cuprates

PHYSICAL REVIEW B VOLUME 45, NUMBER 9 1 MARCH 1992-I

Algorithm for determining LO-phonon-mode parameters from optical-reflectance data: Applicationto high-T, cuprates

J.-G. ZhangNational Renewable Energy Laboratory, Golden, Colorado 80401

G. W. Lehman and P. C. EklundDepartment of Physics and Astronomy, University ofKentucky, Lexington, Kentucky 40506

(Received 3 October 1991)

An algorithm is proposed to derive the longitudinal-optic-phonon-mode parameters from reflectancespectra applicable when the mode damping is moderate to weak. The algorithm, which can easily beperformed using commercial software for a personal computer, leads to a unique set of mode parametervalues that satisfy both the generalized Lyddane-Sachs-Teller relation and the sum rule. As examples,we apply our algorithm to optical-reflectance data on La2Cu04 z and Nd&Cu04 q and obtain values forthe transverse and longitudinal frequencies and associated damping. The results are compared with oth-er methods commonly used in the literature.

I. INTRODUCTION

It is well known that the frequencies of the long-wavelength ( k =0) transverse-optical (TO) and thelongitudinal-optical (LO) modes are not equal in polarcrystals. The optical determination of the TO-mode pa-rameters is relatively easy. However, the quantitativedetermination of the LO-mode parameters from opticaldata is a more complicated issue and many approxima-tion methods have been used in the literature. ' Wepropose in this work an algorithm to compute rigorouslyLO-mode parameters from experimentally determinedTO-mode parameters. Unlike previous methods ourmethod can identify both the LO frequency and dampingconsistent with the generalized Lyddane-Sachs- Teller(LST) relation and optical sum rule.

For many crystals, such as La2CuO4 & andNdzCu04 & discussed in this paper, the phonon self-

energy is a slowly varying function of frequency and theinfrared refiectance spectra R(co) show narrow bands in-dicative of moderate to weak TO-mode damping. In thiscase, as Gervais has indicated in a recent review, thedielectric function E(co) can be expressed as a sum ofLorentz oscillators whose parameters can be used to fitwell the reflection spectra. In this way experimentalvalues are obtained for the TO-mode optical strengths,frequencies, and damping constants. This fitting pro-cedure, for example, has been used in our previous studyof LaqCu04 ~ and NdqCu04 ~. ' As is well known, theq-0 LO modes are not infrared active. Nevertheless,the LO frequencies and damping constants affect the op-tica1 strength of the associated TO mode. Our algorithm,which can be run quickly on a personal computer (PC),allows this LO-mode information to be computed directlyfrom the experimental TO-mode parameters.

The most direct method to study LO phonons is vianeutron-scattering data. Unfortunately, technicaldiSculties arise in obtaining sufficient quality neutrondata as the LO frequency increases, and in these cases,

optical determination of LO-mode parameters may be theonly open avenue. Several schemes ' ' have been com-monly used to obtain LO-mode information from opticaldata, such as assigning the LO frequencies QLz ~

to peakpositions in the imaginary part of ( —1/e), where thedielectric function e(co) is first determined from aKramers-Kronig transformation of R (co), " or identify-ing fl„o, with the zero crossings in the real part of e(to),i.e., e, . Chang et al. ' suggested a procedure to identifythe LO modes with the frequencies corresponding to theminima of the modulus of the complex dielectric function~e~. None of these graphical methods obtain the LOdamping parameters. A product form of the dielectricfunction, which contains LO damping, is also often usedto fit the reflectance directly. However, if the LO and TOparameters are treated as independent in an unrestrictedfitting procedure, the final values need not satisfy the sumrule or the general LST relation. Below, for the specificexamples of the high- T, cuprates La2Cu04 & andNd2CuO~ &, we present results for the TO- and LO-phonon-mode parameters obtained from our optical datausing our algorithm and compare these results to thosewe obtain using the alternate methods described above.

