eiji kaneshita, masanori ichioka and kazushige machida- phonon anomalies due to stripe collective...

8/3/2019 Eiji Kaneshita, Masanori Ichioka and Kazushige Machida- Phonon anomalies due to stripe collective modes in high T

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rXiv:cond-mat/010

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Phonon anomalies due to stripe collective modes in high Tc cuprates

Eiji Kaneshita, Masanori Ichioka and Kazushige MachidaDepartment of Physics, Okayama University, Okayama 700-8530, Japan

(Dated: February 6, 2008)

Phonon anomalies observed in various high Tc cuprates by neutron experiments are analyzedtheoretically in terms of the stripe concept. The phonon self-energy correction is evaluated bytaking into account the charge collective modes of stripes, giving rise to dispersion gap, or kink and

shadow phonon modes at twice the wave number of spin stripe. These features coincide preciselywith observations. The gapped branches of the phonon are found to be in-phase and out-of-phaseoscillations relative to the charge collective mode.

The pairing mechanism of high Tc superconductorshas been still highly debated since its discovery over adecade ago. The debates are mainly centered on howto describe the normal state above its superconductingtransition Tc. Recently, a remarkable series of neutronexperiments on various cuprates has been done to re-veal unequivocally the phonon dispersion anomalies atthe particular reciprocal point. The phonon anomalies

differ slightly in their appearance, ranging from a sharpdispersion jump to the dispersion kink. This dependson the materials; La2xSrxCuO4 (LSCO)[1, 2, 3] orYBa2Cu3O6+x(YBCO)[4, 5, 6, 7] or the focused phononmodes such as breathing mode, or bond-stretching mode,etc. This ubiquitous phenomenon for which proper ex-planation has not been given so far is interesting becausenot only it gives a clue to pairing mechanism in high Tccuprates, but more importantly it gives a key to betterunderstanding the normal state properties with full ofmysteries.

It is known now that in underdoping La2xSrxCuO4the incommensurate static magnetic peaks at Q are ob-

served around (, ) point in two dimensional CuO2plane (here the lattice constant a is unity) by neutronexperiments[8]. As doping proceeds with x, the order-ing vector Q rotates precisely at the insulator-metaltransition (x = 0.05) by 45 around (, ) point, cor-responding to spin modulation change from the diago-nal stripe Q = (2(12 ), 2(

12 )) to the vertical

stripe Q = (2(12 ), 0) or (0, 2(12 )) where x.

In higher doping region the incommesurability almostsaturates at a particular value 18 . The same 45

rotation of the magnetic superspots is also observed[9]in (La, Nd)2xSrxCuO4, corresponding to the metal-insulator transition, further lending this superspot ro-

tation to the universal mechanism to drive a metal-insulator transition upon doping. In this material[10] andLa2x(Sr,Ba)xCuO4[11] the concomitant static chargesuperspots at 2Q are found by neutron scatterings.

These characteristics coincide nicely with the incom-mensurate spin density wave or stripe picture[12, 13]where a solitonic midgap state accommodates excessholes completely (partially) in an insulating (metallic)state, giving rise to a microscopic charge and spin seg-regation characteristic to topological doping. The par-

tially occupied midgap state is observed directly by re-cent angle-resolved photoemission experiment[14] near(, 0) point as a parallel Fermi surface sheet which isresponsible to metallicity of doped CuO2 plane[15].

As for YBCO, although experimental verification oftrue static magnetic order comes only from SR[16]which reports a similar phase diagram to LSCO in dop-ing vs. static magnetic ordering, there are several com-

mon characteristics mentioned above[17], including neu-tron scatterings[18]. Thus it is expected that the stripepicture may be applicable to YBCO[19]. Numerous otherexperiments such as wipe-out effect for charge order byNMR[20], detailed transport measurements[21], etc. andtheories[22], collectively point to support the idea thatthe stripe concept may be applicable to doped CuO2plane in general.

