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VOLUME 19,NUMBER 20 PHYSICAL REVIEW LETTERS 13NOVEMBER 19674R. W. Motley and A. Y. Wong, in Proceedings of theSixth International Conference on Ionization Phenomena

in Gases. Paris, France. 1963, edited by P. Hubert(S.E.R.M.A. , Paris, France 1964), Vol. 3, p. 133.P. F. Little and H. G. Jones, Proc. Phys. Soc. (Lon-don) 85, 979 (1965).

A. Y. Wong, Phys. Rev. Letters 14, 252 (1965).TP. J. Barrett and P. F. Little, Phys. Rev. Letters14, 356 (1965).G. M. Sessler, Phys. Letters 16, 277 (1965).~I. Alexeff and W. D. Jones, Phys. Rev. Letters 15,286 (1965).10W. D. Jones and I. Alexeff, in Proceedings of theSeventh International Conference on Ionization Phenom-ena in Gases, Belgrade, 1965 (Gradjevinska Knjiga

Publishing House, Beograd, Yugoslavia, 1966).~~F, W. Crawford and R. J.Kuhler, in Proceedingsof the Seventh International Conference on IonizationPhenomena in Gases, Belgrade, 1965 (GradjevinskaKnjiga Publishing House, Beograd, Yugoslavia, 1966),p. 326.G, M, Sessler, Phys. Rev. Letters 17, 243 (1966);Phys. Letters 22, 363 (1966).~3R. W. Gould, Phys. Rev. 136, A991 (1964).E. A. Jackson, Phys. Fluids 3, 789 (1960).~5G. M. Sessler, Phys. Fluids 7, 90 (1964).1 N Sato and Y. Hatta, J. Phys. Soc. Japan 21, 1801(1966).~7S. C. Brown, Basic Data of Plasma Physics (JohnWiley@ Sons, Inc. , New York, 1959).

ELECTRIC FIELD DEPENDENCE OF OPTICAL-PHONON FREQUENCIESJ.M. Worlock and P. A. FleuryBell Telephone Laboratories, Holmdel, New Jersey(Received 27 September 1967)

In electric-field nduced Raman-scattering experiments on the cubic perovskiteSrTi03, a striking electric field shift and splitting of the soft optical phonon mode isobserved. Experiments were done at temperatures ranging from 8 to 250oK and forelectric fields between 0.2 and 12 kV/cm. We interpret the temperature and electricfield dependence of the phonon frequency using Devonshire model of ferroelectricity andthe Lyddane-Sachs- Teller relation.In Cochran's theory of ferroelectricity, ' soft-

phonon modes are of central importance. Asthe transition temperature is approached fromabove (in the paraelectric phase) the phononfrequency tends toward zero. We have previ-ously reported studies of the soft-phonon modesin KTa03' and SrTiO3 in which an electricfield was employed to induce Raman scatter-ing from these odd-parity phonons. In thisLetter we report the observation of strikingelectric field dependence of some optical-pho-non frequencies in SrTiO, . In particular, thelowest lying transverse optical phonon has beenobserved to shift in frequency by 400% and tosplit into two components polarized paralleland perpendicular to the applied field. Schau-fele, Weber, and Silverman4 have recentlyobserved a small electric-field shift in thesoft mode frequency at 77'K.

We have examined the induced Raman scat-tering at a variety of electric fields (between0.2 and 12 kv/cm) and at temperatures between8 and 250'K with the following general results:(1) With very small applied fields the soft-modefrequency varies with temperature from 11