II. LATTICE VIBRATIONAL CONTRIBUTIONTO THE DIELECTRIC FUNCTION e(q, co)

For a homogeneous crystal with at least orthorhombicsymmetry, the response to a long-wavelength (q -0) pho-ton can be described by a generalized dielectric functione=e(O, to), which is written as the sum of dampedLorentz oscillators, one oscillator for each distinguish-able TO phonon at q =0

2PJ

&—0 CO CO& +ECOQ .

where e& and ez are the real and imaginary parts ofe(O, Co), respectively. e is a constant that approximatesthe high-frequency electronic contribution to the dielec-

1992 The American Physical Society

45 ALGORITHM FOR DETERMINING LO-PHONON-MODE PARAMETERS FROM. . . 4661

V D=eV E=e( —ik) E= ie—(kl+k~).E=O, (2)

where the charge density p=0, k is the wave vector ofthe electromagnetic field E, and k~~ and k~ are the com-

tric function, copj coj and y. are the optical strength, re-

storing force, and damping of the jth TO mode, respec-tively. %'e can include the possibility of a free-carriercontribution (in the Drude approximation) by setting

coo =0 for the j =0 mode; then ~ o and yo are the plasma

frequency and damping for free electrons, respectively.We refer to Eq. (1) as the "sum" form of the dielectricfunction.

Using the Maxwell equation V.D=p, the linear rela-tionship D =eE, and the plane-wave form for E, we have

ponents parallel and perpendicular to the electric vector,respectively. This equation has two roots. If e(co)=0,then k~~ need not be zero, and the wave could propagatealong the direction of E. Therefore, the longitudinal-mode frequencies [co& J are those points in the complexco plane where e(co) =0 If. e(co)%0, then we have to haveV' E=O or k~) 0 i.e.; the wave vector k is perpendicularor transverse to E. The complex poles tcorj J of e(co) aredefined as the transverse-optical-mode frequencies, whereIm(cori ) is related to the mode damping.

The sum form may be reexpressed in a "product"form, which allows the introduction of the LO-mode pa-rameters in a straightforward way. First, we note thatthe sum form of the dielectric function in Eq. (1) can alsobe written as'

e g (co co +indy ) g g co;(co co +!coy )j=0 i =0 j=Oj pi

e(co}=(co coj—+ivor }

j=0

(3)

The complex LO-mode frequencies co& will therefore bethe roots of the numerator N (co) in Eq. (3), which is a po-lynomial in co of degree 2(n +1}.We make the substitu-tion co=is in N (co }and note that the resulting polynomialin the complex variable s has purely real and positivecoefficients. All the roots of N(is} must then occur incomplex conjugate pairs, i.e., (s,s'}. Therefore, theroots of N(co) also must occur in pairs as (apz, —co&' ),and N(co) can also be written as

E'O

J =0 COJ

j=O i =j+1r L rLi'

(7)

(8)

N(co}=c g (co co~ )(—.co+co. ~ ),j=0

(4)

where c is a scale factor. Equation (3) can therefore bewritten as

l~c l2+iK'i )

& =0 (co coj + icor& )(5)

where lcozil is the modulus of cozen, rzj. = —21m(co& ) andc =e„We refer to. Eq. (5) as the product form of e(co).Substituting the roots cozk into N(co), it can be seen thateach complex LO-mode frequency ~~k is related to allthe lattice vibration parameters (coj., rJ,co~i ) via

= g (co co +ice „y ). —(6). .

j=0

From Eq. (6) it can be seen that it is not appropriate tocarry out independent adjustment of the parameters inthe product form (5) in order to fit the optical data.Comparing the coefficients in the numerators of Eqs. (3)and (5), we can obtain several useful relations

j=0

CO2+ PJJ

n —1 n+g g rr;.j=0 i =j+1

(9)