Here we are going to demonstrate that the phononanomalies observed in various cuprates are a direct man-ifestation of existence of the low-lying collective chargeexcitations associated with stripes. Physically, the -phase shifting domain walls accommodate excess holes

and the remaining antiferromagnetic regions are intact.The domain wall motion, or the charge collective oscil-lation, is strongly coupled with the particular phononmode, such as breathing or bond-stretching modes. Bothelectron and phonon systems are mutually influenced bythe others to lower the total energy. The spin and chargeinhomogeneity is a universal phenomenon in doped CuO2plane, out of which high Tc superconductivity is created.This process is still beyond our understanding at the mo-ment, but this identification, we believe, is important to-wards this step. This is particular true for YBCO wherethere is no defining experiment to uncover the chargecollective modes which has been elusive so far. Since

the stripe concept, although originally derived from meanfield approximation[12, 13], may be valid for describinga doped Mott insulator, or at least, the stripe is a conve-nient vehicle to accommodate excess holes, it is worth-while to pursue this identification.

We start with the Hubbard model H= i,j,ti,jci,cj,

+Uinini to describe a stable stripe and its ex-citations. The mean-field ground state[12, 15] andrandom phase approximation (RPA)[23] yield a richstructure for individual and collective excitation spec-
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(0,0) (0,)2Q

0.1

0.2

(-/2,/2) (/2,/2)2Q

(a) (b)

FIG. 1: Dispersion curves of the charge collective modearound 2Q. We plot Imnn(q, ) for two paths along 2Q(a) and perpendicular to 2Q (b).

tra of spin and charge channels; namely the dynamicaltransverse (longitudinal) susceptibility xx(zz)(q, ) =Sx(z);Sx(z)q,, and the dynamical charge suscep-tibility nn(q, ) = n;nq,. The RPA equationfor SS = S;S is written as SS(r1, r3, )

=

0 (r1, r3, ) +U

r2

0 (r1, r2, ) SS(r2, r3, )with non-interacting susceptibility 0 . After Fouriertransformation to k-space, this is reduced to a matrixequation of 0 and SS . By solving it, we obtainxx(q + l1Q,q + l2Q, ). There, q + l1Q and q + l2Qare, respectively, outgoing and incoming wave vectors.They can change by (l1 l2)Q through the Umklappprocess, since the underlying electronic state is spatiallymodulated with the periodicity Q. In a similar man-ner, we calculate n;n and n;n, and obtainzz(q + l1Q,q + l2Q, ) and nn(q + l1Q,q+ l2Q, ).

The spin modes xx(zz)(q, ) xx(zz)(q,q, ) giverise to collective excitation at the ordering vector Qand its odd-harmonic points as a Goldstone mode. Thecharge mode nn(q, ) nn(q,q, ) exhibits the collec-tive excitations at q = 2Q, 4Q . There, xx gives spinwave, when we assume that the magnetization points tothe z axis. The translational phason modes appear innn (zz), which correspond to the motion of the charge(spin) stripe[23]. Among various individual and collectivemotions associated with stripes, the phason mode in thecharge oscillation is directly coupled to the phonons andmost relevant to the phonon renormalization. In Fig.1,Imnn(q, ) is shown around q = 2Q = (0,

2 ) along the

qy- and the qx-directions. This particular charge mode

with the anisotropic velocities is situated at 2Q. Themotion parallel (perpendicular) to 2Q vector correspondsto the compression (meandering) motion of stripes withhigh (low) velocity. In the following we illustrate typ-ical results for the metallic vertical stripe [15] which isstable with Q = (, 34 ) for the electron filling per site

n = 0.82 and the parameters are fixed as Ut

= 4.0 andt

t= 0.2 (t(t) is the (next) nearest neighbor hopping

integral). The main features of the results do not muchdepend on the choice of these values. We note that t is

(0,0) (0,)2Q (0,0) (0,)2Q

0.06

0.08

0.1(a) (b)

FIG. 2: Renormalized phonon spectral functions for |g|2 =0.001. (a) is logarithmic plot to emphasize the small inten-sity of the shadow bands. (b) is contour plot shown for highintensity. The simple sinusoidal form is assumed for the un-perturbed phonon (A = 0.01, B = 0.08, k = 0)[24].

