this notation considerably because only a few of these occupation numbers change in the scattering process. For example, the initial state consists of the electronic and phonon system in their ground CMPstates and one photon in state Other k L l L. The final state consists of the electronic system in the ground state, one photon in state k S l S, and one phonon in state qm, so that we can express these states as i f m { N } { N } { N },, l nk qm k = 0010,,, ; E = hw L S i L = 010,,, 1; E = hw + hw qm L S f S qm (R.12)
The matrix element W fi III The transition between these states is caused by the interaction hamiltonian H int. We will take H int = H el + H er, where H el is the electron-phonon interaction and H er the electron-photon interaction. The electron-phonon interaction can be written as nn + H = M b + b c + c el  kqm qm -qm k -q, ns kn s kqnn ms CMP ( ) Other (R.13) Explicit expressions for M kqm will be discussed below. The electron-radiation hamiltonian arises from the A p coupling term in the interaction of an electron-system with an electromagnetic wave. It has a form similar to R.13: H er e = Ê ˆ Ë m Ê Á Ë 2p 2 Vn w K K 12  kknn ls ˆ nn ( + + e p a + a- ) c -, c l kk Kl Kl k K ns kn s (R.14) where p kk is the matrix element of the momentum operator between the periodic parts of the Bloch wave functions for states nk and n (k-k)
The matrix element W fi IV When applied to the specific problem at hand, the notation in these expressions can be simplified. We first limit the summations over electronic bands n, n to a single empty conduction band and a singly fully occupied valence band. We assume parabolic bands. CMP E t k Other 2me + h2 2 E t k - h2 2 2m h
The matrix element W fi V We keep the notation c, c + for annihilation/creation operators of electrons in the conduction band, and we introduce v, v + as annihilation/creation operators for electrons in the valence band. We also limit ourselves to the case when a photon of frequency w L is annihilated, a phonon is created, and a photon of frequency w S is created. We then rewrite the electron-phonon hamiltonian as CMP Other ( ) Â el qm kqm k -q, ns kn s kqm k -q, ns kn s kqms + cc vv H = b M c + c + M v + v (R.15) Notice that we have limited ourselves to intraband terms of the form c + c and v + v. We have not included interband terms of the form c + v, etc. We will see later that the interband contribution is negligible.
The matrix element W fi VI For the electron-radiation hamiltonian we further assume that the incident and scattered light polarizations are parallel to the same axis z. In nanotubes z must be the axis of the tube. Other directions give zero. We can now write with: H H e CMP Other cv = Ê 12 ˆ Ê 2p ˆ pz a c + v v + c Ë S - S - S m Á Â + 2,,, Ë Vn w + + er kk K k K s ks k K s ks S S ks 12 - er e = Ê ˆ Ë m Ê Á Ë 2p 2 Vn w L L H = H + H + - er er er, ( ) ˆ Â p vc z a c + v v +, c L + L, + + L, ks ( ) s s s s kk K k K k k K k Here we have neglected intraband terms. This is because intraband matrix elements of the momentum operator are very small for small wave vector transfers (by virtue of Green s theorem for periodic functions, see Ashcroft-Mermin, Appendix I). (R.16) (R.17) (R.18)
The matrix element W fi VII Since each application of H er can change one photon state (the interaction is linear in the photon creation and annihilation operators) and each application of H el can only change one phonon state, we can only transition from the initial state to the final state by two applications of H er and one application of H el. In other words, we need to go to at least third order in H int : W fi fh H H i = Â int m m int n n int ( E - E ) mn m i ( En - Ei ) CMP Other fh H H i = Â int m m int n n int mn E - hw E hw ( ) ( ) m L n - L (R.19) Notice that the initial state i> contains a filled valence band and an empty conduction band. It is then clear that only the c + v terms in R.17 and R.18 can contribute. (This is of course a manifestation of Pauli s exclusion principle.) Of these, only R.18 can lead to a resonant enhancement by annihilating the incoming laser photon and creating an electronic excitation of equal energy, so that the rightmost denominator in R.19 becomes small.