cm ' at 8'K to 85 cm ' at 250'K, in good agree-ment with the predictions of the Lyddane-Sachs-Teller (LST) relation inserting Weaver's val-ues of the dielectric constant. ' (2) At low tem-peratures (&55 K) the spectrum of the field-induced scattering exhibits a field-dependentstructure. Figure 1, taken at 8'K, shows thatas the field is increased not only does the fre-quency of the sof t mode shift from 11 to 45 cmbut also three additional peaks become visible.Those labeled 8 and C are identified as com-ponents of the soft TO mode, polarized perpen-dicular and parallel to the applied field, respec-tively. Peaks A and B are not components ofthe soft mode, and appear as well in the intrin-sic spectrum of SrTiO, [see Fig. 1(b)]. (3) Theeffects of electric field become increasinglystrong as the temperature is lowered. Thisis true for the efficiency of the induced Ramanscattering as well as for the shifting and split-ting of phonon frequencies. (4) In contrast. tothe case of KTaOwe have observed in SrTi03induced Raman scattering from the other TOmodes at 170 and 550 cm '. We shall not dis-cuss these results here except to say that these

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VOLUME 19,NUMBER 20 PHYSICAL REVIEW LETTERS 13 NOVEMBER 1967mode frequencies exhibited neither tempera-ture nor electric field dependence, and thatthe frequencies we observe are in good agree-ment with values obtained by other methods. '~'We shall show below that our observationson the soft mode can be quantitatively account-ed for in terms of a, combination of simple ther-modynamic theory and a slightly genera, lizedLST relation.The details of the experiment were essential-ly as in the earlier work' on KTa.O, with the

400 V/cm4000 V/cm

1600

exception of the method of applying the electricfield. To facilitate accurate measurementsof field strength and to overcome possible prob-lems with sample heating at low temperatures,we imposed repetitive square voltage pulsesand synchronously detected the field-inducedscattered light using a. gated boxcar integra-tor. This method permitted discriminationagainst the strong background spectrum of in-trinsic scattered light. By the intrinsic spec-trum we mean the spectrum obtained by usu-al Raman techniques with no appli. ed fields.Both the pulse width (~200 psec) and the rep-etition rate (10 to 300/sec) were varied so asto prevent sample heating even when the peakfield strength was increased substantially.Boxcar gate widths were typically one-fourththe voltage pulse width.Cochran' and Anderson' first pointed out theimplications of soft-phonon modes for the fer-roelectric phase transition. The fundamentalrelation is that due to LST9:

A BI I I a I a I a I a

a I I I s I a I a I sIO 20 30 40 50PHONON FREQUENCY (cm-I )

FIG. 1. (a) Electric-field nduced Raman spectrumin SrTi03 at 8'K. The spectra for different appliedfields exhibit the large shift of the soft-phonon modecomponent C polarized paralled to the electric field,as well as the lesser shift for the soft-mode compo-nent B polarized perpendicular to the field. The com-ponents of the Raman-scattering tensor observed to benonzero are o.~z for peaks A and B, and n~z for peaksC and D. The electric field is applied in the x direc-tion. The induced scattering intensity generally in-creased as the field increased and the vertical scalesare not the same for spectra obtained at different fieldstrengths. (b) The portion of the intrinsic (zero-field)spectrum of SrTi03 taken at 8'K and with the same slitsettings as in (a). The sharp features A and D appearin both the induced and intrinsic spectra. In the intrin-sic spectrum the nonzero Raman tensor componentsare nzz for A and nzz for D.

Here ~1~ and vT~ refer to the longitudinal andtransverse optical phonon frequencies, ' e isthe high-frequency dielectric constant, andthe low-frequency dielectric constant. Un-der the assumption that only the soft-mode fre-quency, ~, varies with temperature and thatis insensitive to temperature, Eq. (1) re-duces to &us'e =const. (independent of T). Ourresult at very low fields on the soft-mode fre-quency in SrTi03 agree with this simplifiedform when the measured values of the dielec-tric constant' are inserted: ~z'e =3.0X10' cm ',over the temperature range from 8 to 250'K.To understand the behavior of the soft-modefrequency under the influence of electric fields,we assume that the LST relation remains val-id in the presence of a field and use a simplephenomenological thermodynamic argumentto obtain the field dependence of the dielectricresponse. Expressing the crystal's free ener-gy per unit volume, A, in terms of the temper-ature, T, and the polarization, I.', we haveignoring tensor indices, ' and assuming e1

+-,](T)P'+ ~ ~ ~ (in mks units). (2)1177

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VQLUME 19,NUMBER 20 PHYSICAL REVIEW LETTERS 13NovEMBER 1967

(for small fields),(e,e) ' = -y(T) +3[((T)E']' +~ ~ ~

(for large fields).