(0,0) (0,)2Q

0

0.1

0.05 1 1

3

45

2

FIG. 3: Schematic figure for relevant modes. 1: Charge col-lective mode. 2: Unperturbed phonon. 3,4, and 5: Phononsshifted by 2Q, 4Q and 6Q via Umklapp processes.

an order of 0.5eV and the phonon energy 80meVfor the breathing mode or bond-stretching mode whichis coupled strongly to the electronic state. We focus on

an energy range of typically

t 0.1 in the following. Allthe energy is scaled by t.In order to study the effects of the stripes on the

phonon Greens function D(q+ l1Q,q+ l2Q, in) withinRPA, we consider the following standard electron-phononHamiltonian: H = k,q,gc

k+q,ck,(bq + b

q) with g

being the coupling constant. Since the electron bubblein the phonon self-energy correction is nothing but thedynamical charge susceptibility mentioned, we immedi-ately obtain an equation in a matrix form for the thermalGreens functions m{lm|g|2D0(q+ lQ, in)nn(q+lQ,q +mQ, in)}D(q +mQ,q +mQ, in) = D0(q +lQ, in)lm, where the unperturbed Greens function

D0(q, in) = 2q/(2n+2q), and q is the unperturbedphonon dispersion. Diagonalization of this matrix equa-tion yields the renormalized phonon Greens function.

We show the renormalized phonon spectral function 1

ImD(q,q, in + i) in Fig.2, where we assume

a simple sinusoidal form[24] for q as a typical phonon.It is seen from Fig.2 that the dispersion exhibits severalcharacteristic modifications: (A) The discontinuities ofthe dispersion occur at various points, notably near 2Qwhere the dispersion cone of the collective charge mode


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FIG. 4: Schematic figure for two in-phase (lower energy) andout of phase (higher energy) oscillation patterns correspond-

ing to the two branches above and below the gap at 2Q. Thevertical bars show the charge domain walls of the stripes.

shown in Fig.1 and the phonon mode intersect. (B) Theappearance of the downward hybridized new phonon dis-persions. (C) The flat dispersion appears above the gapat 2Q. (D) The shadow band which is a replica of theoriginal mode is seen.

These features come from the two physical reasonsschematically illustrated. in Fig.3: The hybridization ofthe charge collective mode centered at 2Q denoted as 1with the original phonon 2 in Fig.3 and the band-folding

due to the periodic structure with the charge orderingvector 2Q, which amounts to shifting the original dis-persion 2, resulting in phonon dispersions; curve 3 5 inFig.3. The hybridization of these dispersions gives riseto the reconnected phonon modes in Fig.2.

These main four features (A-D) of the dispersion mod-ification become more evident when the electron-phononcoupling |g|2 increases, or the unperturbed dispersion issituated at lower energy side. The gap size at the discon-tinuity increases as the original phonon band is widen. Itis noteworthy that the discontinuity of the dispersion, ora kink feature and associated new lower mode branching-

out from the disconnected lower phonon are reminiscentof the observation on O6.35 and O6.60 in YBCO by Mookand Dogan[4] (see their Figs.1(d) and (e)). Note also thatsince the discontinuity point occurs at the intersection ofthe charge collective mode at 2Q and phonons, the ob-served dispersion anomaly should change with doping; Asdoping increases the 2Q vector increases. This is indeedobserved by neutron experiment[6] and also consistentwith the doping dependence of the fundamental vectorQ of the magnetic peaks[18]. These constitute an inter-nal consistent picture for the stripe origin.

The physical implication of the phonon gap formationis understood by analyzing the phonon oscillation pat-

terns of the upper and lower phonon branches. Accord-ing to linear response theory, the oscillation patterns aregiven by (r, t) = l

1hD(q + lQ,q, )eilQrhqeiqrit

where (r, t) is the lattice displacement and hq is the ex-ternal field with the wave number q and the frequency .At the wave number 2Q = (0, 2 ), the oscillations corre-sponding to the upper ( = 0.08) and lower ( = 0.075)branches shown in Fig.2 are schematically displayed inFig.4. The four site period oscillations are either in phaseor out of phase with the charge oscillation in the electron

(0,0) (0,)2Q

0.06

0.08

0.1

(-/2,/2) (/2,/2)

(c) (d)

(a) (b)

0.06

0.08

0.1

2Q

FIG. 5: Renormalized phonon spectral functions for twoperpendicular paths. |g|2 = 0.001. (a) and (c) (0, qy) for0 qy . (b) and (d) (qx,

2) for |qx|

2. The unper-

turbed phonon with a steep slope (A = 0.01, B = 0.08, k =0.9999)[24] is assumed. (a) and (b) are logarithmic plots. (c)and (d) are contour plots.

system which is the phason motion of the stripes.