The matrix element W fi VIII Maximum resonant enhancement is then obtained if the energy E m is comparable to the energy E n. When the phonon energy is small compared to the separation between the conduction and valence bands, this condition is met when the middle operator causes an intraband transtion. Thus the most resonant contribution to the Raman cross section arises from W fi + - fher m m HeL n n HeR i = CMP Â Other E - E E E mn ( ) ( ) m i n - i (R.20) By inserting R.15, R17, and R18 in R.20 we obtain sums over four indices k,k, k and q. (And also over the spin index s, but this gives trivial factors of 2). There are two types of terms that yield nonvanishing contributions:
The matrix element W fi IX CONDUCTION BAND TERMS These terms contain matrix elements of the form + + 0v c c c c + v 0 n ( W ) + 1 S q k - K k k - k k + K k qm where 0> refers to the electronic ground state, and n(w) is the phonon occupation number at temperature T. We assume that there is one incident photon and one scattered photon, so the application of the photon creation/annihilation CMP operators gives just 1. Other These elements are zero except when k = k + K L, k = k + K S, and q = K L - K S. When these conditions are satisfied the matrix element is equal to +1.The last condition is clearly the manifestation of crystal momentum conservation. VALENCE BAND TERMS These terms contain matrix elements of the form + + 0v c v v c + v 0 n ( W ) + 1 S q k - K k k - k k + K k qm These elements are zero except when k = k + K L - K S, k = k + K L, and q = K L - K S. When these conditions are met the matrix element is equal to -1. (Using commutation properties of Fermion creation/annihilation operators.) L L (R.21) (R.22)
The matrix element W fi X We thus end up with a single summation over the index k.we now take into account the fact that the relevant light wave vectors are negligible relative to a reciprocal lattice vector in a typical crystal. Thus we can set K L = K S = 0 in all matrix elements and energies. We thus obtain W fi e = Ê ˆ Ë m cv vc Ê 2ph ˆ p p M cc M vv z z m m Á Ê 1 ˆ - Á Â, k, k k0 k0 Ë VnLnS CMP Ë wlws ks ( Ek - hw Other S ) Ek - hws 2 12 where E k is the energy of the intermediate state given by E k We now recognize that the states closest to to k = 0 will make the dominant contribution because they are closest to the singularity in the density of states. We can then replace the momentum and phonon matrix elements for their values at k = 0 and take them out of the sum. We then obtain ( ) ( ) 2 2 2 hk hk hk = Et + + = E + ( m ) = m + m 2m 2m 2 m ; with * * e h - 1-1 - 1 t e h (R.23) (R.24)
W fi  ks where we have defined The matrix element W fi XI e = Ê ˆ Ë m Ê 2ph ˆ Á Ê 1 Á Ë Vn n Ë w w 2 12 L S L S 1 ( Ek - hw E CMP S ) k - ( hw ) cv P p p z = ( ) vc z, 0 z, 0 * S ˆ 2 Pz M cc vv m - Mm ( ) Other. The last factor can be converted into an integral: 1 r EdE ( - ) Ú k E hw E - hw ( E - hw + ih) E - hw + ih  s k ( ) = ( ) ( ) S k S S L (R.25) (R.26)
The matrix element W fi XII where we have added a phenomenological broadening parameter that represents the lifetime of the excited states. For a one-dimensional solid: Other where the first factor of 2 takes into account the spin degeneracy. Using R.24 de Hence, using R.27 we obtain h = kdk = m L rkdk = 2 Ê d k (R.27) Ë 2 p ˆ h m 2 2 * * ( ) * È2m E - E Í Î 2 h L m r( E) = Ê * ˆ È Ë ph Í Î2 E - ( ) E t t 12 12 dk (R.28) (R.29)