III. ALGORITHMFOR EXTRACTING LO-MODE PARAMETERS

Gervais has discussed the case of the broad phononbands in the optical reflectance, such as observed instrontium titanate, and has suggested that the peakbroadening is related to the frequency dependence of thedamping constant or phonon self-energy. In this case thedielectric function derived from a semi-quantum-mechanical treatment including the phonon self-energyhas the product form [Eq. (5)], except that the TO- andLO-mode parameters are independent of each other.Normally, however, the phonon bands are reasonablynarrow and can be well fit by the sum form of the dielec-tric function, indicating that the mode damping is essen-

Equation (7) is the generalized LST relation, where wehave included the Drude term (co0=0}. This relationconnects the modulus of the longitudinal-mode frequen-cies with the TO-phonon frequency e and TO-modestrength (or infrared activity). Equation (8) gives the sumrule on the transverse and longitudinal damping con-stants. Finally, Eq. (9) gives the sum rule for the modulusof the longitudinal-mode frequencies, and Eq. (9) correctsan error in the literature. '

4662 J.-G. ZHANG, G. W. LEHMAN, AND P. C. EKLUND 45

CO—+ CO

2TJ J

7J4

. Vj2

2(10)

i.e., the jth complex TO-mode frequency depends only onthe jth restoring force and jth damping constant. FromEq. (10) it can be seen that the phonon frequency is equalto the modulus of coT/, i.e., ~ coTJ ~

=co, .For the simplest case of an insulator with one TO

mode, one obtains

L12

COp 1+CO 1

1/2Vl . Yl

4 2

and2

~~1.&~= +~~Lt (12)

tially independent of frequency. In this case the TO- andLO-mode parameters must be adjusted to satisfy Eqs.(7)—(9). We next show the procedure to accomplish this.First, the sum form (1) is used to fit R (co), and preferablye(co) [as determined from a Kramers-Kronig transforma-tion of R(co)], one arrives at the set of phonon-mode pa-rameters Ice~, y, ]. The TO- and LO-mode parametersare then numerically obtained from Icoj., yj ]. The com-plex transverse-mode frequencies coT are easily obtainedfrom the poles of (5) as

r 1/2

2

E~ COjpJ(15)

For the longitudinal modes QLj, we obtained a simple re-sult only when one phonon is present, i.e., for n =0,QL& =~~0/e„. When n ~ 1 the expression for AL is verycomplicated, but we can prove that the minimum valuecorresponding to Qz, is

tained from optical reAectance using our procedure andalgorithm to those obtained by methods commonly foundin the literature.

Chang et al. ' proposed to identify the "real" part ofthe TO and LO frequencies on the following basis. TOfrequencies QTj are identified with real frequencies whichcorrespond to the maximum in ~e(~)~, where

~6( co ) ~ ( E)+ 'E2 )

' . We refer to this approach as the~e(co) ~,„method. Second, LO frequencies Qi areidentified with real frequencies which correspond to mini-ma in ~e(~)~. We refer to this approach as the ~e(co)~

method.In many cases, for example, in the La2Cu04 z and

Nd2Cu04 & crystals studied in this work, we have

yj/co~ &&1 (except the Drude term ya/cu0). In this limit

it can be shown that QT = ~coT, ~=co, to a good approxi-

mation. The corresponding maximum value for ~e(co) ~is

However, analytical solutions for the coLk are very com-plicated when more than one TO mode is present, andnumerical methods are required. We next describe an al-gorithm to calculate the coL in this case. Considering thenumerator N (co) in Eq. (3) to be a 2(n + 1)th degree poly-nomial in co, we have

2(n+1) 2(n+1)N(co) =e„g A co =e„g (co —

conj ) . (13)m=0

The coeScients A can be obtained exactly by notingthat

(14)To see that this expression is correct for the kth term inthe sum [Eq. (13)], take N(co)=e„co" and perform thesum over p in Eq. (14). For this case we can show thatA =5 k for k =0, 1, . . . , 2(n+1); therefore, it followsthat Eq. (14) is exact for any polynomial of 2(n + 1)th de-gree in co. The factorization indicated in Eq. (13) is per-formed by synthetic division, finding one root at a time.This algorithm is easily programed on a small computer.We have also noted that the 3, and thus the complexroots of the dielectric function, can also be obtained bycommercial software, such as MATHEMATICA (WolframResearch, Inc.},which runs on a PC.