We show the renormalized dispersions parallel to 2Qin Figs.5(a) and (c) and perpendicular to 2Q centeredat 2Q in Figs.5(b) and (d). Here we assume the unper-turbed phonon with a steep slope at (0, 2 ), simulatingthe breathing mode in LSCO with the energy 80meV.Noticeable features of the modification are: (A) Ratherlarge dispersion discontinuity observed at 2Q is seen inFigs.5(a) and (c). (B) It is noted that the shadow modewhich is a replica of the original band can be seen inFigs.5(a) and (b). (C) The anomaly continues to ex-tend to the perpendicular direction seen from Figs.5(b)and (d). These features (A-C) coincide with the neutronexperiment[1] on LSCO with x = 0.15 where at (2 , 0) inthe (qx, 0) scan they observe a sharp drop of the bond-stretching mode (see their Fig.2) and this anomalous be-havior continues to the perpendicular direction (see theirFig.4). As they emphasize, the phonon anomaly occursnot because of the simple band-folding effect due to newcharge periodicity with 4 lattice period which would re-

sult in a gap at the new Brillouin zone boundary at (

4 , 0),but the observed gap is at ( 2 , 0).

The anomalous reciprocal regions for the two perpen-dicular directions have different widths as seen from Fig.5because the underlying charge cone is anisotropic as men-tioned, yielding different widths (compare the openingsof the cones at the center in Figs.1(a) and (b) correspond-ing to the cones in Figs.5(a) and (b)). Thus it is possibleto use it as a kind of spectroscopy, namely, to investigatethis anisotropy of the charge collective mode through the


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analysis of the phonon anomaly. In fact according to theabove experiment[1] the width of the anomalous regionin the perpendicular direction is wider than that in theparallel direction. This accords with our result in Fig.5.We can attribute it as arising from the lower velocity ofthe meandering mode than the compression mode[23].

McQueeney, et al.[7] report that there is no phononanomaly below x 0.05 in LSCO where the static di-

agonal stripe is stabilized. There are two possibilities:(1) Since the velocity of the collective charge excitationis rather low for the insulating diagonal stripe[23], thephonon mode investigated by them may not be coupledwith this excitation, failing to drive the anomaly. (2)As the second possibility, the collective charge excitationcould be gapped[23]. There is no low-lying excitationcoupled to phonons. In this connection, we point outthat in Cr, where the diagonal stripe in the present termi-nology is realized, neutron experiment to see the phononanomaly at 2Q corresponding to the so-called strain waveor charge order is desirable.

Considering the electron-phonon coupling, we studythe phonon anomalies in the stripe state. The phononanomalies observed mainly by neutron scattering exper-iments on LSCO and YBCO with various doping levelsare accounted for as arising from two physical effects:Hybridization with the charge collective mode at 2Q andband-folding of a phonon band. These give rise to re-connection and gap formation in bands. The underly-ing physics behind the phonon anomalies at the partic-ular reciprocal point, namely 2Q, is the existence of thecollective charge excitations associated with stripes; thetranslational phason and meandering gapless modes withdifferent velocities situated at every even-multiples of the

fundamental vector Q which varies systematically withhole-doping. Thus this also leads to systematic changesof the 2Q vector, a fact that the experiments have found(see Fig.4 in Ref.[6]). In order to confirm our identifica-tion, further neutron experiments are desirable, in partic-ular to find the yet-uncovered lower modes branching-outdownwardly from the phonon discontinuity at 2Q. Sincein principle the phonon anomalies occur for all phononmodes and the observability depends on the electron-phonon coupling constant, lower phonon modes are morefavorable for this purpose.

It might be interesting to point out that similar phononmode anomalies are also found in (La,Sr)2NiO4[25] where

the diagonal stripes are stabilized because as mentionedthis system is an insulator up to higher dopings, thus45 rotation of the superspots occurs. In yet anothernon-cuprate superconductor Ba0.6K0.4BiO3, which is re-lated to charge density waves instead of spin density wavehere the phonon anomaly of the bond-stretching mode

appears at a certain reciprocal point[26]. We believe thatthese anomalies could be analyzed in the same way as inhere, which belongs to future problem.

We thank J. Tranquada, R. McQueeney H. Mook andN. Wakabayashi for their useful discussions and informa-tion.