The matrix element W fi XIII We thus need to calculate the following integral Ú = -12 ( E - Et ) de ( E - hw + ih) ( E - hw + ih) = L ip È Í 1 - Other 1 hw ph Î( hwl - Et -ih) hws - Et -ih These integrals are discussed in the review article by Martin and Falicov. Combining R.30 with R.29, R.25, and R.26, we finally obtain S ( ) 1 2 1 2 e cc vv L m W = ( ) ( )( )( ) * p P - M n fi È Í Î 2 12 2 1 1 2 m V n n w w z m L S L S ph 1 1 - hw - E -ih hw - E -ih ( ) 1 2 1 2 L t S t m (R.30) ( )( )( ) ( W ) + 1 ( ) hw 2 12 (R.31) R. M. Martin and L. M. Falicov,"Resonant Raman Scattering", in Light Scattering in Solids, edited by M. Cardona (Springer Verlag, Berlin, 1975), Vol. 8, p. 79. m
Raman cross section: quantum theory IV so that the cross section finally becomes ds 1 e = Ê ˆ dw 2 Ë mc h 2 F( w ) L 4 2 2 Ê ws S Á Ë w L ˆ Ê n Á Ë n L ˆ L Ê Á Ë W m ˆ 4 P M cc z m M vv 2 * - m m n ( ) ( Wm ) + [ 1] (R.32) with F( w ) = i L Ê Á Ë 1 1 - hw - E -ih hw - E -ih ( ) ( ) 1 2 1 2 L t S t ˆ (R.33)
Raman cross section: quantum theory V The measured photon count rate at the detector is proportional to the cross section, but the proportionality factor is usually left undetermined, since the measurement of an absolute Raman cross section requires a careful calibration with a known standard. [Notice that we defined the cross section in terms of power. We can define a photon number cross section which is equal to R.1 times (w L / w S )] If no absolute cross section is measured, only the function F(w L ) is needed to analyze the experimental results, assuming that the approximation of a single conduction and a single Other valence band is valid. This is the case for armchair tubes. From fits of F(w L ) to the experimental data it is possible to determine the transition energy E t and the lifetime of the excited state. In order to study the properties of this function we define Fx ( ) = i Ê Á Ë 1 1 - x -1 -ied x - D -1 -ied ( ) ( ) 1 2 1 2 ˆ (R.34)
Raman cross section: quantum theory VI There are two singularities associated with the incoming laser photon or the scattered photon having the same energy as the transition energy. Notice that the imaginary part has the expected asymmetric shape of a one-dimensional density of states, but the real part has the opposite shape, leading to symmetric intensity profiles. Intensity Imaginary part D = 0.1 e = 0.01 Other Real part Real F, Imag F 0.95 1.00 1.05 1.10 Normalized laser frequency x L 1.15
Raman cross section: quantum theory VII A more realistic value for D in carbon nanotubes is D = 0.01. As the broadening of the electronic states increases, the intensity profile evolves to a single peak Intensity D = 0.01 e = 0.10 Other Intensity D = 0.01 e = 0.01 0.990 1.000 1.010 1.020 Normalized laser frequency x L Intensity D = 0.01 e = 1.00 0.990 1.000 1.010 1.020 Normalized laser frequency x L 0.96 0.98 1.00 1.02 1.04 1.06 Normalized laser frequency x L
REP for 200 cm -1 RBM E 11 = 1.915 ev Intensity h = 60 mev 1.70 1.80 1.90 2.00 2.10 2.20 Photon energy (ev) M. Canonico et al., Phys. Rev. B 65, 201402 (2002)
Trigonal warping in carbon nanotubes I The figure shows the 1D density of states for several metallic nanotubes of approximately the same diameter, showing the splitting of singularities (increasing from zero in armchair tubes to a maximum value in zig-zag tubes) due to trigonal warping in the band structure of graphene. The associated optical transitions are very close to each other and therefore one has to include two conduction bands and two valence bands in the calculation of Raman cross sections. Other From R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Phys. Rev. B 61, 2981 (2000)