IV. APPLICATION OF THE ALGORITHMTO La2Cu04 g AND Nd2CuO4

In this section we compare the results for the LO- andTO-mode parameters in La2Cu04 & and Nd2Cu04 &

ob-

2Q)pj

Another method often used to determine thetransverse-mode frequency QT is to identify it as themaxima in the imaginary part of dielectric function E2(Refs. 2 and 15) [ez(co)~,„methods]. Rewriting both E,and e2 in product form (5) and in the limit y /co «1(j&0), we find that both E2(co) and ~e(co)~ have a max-imum at QT =co and so these methods are essentiallyequivalent.

Some authors used "Drude's method" ' ' to determinethe real longitudinal frequencies QL, where the QL areidentified with zeros in the real part of dielectric function[e,(co)~„„method]. The same value is obtained from~e(co}~;„if we assume e2=0 (or y, =0 j%0) in the vicini-

ty of QL . It should be noted, however, that it is not al-ways possible to find the zero-crossing frequency of e, (co)corresponding to every longitudinal mode. This will beseen from the case of La2Cu04 & and Nd2Cu04 &. Bark-er adopted a procedure to identify QL from e, even inthe absence of zero crossing. He suggested setting thedampings to zero and then identifying the longitudinal-mode frequency from zero crossing of the fitting curvefor e, . Even in this case, we still have to remove the extrazero crossing arising from a factor of (co —co,-') in thenumerator of E'1.

Another method used in the determination of the LOmode is to identify the peak positions in the imaginarypart of (

—I /e), or energy-loss function, " as QL . If wetake the Ql obtained in the e, (co) ~„„method as a first-order approximation, then we have ~e(QI~)~=e'2(QLJ) [this is not a minimum value of ~e(co)~]. At


this frequency, then, we have Im( —I/e)=1/~e~ .Ap-parently, the Im( —I/e)~, „method is equivalent to thee(ro) ~;„method in the limit of y~ /coj. ((1 (jAO).

Turning to specific examples, we have studiedNd2Cu04 & and La2Cu04 & single crystals, which pos-sess tetragonal symmetry at high temperature ( &400 K),but exhibit a small orthorhombic distortion at room tem-perature. The details of the optical measurements them-selves can be found in Refs. 6 and 15. In our previous pa-pers, ' we have analyzed the lattice vibrations in

La2Cu04 & and Nd2Cu04 & by fitting the Kramers-Kronig-derived functions E'& 6'2 and the reflectivity R tothe sum form of e(co). The LO-mode frequencies foundin these studies were obtained by the Im( —I/e)~method discussed above. In this paper the sum form ofe(co) is again used to fit the experimental data (e„e2,R)and obtain the TO-mode parameters (co&~, yTJ, and ro&J ),but our algorithm is then used to derive the LO-mode pa-rameters from these parameters. In the following discus-sion, we separate the real and imaginary parts of thecomplex TO (LO) frequencies coTJ. (cuLJ ) into a real fre-

quency Qz;= ~roT ~ (QL =~roL ~) and the correspondingdamping parameter y TJ (yLJ ) as defined above.

In order to reach an objective judgment on thedifferent fitting procedures, commercial software such asIGOR by Wavemetrics, Inc. was used to perform theleast-squares fitting automatically. This program uses theLevenborg-Marquardt algorithm for gradient least-squares searching in a multidimensional parameter space.All of the phonon parameters (12—20) are adjusted simul-taneously to obtain the best fittings. The sum of least-squares differences between the fitting curve and experi-mental data is less than 10 in all cases discussed here.