[1] R.J. McQueeney, et al., Phys. Rev. Lett. 82, 628 (1999).[2] R.J. McQueeney, et al., Phys. Rev. Lett. 87, 077001

(2001).[3] L. Pintschovius and M. Braden, Phys. Rev. B60, R15039

(1999).[4] H.A. Mook and F. Dogan, Nature 401, 145 (1999).[5] Y. Petrov, et al., cond-mat/0003414.[6] H.A. Mook and F. Dogan, cond-mat/0103037.[7] R.J. McQueeney, et al., cond-mat/0105593.[8] M. Matsuda, et al., Phys. Rev. B62, 9148 (2000). S.

Wakimoto, et al., ibid. B 60, 769 (1999) and cited paperstherein.

[9] S. Wakimoto, et al., cond-mat/0103135.[10] J.M. Tranquada, et al., Nature 375, 561 (1995).[11] M. Fujita, et al., cond-mat/0107355.[12] K. Machida, Physica C158, 192 (1989). M. Kato, et al.,

J. Phys. Soc. Jpn., 59, 1047 (1990).[13] H. Schulz, Phys. Rev. Lett. 64, 1445 (1990). J. Zaanen

and O. Gunnarsson, Phys. Rev. B 40, 7391 (1990). D.Poilblanc and T.M. Rice, ibid. B 39, 9749 (1989).

[14] X.J. Zhou, et al., Science 286, 268 (1999).[15] K. Machida and M. Ichioka, J. Phys. Soc. Jpn., 68, 2168

(1999). M. Ichioka and K. Machida, ibid. 68, 4020 (1999).These show that the metallic stripe state is also describedby the mean field theory.

[16] Ch. Niedermayer, et al., Phys. Rev. Lett. 80, 3843 (1998).

[17] It is noteworthy that by applying stress the intensitiesof four incommensurate spots near (, ) changes intotwo pair of spots with unequal intensity, implying a one-dimensional stripe originally envisaged theoretically[12,13].

[18] P. Dai, et al., Phys. Rev. B 63, 54525 (2001).[19] Note that static charge order is observed in YBCO6.45

which is referred to in Ref.[6].[20] A.W. Hunt, et al., Phys. Rev. Lett. 82, 4300 (1998). Also

see P.M. Singer, et al., cond-mat/0108291.[21] Y. Ando, et al., cond-mat/0108053.[22] See for example, F. Becca, et al., cond-mat/0108198.[23] E. Kaneshita, et al., J. Phys. Soc. Jpn., 70, 866 (2001).

Also see, M. Ichioka, et al., ibid 70, 818 (2001).[24] We assume the dispersion for the unperturbed phonon:

q = A2 (1sn(qx+K))sn( qy

qx 2K)+ A2 (1+sn(qx+K))+B for |qx| |qy| and qx qy for |qx| < |qy | where sn(q)is the elliptic function and K complete elliptic integralwith modulus k.

[25] J.M. Tranquada, et al., cond-mat/0107635.[26] M. Braden, et al., cond-mat/0107498.
http://arxiv.org/abs/cond-mat/0003414http://arxiv.org/abs/cond-mat/0105593http://arxiv.org/abs/cond-mat/0103037http://arxiv.org/abs/cond-mat/0105593http://arxiv.org/abs/cond-mat/0103135http://arxiv.org/abs/cond-mat/0107355http://arxiv.org/abs/cond-mat/0108291http://arxiv.org/abs/cond-mat/0108053http://arxiv.org/abs/cond-mat/0108198http://arxiv.org/abs/cond-mat/0107498http://arxiv.org/abs/cond-mat/0107498http://arxiv.org/abs/cond-mat/0107635http://arxiv.org/abs/cond-mat/0107498http://arxiv.org/abs/cond-mat/0107498http://arxiv.org/abs/cond-mat/0107635http://arxiv.org/abs/cond-mat/0108198http://arxiv.org/abs/cond-mat/0108053http://arxiv.org/abs/cond-mat/0108291http://arxiv.org/abs/cond-mat/0107355http://arxiv.org/abs/cond-mat/0103135http://arxiv.org/abs/cond-mat/0105593http://arxiv.org/abs/cond-mat/0103037http://arxiv.org/abs/cond-mat/0003414

eiji kaneshita, masanori ichioka and kazushige machida- phonon anomalies due to stripe collective...

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