1. Physics and Astronomy Electron-phonon interaction and Resonance Raman scattering in one- dimensional 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

2. Physics and Astronomy Infrared absorption and Raman scattering

3. Physics and Astronomy Energy units for optical spectroscopy k  wave vector; 1/ = wavenumber 1/  energy 1 eV = 8067.5 cm -1 300 cm -1 = 37.2 meV

4. Physics and Astronomy Conservation rules for a Raman process BACKSCATTERING q phonon KSKS KLKL Raman Brillouin (cm -1 ) 300

5. Physics and Astronomy Phonons in crystals Raman-active

6. Physics and Astronomy What determines Raman intensities? R

7. Physics and Astronomy Why a frequency shift? If P(t,t’) = P(t-t’)  P(  ’) = P(  )  (  ’)  M(  ) = P(  ) E(  ) If a phonon is present: In QM this is just energy conservation!

8. Physics and Astronomy The carbon nanotube family (n,n) Armchair (n,0) Zigzag (n,m) Chiral

9. Physics and Astronomy The electronic structure of graphene V pp  = 3.1 eV

10. Physics and Astronomy Nanotubes: folding the graphene band structure (11,8) (15,0) (10,10)

11. Physics and Astronomy Nanotubes: electronic density of states (11,8)(15,0)(10,10)

12. Physics and Astronomy Vibrational modes in nanotubes

13. Physics and Astronomy The radial breathing mode in carbon nanotubes (zigzag) (armchair) J. Kürti et al. Phys. Rev. B 58, R8869 (1998)

14. Physics and Astronomy Why 1/r dependence? Let us consider an atom of mass M, The radial force is F r = 2F t cos  = F t a C-C /r 0. But F t = K s  a C-C. On the other hand,  a C-C = 2u r cos  u r a C-C /r 0. Hence Therefore

15. Physics and Astronomy Raman intensities in carbon nanotubes A.M. Rao et al., Science 275, 187 (1997)

16. Physics and Astronomy Resonance Raman excitation profiles

17. Physics and Astronomy M. Canonico et al., Phys. Rev. B 65, 201402 (2002)

18. Physics and Astronomy Resonance Raman scattering Light Phonon

19. Physics and Astronomy Raman cross section: quantum theory I CMPOther The scattering cross section is defined as (R.1) (R.2) The incident power per unit area is where N L = number of incident photons per unit volume and c/n L is the speed of light. The radiated power is (R.3)

20. Physics and Astronomy Raman cross section: quantum theory II CMPOther For an arbitrary scattering wave vector k S the transition probability is given, according to Fermi’s golden rule, by where W fi is the matrix element of the transition operator. Combining R.4 with R.1-3, we obtain (R.4) where dw/dt (d  ) is the transition probability per unit time from a state with a photon (  L, k L, L ) to a state with a photon with polarization S and a wave vector k S within the solid angle d  plus a phonon of frequency  ph and wave vector q. (R.5)

21. Physics and Astronomy Raman cross section: quantum theory III CMPOther Inserting this into R.5, we finally obtain The key quantity that contains resonance effects is the transition matrix element W fi The sum over k S can be transformed into an integral: (R.7) (R.6)

22. Physics and Astronomy The matrix element W fi I CMPOther A harmonic vibrational hamiltonian (phonons): A one-electron (band structure) hamiltonian of the form 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: (R.10) (R.9) A free radiation term (photons) of the form (R.8)

23. Physics and Astronomy The matrix element W fi II CMPOther 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 states and one photon in state k L L. The final state consists of the electronic system in the ground state, one photon in state k S S, and one phonon in state qm, so that we can express these states as Therefore the quantum states of the system are characterized by three sets of occupation numbers: (R.12) (R.11)

24. Physics and Astronomy The matrix element W fi III CMPOther 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 (R.14) (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: 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)

25. Physics and Astronomy The matrix element W fi IV CMPOther 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 con- duction band and a singly fully occupied valence band. We assume parabolic bands.

26. Physics and Astronomy The matrix element W fi V CMPOther 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  L is annihilated, a phonon is created, and a photon of frequency  S is created. We then rewrite the electron-phonon hamiltonian as (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.