When the sum form of the dielectric function is used,IGOR can easily converge to a unique set of parametersfor the TO modes, and then these parameters are used todetermine LO-mode parameters by the algorithm pro-posed in this work. As a comparison, we also used theproduct form of the dielectric function [Eq. (4)] and allowindependent adjustment of all of 4(n + I) parameters tofit the reflectance spectra. For the product form IGORalso returned with good least-squares fits to the data, tak-ing noticeably longer with more parameters, as expected.We did note that the "best-fit" parameter values for theproduct form, particularly the LO damping parameters,were somewhat different from run to run of IGOR.Representative results from these fitting exercises for sumand product fitting are gathered together in Tables I, II,and III for La2Cu04 & (EJ.c), LazCu04 s (E~~c), andNd2Cu04 &, respectively. The differences between thephonon-mode parameters obtained by these two methodsare listed as "5, " in the corresponding tables.

Comparing the parameters obtained from representa-tive sum and product fitting, we find that the real TO fre-quencies 0z~

=~co rj ~

obtained are nearly equal (thedifferences are less than 1%). However, the LO-mode pa-rameters and some TO damping parameters yT- deter-TJmined by these two methods were not found to be in goodagreement. Furthermore, the product-form parameterswere found not to satisfy the LST relation [Eq. (7)] or thesum rules [Eqs. (8) and (9)]. In many trials, IGQR re-

turned negative values for the damping constants, e.g.,yl 3 in the case of La2Cu04 & (E~~c ) and y T4 in the

case of Nd2Cu04 z. For Nd2Cu04 & we also noted thatthe product fitting returned QL3) QT4, which is physical-ly unacceptable. In addition to these unphysical parame-ter values, we found that some of the mode frequenciesobtained by product fitting varied by more than 200cm ' from one trial run to another and dependsignificantly on the initial value of the fitting parameterschosen in Eq. (5). If the initial values of the mode param-eters in the product fitting were chosen to be within 5%of the those obtained in the sum fitting, the product-formfitting parameters usually converged to the same parame-ters as obtained in the sum-form fitting. However, whenthe initial parameters in the product fitting are chosenmore than 5% different from those obtained from sum-form fitting, the final values of the least-squares parame-ters were much different and even unphysical in somecases, even though the reflectance curves calculated fromthe two sets of parameters were indistinguishable visual-ly. These results indicate the utility of our procedure pro-posed here.

1.0

0.8 em

E 0.6

g ~

s II

~ ~

g I

~m~r

er mr mom

40(0)

Im(-1le) La CLQ4, E & c - 2p

~ mr ~ mrmrmrmr Q rmrmrmtmrmrmrormrmrmrm-memem mmemr

O0.4—

OP

--20

0.2— --40

0.0200 400 600

Energy (cm'}

I

800-60

i 000

30-

eS'

o 20-

I

I

I

III

I

La Cu04, E&c

~ eeeeer

(b)

10-

I

200I

400I

600 800 1 000

Energy (cm )

FIG. 1. (a) Functions I/~e~, Im( —I/e) and e, vs energy forLa2Cu04 z (Etc). The dielectric functions e& and e2 were ob-tained from a Kramers-Kronig transformation of reflectancedata. (b) Functions ~e~ and e, vs energy for La2Cu04 s (Elc).The dielectric functions e, and e2 were obtained from aKramers-Kronig transformation of reflectance data.

J.-G. ZHANG, G. %'. LEHMAN, AND P. C. EKLUND 45

TABLE I. LO-, TO-phonon-mode, and Drude parameters for La2Cu04 z (Elc). All parametervalues are in cm

Phononparameter

~2u, 1 TlnL17 Tl

VL1~ 2u, 2 +T2

QL2

yT2

VL2

A2„3 QT3

nL3rT37L3

( ')67p 'T

Sumfitting

14829443

140359418

48140669700

5674

3766

0.95

Productfitting

14831643

143359447

47227675713

56103

6,p

022030

291

876

130

29

Max in

141

359

661

Max in

145

359

Min in

294

710

Max inIm( —1/e)

