Collective Modes and Electronic Raman Scattering in the Cuprates

2 Pages·0.07 MB·English

by  F. Venturini

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0 Collective Modes and Electronic Raman Scattering in the Cuprates
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n F. Venturini1, U. Michelucci2, T.P. Devereaux3, and A.P. Kampf2
a 1Walther-Meissner-Institut, Walther-Meissner-Str. 8, 85748 Garching, Germany
J 2Theoretische Physik III, Universit¨at Augsburg, 86135 Augsburg, Germany
9 3Department of Physics, University of Waterloo, Waterloo, Canada, N2L 3G1
1

] WhilethelowfrequencyelectronicRamanresponseinthesuperconductingstateofthecupratescanbelargely
n understood in terms of a d-wave energy gap, a long standing problem has been an explanation for the spectra
o observed in A1g polarization orientations. We present calculations which suggest that the peak position of the
-c observed A1g spectra is dueto a collective spin fluctuation mode.
r
p
u
s 1. INTRODUCTION kz can be 0 or π, for bonding or anti-bonding
. bands of the bilayer, respectively.
t
a In spite of the considerable efforts to explain
The spin susceptibility (χ ) is modeled by ex-
m the experimental Raman spectra of cuprate su- s
tendingtheweakcouplingformofadx2−y2 super-
- perconductors,theA1g superconductingresponse conductor to include antiferromagnetic spin fluc-
d
is not yet completely understood. It has been
n tuations by an RPA form with an effective in-
o shown that the theoretical description of the teraction U¯; i.e. χs = χ0/(1−U¯χ0) where χ0
c A1g Raman response was very sensitive to small is the simple bubble in the d-wave state. This
[ changes in the Raman vertex harmonic represen-
formofthespinsusceptibilityismotivatedbythe
tations, yielding peak positions varying between
1 fact that it contains a strong magnetic resonance
v ∆ and 2∆ [1]. However, the data show peaks peakatq=Q=(π,π,π)whichwasproposed[4]
7 consistentlyslightlyabove∆forbothYBCOand
to explain the INS resonance at energies near 41
6 BSCCO.
meV in YBCO [2] and BSCCO [3].
2
Inthispaperwepresentcalculationssuggesting
1 The Ramanresponsefunctioninthe supercon-
0 thattheA1g peakpositionislargelycontrolledby ducting state is evaluated using Nambu Green’s
0 a collective spin fluctuation (SF) mode near 41 functions. The spin fluctuations contribute to
0 meV,consistentwith inelastic neutronscattering
the Raman response via a 2-magnon process as
t/ (INS) observations [2,3]. We show that the A1g shown in Fig. 1 [5] where a schematic represen-
a responseisstronglymodifiedbytheSFtermand
m tation of the Feynman diagrams of the SF and
is not sensitive to small changes in the Raman
the bubble contribution is plotted. For the elec-
- vertex. The experimental peak position is well
d tronic propagators we have used the bare BCS
n reproduced by our model whereas the B1g and Green’s functions and a d-wave superconducting
o B2g responseremainessentiallyunaffectedbythe gap ∆k =∆0[cos(kx)−cos(ky)]/2.
c SF mode.
The total Raman response is calculated in the
:
v gauge invariant form which results from taking
i
X into account the long wavelength fluctuations of
2. MODEL CALCULATION the order parameter [1]. The total Raman sus-
r
a ceptibility is thus given by
The CuO2 bilayer is modeled by a tight bind-
ing band structure with a nearest (t) and a next χ (q=0,iΩ)=χ (0,iΩ)− χ2γ1(0,iΩ) (2)
nearest neighbor hopping (t′) parameter and an tot γγ χ11(0,iΩ)
inter-plane hopping given by [4]
where χ (q=0,iΩ) is determined according to
ab
Fig. 1. The analytical continuation to the real
2
t⊥(k)=2t⊥cos(kz)[cos(kx)−cos(ky)] . (1) axis is performed using Pad´e approximants.
2
χ 12
= +
ab a b a b
10 B
1g
u.) 8
a.
Figure 1. Feynman diagram considered for the Ω) ( 6
pglayrtainclde-shoolliedalinndesSrFepcroensternitbupthiootnosn.,DSaFshaendd,ewleigc-- χ’’ ( 4 A1g
tronic propagators, respectively. The solid circle
2
marks the coupling U¯ for the electron-SF vertex.
0
0 100 200 300 400 500 600 700
Ω (cm-1)
We have used several different forms for the
Raman vertex γ which possess the correct trans- Figure 2. Comparison of the A1g and B1g total
response with Bi-2212 data taken from Ref. [6].
formation properties required by symmetry. Our
calculations show that the SF term yields van- The values of the parameters are ∆0/t = 0.25,
ishingly small corrections to the response in the t′/t = 0.45, t⊥/t = 0.1, hni = 0.85, U¯/t = 1.3,
t=130 meV.
B1g and B2g channels, but contributes substan-
tially to the A1g channel. The shape of the to-
tal response in the A1g geometry is mainly de-
pendent on the value of the effective interaction From this workwe conclude that including the
U¯. Variations of U¯ change the relative magni- SFcontributionintheRamanresponsesolvesthe
tude of the two diagrams summed in Fig. 1, previously unexplained sensitivity of the A1g re-
changing the position of the peak in A1g geome- sponse to small changes in the Raman vertex.
try. Importantly, we find that the A1g response Whereas the SF (two-magnon) contribution con-
shows little dependence on the form used for the trols the A1g peak, the B1g and B2g scattering
vertex: cos(kx)+cos(ky),cos(kx)cos(ky), or the geometries are essentially unaffected and deter-
vertexcalculatedinaneffective massapproxima- mined by the bare bubble alone.
tion. Theseresultscanbeexplainedbysymmetry We would like to thank R. Hackl for numer-
reasons given that the SF propagator is strongly ous discussions. One ofthe authors (F.V.) would
peaked for Q momentum transfers. liketothanktheGottliebDaimlerandKarlBenz
Foundation for financial support.
3. COMPARISON WITH DATA
REFERENCES
We compare the calculated Raman response
1. T.P.Devereauxetal.,Phys.Rev.B51,16336
with the experimental spectra of an optimally
(1995); ibid. 54, 12523 (1996).
doped Bi-2212 sample [6] in Fig. 2. Adding the
2. H.F. Fong et al., Phys. Rev. Lett. 75, 316
SF contribution leads to a shift of the peak posi-
tionfromnear∼∆0 forU¯ =0tohigherfrequen- (1995); Nature 398, 588 (1999).
3. H.A. Mook et al., Phys. Rev. Lett. 70, 3490
cies, allowing a better agreementwith the exper-
(1993).
imentalrelativepositionsofthe peaksinA1g and 4. N.Bulutet al.,Phys.Rev.B53,5149(1996).
B1g geometries. For the fit we have adjusted t to
5. A.P.KampfandW.Brenig,Z.Phys.B-Con-
achieve a good agreement with the B1g channel,
obtaining t = 130 meV, and then adjusted U¯ to densed Matter 89, 313 (1992).
6. T.P. Devereaux et al., Phys. Rev. Lett. 72,
match both the A1g peak position as well as the
3291 (1994).
peak in the SF propagator to be consistent with
the INS peak at 41 meV.