Electronic Excitations and Phonon Renormalization Effects - American

Chapter 13

Raman Scattering from the Superconducting Phase: Electronic Excitations and Phonon Renormalization Effects 1

1

1

2

2

V. G . Hadjiev , T. Strohm , M . Cardona , Z. L . Du , Y . Y. Xue , and C. W. Chu 2

Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: September 2, 1999 | doi: 10.1021/bk-1999-0730.ch013

1

Max-Planck-Institut Für Festkörperforschung, Heisenbergstrosse 1, D-70569 Stuttgart, Germany Department of Physics and the Texas Center for Superconductivity, University of Houston, Houston, T X 77204-5932

2

The Raman scattering from (Cu,C)Ba Ca Cu O reveals strong superconductivity-related effects. In the Raman spectra of this compound we observed most of the characteristic changes expected in going from the normal to the superconducting state. These are a redistribution of the low frequency electronic excitations, the appearance of a Raman peak due to the scattering via electronic pair breaking, and especially, strong superconductivity-induced phonon self-energy effects. The A phonons at 235 cm and 360 cm , which involve the vibrations of the plane oxygen with some admixture of Ca displacements, exhibit a strong renormalization below T . Particularly remarkable is the phonon intensity increase of up to two orders of magnitude when the material becomes superconducting. 2

3

4

x

-1

-1

1g

c

Raman spectroscopy is known to be a powerful technique for the characterization and basic studies of high T superconductors (HTCS, see, e.g., (/)). The low-energy Raman response of these materials is mainly due to the excitation of electrons occupying states near the Fermi surface (FS) and to the scattering of light by optical phonons which proceeds via electron-phonon interactions. The opening of the superconducting gap results in a redistribution of electronic states and excitations in the immediate vicinity of the FS. This leads to the appearance of an electronic Raman peak below T , caused by quasiparticle creation through pair-breaking, and to phonon renormalization by the superconductivity-induced phonon self-energy. Information on the anisotropy of the superconducting order parameter (gap) can be inferred from the electronic peak that develops below T (2-4), provided the peak is discernible (5). The asymptotic behavior of this peak for ω -» 0 also contains symmetry information (5, 4). On the other hand, those phonons that are strongly coupled to electrons c

c

c

180

© 1999 American Chemical Society

In Spectroscopy of Superconducting Materials; Faulques, E.; ACS Symposium Series; American Chemical Society: Washington, DC, 1999.

181 occupying states near the FS can be very sensitive to the opening of the superconducting gap. This results in a change of the phonon renormalization (selfenergy) induced by the superconducting transition (6-8). The real part of this self energy represents the contribution of the electron-phonon interaction to the phonon frequency and its imaginary part the contribution to its linewidth. For conventional superconductors this self-energy effect is small and usually gives a change of a few tenths of a wavenumber (9). In HTCS, the phonon self-energy effects below T may become sizable (~ 10 cm" ) as is well known since the early measurements of Y B a C u 0 ^ (10). Coupling of the phonons to the pair breaking excitations leads also to admixture of the corresponding Raman transition amplitudes that results in changes in the phonon intensities when the temperature is lowered below T (11,12). We illustrate here the superconductivity-induced effects in the Raman spectra of HTCS as exemplified by a recent study (13) of (Cu,C)Ba Ca Cu O ((Cu,C)-1234). This compound exhibits a remarkably strong modification of its Raman spectra below T (13) similarly to H g B a ^ C u A UU2). c

1


2

3

7

c

2

3

4

x

c

Raman scattering from solids (theory) Electronic Raman scattering efficiency. In a Raman experiment one often measures the scattering intensity I , which is given by the energy flux of electromagnetic (light) wave scattered into the solid angle Ω subtended by the spectrometer within a frequency interval m + Sm . In cases when a multichannel detector is used, δω is given by the frequency interval dispersed over one or several pixels of the detector. The Raman intensity can also be defined in terms of the number of scattered photons. In this case it is proportional to the scattering cross section d a/dCldco and the generalized dynamic structure factor of the sample S (q,œ) at a temperature Τ for a given Raman shift ω and a wavevector transfer q s

0

s

s

5

2

s

T

'

d

'°

s

Here ω

η

ω , and ω = m - ω 8

t

Β

·„ ^(ς,α>).