290

452

698

Zero in

282

We next turn to the discussion of the phonon parame-ters in La2Cu04 z and Nd2Cu04 z determined by vari-ous graphical methods and compare these parametervalues with those obtained by the sum and productmethods discussed above. In Figs. 1(a) and 1(b), we plotthe dielectric functions for La2CuO~ s (Elc) commonlyused to determine graphically the LO- [Fig. 1(a)] and TO-[Fig. 1(b)] mode frequencies. The peak positions in Fig.1(a) of 1/~e~ and Im( —I/e) locate approximately theLO-mode frequencies QL, as do the zero crossings of e, .In Fig. 1(b) the peak positions of ~e~ and e2 have beenused to determine TO-mode frequencies 0&. The A&-.

and QLJ obtained by these graphical methods and thoseobtained by the sum and product fittings are gathered inTable I. The transverse frequencies QTJ obtained by

E2~,„methods are in good agreement with those ob-tained by sum fitting (the difFerences are less than 3cm '). The difFerences between Qr obtained by the~e~,„method and sum fitting are also relatively small( (8 cm ').

Some of the QL. obtained graphically differ noticeablywith those obtained by sum fitting. For example, (1) us-ing ~e~;„, QL2 and QL &

differ by —10—40 cm ': (2) usingIm(l/e)~, „, QI, and QL3 differ only by 1 —4 cm ' but

QL2 was found to exhibit a significant difference of —30cm '; and (3) using e,(ro)~„„,QL, and QL2 difFer by —10cm '. Note that the e, (co)

~ „„,method cannot identify

QI3 (i.e., there is no zero crossings associated with thismode because of a nearby electronic contribution).

TABLE II. LO-, TO-phonon-mode, and Drude parameters for La2CuO~ s (E~~c). A11 parametervalues are in cm

Phononparameter

Sumfitting

Productfitting ~sp

Max in Max in Min in Max inIm( —1/e)

Zero ln

Eu, 1 +TlQL1

YT1

YL1

E., 2 T2QL2

yT2

E„3 QT3

L3yT3

E., 4 &r4AL4

QT4

3 I4

e =3.6

234314

8113

314351

1223

354456

2059

516579

3249

234314

8113

315357

12—26'354448

—20'69

516573

3249

00001

60

4908

40100600

225

316

349

510

230

316

345

510

312

341

466

583

312

342

461

575

464

581

'Negative values for damping parameters are acceptable mathematically, but are unacceptable physical-ly.


2.5

1.5-E

01P—4)

0.5—

r,J

~ I

2 0A —..----sI

II

IIII

La G.D, E //c

~%

r' ~a

30

10

0 Pj

--10

--20

0.6

0.5—

4lp 4—

I

E0.3—

O

—0.2—47

0.1—

Nd2Q. D4 ~

(a )— 100

--100

--200

0.0

40

200I

400 600

Energy (cm ')

-30800 1000

a aa ~ a ~ aaaaaaaaaaaaaaaaa a

0.0

200

I

200 4oo 6ooEnergy (cm')

I

800-300

1000

30-

cS'

o 20

10—

E//c150—

cS'

a 100-

50-

'Ife'II ~

'II

/ ~

I ~

~ sg ~~ ~

~ I~ II

Nd2CUO4,

~ a aoaa2

I

200I I

400 600 800 1000

Energy (cm')

FIG. 2. (a) Functions I/~ e~, Im( —I/e), and e, vs energy foraz u04 s (E~~c). The dielectric functions e, and ez were ob-

tained from a Kramers-Kronig transformation of reflectancedata (b) Fu.nction ~e~ and e, vs energy for LazCu04 s (E~~c).

1The dielectric functions e and e wer bt'

d fwere o tanned from aKramers-Kronig transformation of reflectance data.

~ ace a ~ a0 I

0I

200 400 600 800 1000I

Energy (cm )

FIG. 3. (a) Functions I/~e~, Im( —I /e) andu 4 q (Elc). The dielectric functions el and e2 were ob-

tained from a Kramers-Kronig transformation of reflectancedata (b). Functions ~e~ and ez vs energy for Nd CuO (EJ. )

e selectric functions el and e2 were obtained from aKramers-Kronig transformation of reflectance data.