27. Physics and Astronomy The matrix element W fi VI CMPOther 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 (R.17) (R.16) with: 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.18)

28. Physics and Astronomy The matrix element W fi VII CMPOther Since each application of H eR can change one photon state (the interaction is linear in the pho- ton creation and annihilation operators) and each application of H eL can only change one pho- non 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 : (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.

29. Physics and Astronomy The matrix element W fi VIII CMPOther Maximum resonant enhancement is then obtained if the energy E  is comparable to the energy E. 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 (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 , but this gives trivial factors of 2). There are two types of terms that yield nonvanishing contributions:

30. Physics and Astronomy The matrix element W fi IX CMPOther CONDUCTION BAND TERMS These terms contain matrix elements of the form where |0> refers to the electronic ground state, and n(  ) is the phonon occupation number at temperature T. We assume that there is one incident photon and one scattered photon, so the applica- tion of the photon creation/annihilation operators gives just 1. 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 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.) (R.21) (R.22)

31. Physics and Astronomy The matrix element W fi X CMPOther 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 where E k is the energy of the intermediate state given by 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 (R.23) (R.24)

32. Physics and Astronomy The matrix element W fi XI CMPOther where we have defined. The last factor can be converted into an integral: (R.25) (R.26)

33. Physics and Astronomy The matrix element W fi XII Other where the first factor of 2 takes into account the spin degeneracy. Using R.24 Hence, using R.27 we obtain where we have added a phenomenological broadening parameter that represents the lifetime of the excited states. For a one-dimensional solid: (R.27) (R.28) (R.29)

34. Physics and Astronomy The matrix element W fi XIII Other 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 We thus need to calculate the following integral (R.30) (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.

35. Physics and Astronomy Raman cross section: quantum theory IV with so that the cross section finally becomes (R.32) (R.33)

36. Physics and Astronomy Raman cross section: quantum theory V Other 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 (  L /  S )] If no absolute cross section is measured, only the function F(  L ) is needed to analyze the experimental results, assuming that the approximation of a single conduction and a single valence band is valid. This is the case for armchair tubes. From fits of F(  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 (R.34)

37. Physics and Astronomy Raman cross section: quantum theory VI Other 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.

38. Physics and Astronomy Raman cross section: quantum theory VII Other A more realistic value for  in carbon nanotubes is  = 0.01. As the broadening of the electronic states increases, the intensity profile evolves to a single peak

39. Physics and Astronomy M. Canonico et al., Phys. Rev. B 65, 201402 (2002)

40. Physics and Astronomy Trigonal warping in carbon nanotubes I Other The figure shows the 1D density of states for se- veral metallic nanotubes of approximately the same diameter, showing the splitting of singularities (in- creasing 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 cal- culation of Raman cross sections. From R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Phys. Rev. B 61, 2981 (2000)

41. Physics and Astronomy Trigonal warping in carbon nanotubes II Other Because the two transitions can interfere, it is critical to know the sign and magnitude of the prefactor 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. 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 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 eigen- vector. (R.36) (R.37)

42. Physics and Astronomy Trigonal warping in carbon nanotubes III Other where l is the cell index,  the basis index,  a cartesian index, and M(  ) the mass of atom  within the unit cell. For a periodic solid, where N c is the number of unit cells and e is a unit vector normalized to unity: The normal coordinate can be written in terms of the mode eigenvectors  and the atomic displacements u as (R.38) (R.39) (R.40)

43. Physics and Astronomy Trigonal warping in carbon nanotubes IV Other where R is the nanotube radius, and we assume q RBM = 0, the corresponding change in the normal coordinate is 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 If we give all atoms a displacement (R.41) (R.42) (R.43)

44. Physics and Astronomy Trigonal warping in carbon nanotubes V Other where we have invoked the Hellman-Feynman theorem. By comparing R37 with R.15, and using R.41, we can write We therefore need matrix elements of the form (R.44) (R.45) 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.

45. Physics and Astronomy Trigonal warping in carbon nanotubes VII Other

46. Physics and Astronomy Trigonal warping in carbon nanotubes VIII Other

47. Physics and Astronomy Trigonal warping in carbon nanotubes IX Other

48. Physics and Astronomy Trigonal warping in carbon nanotubes X Other 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.

49. Physics and Astronomy

50. Physics and Astronomy 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.