(1)

2

dCldco

ω

are the frequencies of the incident photons, the 2

2

scattered photons, and the excitations in the solid, respectively, and r = e /mc is the Thomson radius. The Raman scattering efficiency, given by the scattering cross 0

2

section determined for a scattering volume equal to unity, {\/V )d a/dQdœ , represents the efficiency per unit pathlength in the sample. Both quantities, the scattering efficiency and the cross section are normalized to the incoming laser flux. When the scattering process is instantaneous, one can define a linear-response function, z (q,t), called the Raman susceptibility. The Fourier transform of the Raman susceptibility, x (q,co), is related to the structure factor S (q,co) through the fluctuation-dissipation theorem (2,4): scai

s

R

T

R

T

S (q û)) = - - ( 1 + η )ΐπιχ (^ω), y

ωΤ

Η


(2)

182 1

where η


ωΤ

= [expfreo/kf) - lj" is the Bose-Einstein thermal occupation factor (Stokes

scattering is assumed). Thus a Raman response from a given material is related to the imaginary part of the Raman susceptibility of those excitations that take part in the inelastic scattering of light. Theoretical models for the Raman response are usually based on calculations of the Raman susceptibility and related vertices for the scattering processes. The Raman scattering is a two-photon process. The Hamiltonian of the system (electrons and photons) contains terms that correspond to two types of scattering mechanisms. The "density-like" term, H , is quadratic in the vector-potential of the light, H M ~ A . It describes non-resonant scattering processes that involve only intraband transitions. The second, "current-like", term H , is linear in A , H ~Ap. This term can lead to resonances and it also accounts for virtual intraband and interband transitions. In order to describe the Raman scattering amplitude, the HM has to be treated in first-order and the term H in second order perturbation theory. Both terms can be combined in the effective Hamiltonian (2,4): M

2

A

A

A

ηM in which the operators A$ and A contain the creation operator of the scattered photons and the annihilation operator of the incident photons, respectively. In equation 3, c* and c are the creation and annihilation operators for Bloch electrons in the band n that is, the sum of their products gives the density of electronic excitations for transitions of electrons from states with a wavevector k to states with k + q, where q = k -k is the wavevector transferred from the light to the sample. The Raman vertex r (q) is a weighting factor in equation 3. It involves all possible intermediate states and energy denominators and therefore can lead to resonance effects. With some simplifications, which seem to agree with available experimental data, one can apply to the Raman vertex calculations the well known effective-mass approximation (see, e.g., (4)). This approximation is valid when (i) the contribution of virtual intraband transitions is negligible with respect to that of the interband transition; (ii) the relevant virtual interband transitions have much larger energy than the incoming and scattered photon energies and one can neglect the terms in the denominators of the Raman vertex that contain ω and ω . In this case the expression for / ( q ) becomes identical to that for the inverse effective mass in the k · ρ theory and in the limit q->0, which is relevant for the experiments, we can write: l

Mq

nk

9

7

s

nk

ι

8

nk


183 that is, the Raman vertex is given by the inverse effective mass tensor contracted with the polarization vectors e and e of the incoming and scattered light, respectively. Equation 4 is particularly useful i f the bandstructure is known or can be calculated within ab initio local-density approximation (LDA) techniques such as the linear muffin-tin orbital (LMTO) method (14,15). Once the Raman vertex is determined, the Raman susceptibility is given by a polarization bubble containing two Raman vertices. Taking into account equation 3, we can express the Raman susceptibility as Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: September 2, 1999 | doi: 10.1021/bk-1999-0730.ch013

7

s

2

% (q-+0,m) = (y n (m)) , R

k

k

(5)

Bz

that is, to calculate the scattering efficiency, which is ~ Im ^ , the Raman vertex has to be squared, multiplied by the k dependent electronic polarizability of the system n (û>) and the result averaged over the Brillouin zone. In equation 5 we omitted the band index η for simplicity. In "clean" metals, that is, in metallic systems with long mean free paths Lk »l, the normal state electronic scattering is dipole forbidden because final states for vertical transitions (q->0) and small energies are not available in the "clean" limit. Usually, various types of defects in the material can induce the Raman scattering (collision limited regime) (16). We note here that there is no satisfactory ab initio theory for the Raman scattering in the normal state of HTCS so far. In the superconducting state, a gap opens in the quasiparticle spectrum and the intraband scattering becomes allowed for q -> 0 even in the "clean" limit (note that ir-absorption remains forbidden in this limit). The electronic Raman scattering for ω below 2Δ corresponds to a photon induced breaking of Cooper pairs into Bogoliubov quasiparticles (bogolons). The Raman susceptibility in the superconducting phase is given by a polarization bubble with two Raman vertices, one of which is renormalized, and propagators for bogolons (2). The vertex renormalization must include the attractive pairing interactions and the repulsive Coulomb interaction. However, for magnitudes of the wavevector transfer q small compared to both the inverse coherence length, ξ~\ and the Fermi wavevector k , the vertex corrections due to the pairing interaction can be neglected. Thus for q « ξ~ ^ the effective kernel, corresponding to U (m) in equation 5 is the Tsuneto function (77), \{ω), which contains the temperature dependent gap function A k

F

0

F

ι

Ρ

k

k

4

( ^

=

^tanhi-^-li

2

2

In equation 6, E = {s -s ) k

k

F

!