TABLE III. LO- TO- ho-phonon-mode, and Drude parameters for Nd C 0 'Evalues are in cm

r 2 u 4 q' lc). All parameter

Phononparameter

SumStting

Product6tting

Max in Max in Min in Max inIm( —1/e)

Zero in

Eu, 1 T1QL1

Tl

HALI

Eu, 2 T2QL2

yT2

3 L2

E„3 QT3

QL3

rT37L3

Eu, 4 QT4

L47 T4

7L4

E'~

co, (cm ')COp 7

1251331623

302332

813

347409

1240

5165823665

71291

0.64

125138

1635

302336

813

347621

123205'516755

—36'976'

05

01204000

2110

32450

17372

911

121

301

347

507

123

302

347

515

138

335

426

608

137

339

616

339

424

585

'These values for dam inp g parameters are acceptable mathematicall b ta y, u are unacceptable physically.

4666 J.-G. ZHANG, G. W. LEHMAN, AND P. C. EKLUND 45

For La2CuO~ s (Eiic), we have used similar graphicalmethods to determine the TO- and LO-mode frequencies.The functions are plotted in Figs. 2(a) and 2(b), and theQT- and QL obtained are listed in Table II. Relative tovalues obtained by the sum fitting method, we find that(1) using the ie(to)i, „and ezra, „methods, the QTi differ

by 2—9 cm '. (2) using Im( —1/e)i, „and ie(ro)i;„ theQL. differ by 1—10 cm '; and (3) using e&(co)i~„, twolow-frequency longitudinal modes cannot be identified.

A similar study was performed on Nd2Cu04 s (Elc).In Fig. 3(a) we compare three approximation methodsused in determination of the longitudinal modes ofNd2Cu04 5 single crystals. The QTJ and QL~ obtainedvia the various graphical methods employing the dielec-tric functions are listed in Table III. QL &

and QLz ob-tained from both the plots of Im( —1/e) and 1/hei areonly 3-6 cm ' larger than those obtained by therigorous sum fitting procedure, and QL3 and QL4 ob-tained from peaks in Im( —1/e) and 1/hei are about 20cm ' larger. Finally, the,e( t)oi „method cannot iden-

tify and QL. In Fig. 3(b), we plot both hei and ez versus

energy for Nd2Cu04 & single crystals. Qz~ for j =1-3obtained by these graphical methods are in excellentagreement with those obtained by sum fitting ( (2 cm ').

However, QT4 is —10 cm ' smaller than those obtained

by the sum fitting procedure.

V. SUMMARY

The transverse- and longitudinal-optical-phononmodes (TO and LO) in a homogeneous crystal, which hasat least orthorhombic symmetry, have been studied. Ac-curate LO- and TO-mode frequencies and damping con-stants can be obtained by fitting directly the reflectancespectra using the sum form of the dielectric function tofirst obtain TO-mode parameters. Next, using the algo-rithm proposed in this work, LO-mode frequencies anddamping are obtained. The LO- and TO-phonon parame-ters obtained in this method are unique and satisfy boththe LST relations and sum rule automatically. In at-tempting to use the product form of the dielectric func-tion to least-squares fit reflectance data, we obtained, inmany cases, phonon-mode parameters which were notphysical, although the reflectance data mere well PtAmong the various graphical methods often used todetermine LO- and TO-mode frequencies, we found itconvenient to obtain approximate TO- and LO-mode fre-quencies from the maxima and minima, respectively, of

(&2+e2)1/2

'A. S. Barker, Jr., in Ferroelectricity, edited by E. F. Weller (El-sevier, New York, 1967).

~A. S. Barker, Jr., Phys. Rev. 136, A1290 (1964).3F. Gervais, P. Echegut, J. M. Bassat, and P. Odier, Phys. Rev.

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cuprates

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