Trigonal warping in carbon nanotubes II Because the two transitions can interfere, it is critical to know the sign and magnitude of the prefactor cc P M - M vv m z ( )( ) 2 * 12 m m in the expression for the transition matrix element. The effective mass can be obtained by fitting a parabolic band to the calculated energy bands. The matrix element of p z can be easily obtained from pseudopotential calculations of the band structure. Other For the electron-phonon matrix elements, we can derive a simple expression for the particular case m = RBM, the radial breathing mode. If we limit ourselves to this mode, the electron phonon interaction is H el H = d where d RBM is the normal coordinate for the RBM and H is the electronic hamiltonian. The derivative means that the atomic positions are displaced by along the phonon mode eigenvector. RBM d RBM (R.36) (R.37)
Trigonal warping in carbon nanotubes III The normal coordinate can be written in terms of the mode eigenvectors c and the atomic displacements u as d  * f c lka f M k u lka = ( ) ( ) ( ) lka where l is the cell index, k the basis index, a a cartesian index, and M(k) the mass of atom k within the unit cell. For a periodic solid, Other 1 c( lka qm) = ka NM el ( q me ) c k iqr ( lk ) (R.38) (R.39) where N c is the number of unit cells and e is a unit vector normalized to unity:  e * ( lka qm ) e ( lka q m ) = d qq d mm ka (R.40)
Trigonal warping in carbon nanotubes IV If we give all atoms a displacement Du( lka ) = DR e( ka RBM ), (R.41) where R is the nanotube radius, and we assume q RBM = 0, the corresponding change in the normal coordinate is Dd = N M D Other R RBMc where we have used R.38, 39, 40 and the fact that the atomic mass is the same for all atoms The electron-phonon hamiltonian thus becomes H el = 1 NM c H R d i h RBM = 2NMw c RBM H ( R b - b + ) RBMRBM (R.42) (R.43)
Trigonal warping in carbon nanotubes V We therefore need matrix elements of the form where we have invoked the Hellman-Feynman theorem. By comparing R37 with R.15, and using R.41, we can write cc vv M - M = -i RBM RBM = i H n R n nk k k = ER h 2NMw c h 2NMw c RBM RBM Ê Ec Ë R Et R Other Ev - R Thus the relevant quantity for the calculation of Raman cross-sections is the derivative of the transition energy relative to the radius of the nanotube. ˆ (R.44) (R.45)
Trigonal warping in carbon nanotubes VII 5 3.0 0 E f 2.0 Energy (ev) -5-10 1.62 ev 1.99 ev Other E f 1.0 0.0 Energy (ev) -1.0-15 -2.0-20 G Z G Z -3.0
Trigonal warping in carbon nanotubes VIII Autovalori a G Transizioni a G 1.5 2.50 1.0 2.25 Energia (ev) 0.5 0.0-0.5 1.62 ev 1.99 ev 1.99 ev Other 1.62 ev 2.00 1.75 1.50 Energia (ev) -1.0 1.25-1.5 5.75 5.8 5.85 5.9 5.95 6 Raggio del nanotubo 1.00 5.75 5.8 5.85 5.9 5.95 6 Raggio del nanotubo (A)
Trigonal warping in carbon nanotubes IX Squared momentum matrix element (1/Å) Square root of effective mass 0.8 1.4 0.6 0.4 1.2 1 0.8 0.6 Other 0.2 0.4 0.2 0 1.62 ev 1.99 ev 0 1.62 ev 1.99 ev
Trigonal warping in carbon nanotubes X We can use this to compute the corresponding Raman cross section. When we reverse the sign of the radial derivative for one of the transitions, we obtain a different profile, indicating that interference effects cannot be neglected. Intensity E 2 -E 1 = 0.37 ev h = 0.07 ev Intensity Other E 2 -E 1 = 0.37 ev h = 0.07 ev de 2 /dr reversed 1.4 1.6 1.8 2.0 2.2 Laser photon energy (ev) 1.4 1.6 1.8 2.0 2.2 Laser photon energy (ev)
REP for 188 cm -1 RBM Intensity (12,6) (15,3); (16,1) (11,8) 1.70 1.75 1.80 1.85 1.90 1.95 2.00 2.05 2.10 Photon energy (ev)
CONCLUSIONS Interference effects are important for the understanding of the Raman cross section in carbon nanotubes. The shape of the Raman cross section as a function of the excitation energy may provide an optical tool to identify the (n,m) values. It is unclear to what extend the inclusion of excitonic effects may affect the simple calculations presented here.