+

1

) .

(6)

2

+A gives the quasiparticle energy dispersion, e k

F

is

the Fermi energy. The Tsuneto function has poles at (B ® A = #2 )^ = > ^ 2g

lg

g

z

Raman response should be affected

by the screening. This fact is most important for the HTCS with a single-sheet FS (one C u 0 plane per unit cell). In these compounds there are only intraband mass 2


185 fluctuations which, presumably, are not strongly k dependent and the scattering related to the average mass ( A ) is completely screened (25). In multilayer systems, like Hg-1234 and (Cu,C)-1234 discussed here, interband fluctuations between the different sheets of the FS become important. These fluctuations give considerable amount of unscreened A scattering (4,25). lg

]g

(iv) The expression for the Raman suceptibility simplifies when averages over the FS instead of the B Z are performed. In this case, the unscreened part of ZR equation 7, at Τ = 0 Κ , can be written as Downloaded by PENNSYLVANIA STATE UNIV on September 13, 2012 | http://pubs.acs.org Publication Date: September 2, 1999 | doi: 10.1021/bk-1999-0730.ch013

M

ΙτηχΙ(ω)

(8) 2

ω^ω

- 4|A |

2

k

where Θ is the step function. This approximation is valid when y and A do not change significantly for small deviations of k perpendicular to the FS. From equation 8 it follows that for a simple isotropic s-gap, A = Δ = const., Imχ (co) = 0 for k

k

k

0

R

ω < 2 Δ , whereas for the d _ gap (it has nodes !), or for one of \d _ i\ type, the 0

xl

yl

xJ

y

scattering continuum extends to zero Raman shift in absence of impurities. We conclude this paragraph with the theoretical predictions for the low frequency behavior of the electronic Raman scattering in superconductors. In clean tetragonal superconductors with a d gap, which seems to be the relevant gap for most x2y2

3

HTCS, the scattering efficiency varies with frequency as ω for the B and as ω for lg

A

lg

and B spectra (3,4). The steeper dependence of the B scattering on frequency 2g

Xg

comes from the fact that the nodes of both the Raman vertex and the gap function coincides. For HTCS with orthorhombic symmetry one must distinguish two cases: that of YBajCujO?, in which the orthorhombicity is determined by the chains along (010), and that of Bi Sr CaCu 0 in which it is determined by a distortion (a 2

2

2

8

superstructure) along (110). In the Y B a C u 0 case the tetragonal B symmetry is 2

3

transformed into A (in the corresponding D g

7

2 h

lg

point group) and a linear appears in

the ω -dependence of the B scattering, while in Bi Sr CaCu 0 the tetragonal B ]g

symmetry remains B of the D lg

2

2

2

8

lg

3

2 h

point group and the ) should be affected by the symmetry (of the phonon!) in that it should be screened for fully symmetric (A or A ) phonons as is the case of the corresponding (A , A ) electronic scattering. Similarly to the electronic Raman scattering, for multilayer superconductors, the screening should usually be reduced. This is supported by the experimental data presented below. κ

k

k


k

ρ

p

5

5

g

lg

g

Xg

Fano effect. It turns out that in most HTCS, the electronic excitations responsible for the phonon renormalization (the phonon self-energy) are also Raman-active and contribute to the electronic Raman scattering. In this case, a quantum interference between the scattering amplitudes of the phonon (a discrete excitation) and the electronic continuum results in an asymmetric phonon lineshape. The spectral distribution of the phonon Raman intensity is given by the well known Fano expression, which is relatively simple for the case of isotropic vertices (23). The corresponding formula that accounts for k -space anisotropy of the vertices, although more realistic for HTCS, is more complicated (Belitsky V . L; Strohm T.; unpublished), and has not yet been used for quantitative analysis of data. Since we use the Fano formula for fitting the experimental data, we briefly present the physical quantities that are involved. For isotropic vertices y = T and g = V we write for the Raman susceptibility k

e

k

(yln (œ)) =T^[R( )-i^)], k

gz

a

and for the phonon self-energy Σ(ω) in the limit q->0


(9)

187 (10)

2

Σ(ω) = (άϊΙ,(ω))

=V {R(a>)-Up(a>)],

Βζ

here ρ(ω) is the density of electronic excitations and R(a>) its Hubert transform (23). We write the simplest form of the Fano expression for the Raman efficiency in terms of real and isotropic vertices (microscopic parameters)


(11) 2

2

ω - ω

Electronic Excitations and Phonon Renormalization Effects - American

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