{5)

(6)The distinction between large and small fieldsis determined by which of the terms in Eq. (4)is dominant. Even for the largest fields encoun-tered in our experiments, terms of higher or-der than P' in Eq. (2) are unnecessary to ex-plain our experimental results. Both limits,Eqs. (5) and (6), have been used to obtain thenonlinear coefficient (T) from the temperaturedependence of &us(E). In the high-field region(easily reached experimentally at 40'K and be-low), combination of Eq. (6) with the LST re-lation predicts

u '(E)+~ '(E=O) =3x10'e x[3((T)E']' (7)S S 0This relationship [that is, a linear dependenceof es'(E) +~+'(E = 0) on E ']has been verifiedexperimentally at temperatures of 8, 27, and40'K. The value of obtained is (1.2+ 0.3)x10in mks units and appears in this region to beindependent of temperature. The value of is larger by about a factor of 2 than that derivedfrom measurements of Rupprecht, Bell, andSilverman on the nonlinear dielectric constantof SrTi03 above 90'K and by about the samefactor than the value found by Itschner fromdirect measurements of the dielectric constantat low temperatures. Possible sources of thisdiscrepancy include inaccuracy in precise de-termination of electric field and differencesin dielectric response to dc and pulsed fields.Figure 2 shows the temperature dependenceof ~~ for various applied fields. While the field-induced frequency shift is indeed dramatic atlow temperatures, the effect diminishes dras-

The derivatives of A are the useful quantities:E = eW/sp = q(T)p

+ (T)P'+ ~ ~ ~ (electric field), {3)1 O'A = y(T) +3((T)p'+ ~ ~ ~

eoE BP (inverse dielectric constant), {4)where e, is the permittivity of the vacuum.If care is taken to handle the approximationsin a. consistent manner one obtains the limit-ing relations

3((T)=)((T) 1+ 3( )E + ~ ~ ~60E'

tically as the temperature is increased; it isnearly unobservable above 80'K. We find aphonon frequency shift of only -1.5 cm ' fora field of 12 kV/cm at 77'K, which disagreeswith the value 4 cm ' of Schaufele, Weber,and Silverman. This discrepancy arises fromtheir interpretation of a peak in the intrinsicspectrum as the zero-field soft mode. Thisalso leads them to conclude that the frequen-cy shift is linear in field, whereas it is in factquadratic, as expected from Eq. (5).Peaks A. and B appear in both the field-inducedspectrum [Fig. 1(a)] and the intrinsic spectrum[Fig. 1(b)]. Their polarization properties areas follows'. For incident light polarized par-allel to the applied field (x direction), lightscattered at peak A has Raman tensor compo-nent o.~z, while peak D has Raman tensor com-ponent n~~. These peaks are unique in thatno other features of the complicated intrinsicspectrum of SrTiO, appear in the field-inducedspectrum. A more detailed discussion of thesepeaks will not be given here, but their narrowwidth and their temperature and electric fieldbehavior argue against their being due to strains,impurities, or two-phonon processes.The large phonon frequency shifts inducedby easily achievable electric fields suggesta number of interesting scientific and techno-logical uses. At low temperatures the linewidthof the soft mode remains quite narrow (&2 cm ')even as the field is raised and the phonon fre-quency is swept from 11 to 45 cm '. This,

60

50IE 40-Z4JOUJu 30OOr 20-

I0i I I I I I I I I I I I I20 40 60 80 100 I20TEMPERATURE (OK)

FIG. 2. Temperature dependence of the soft-modefrequency for various values of applied electric field.Only the C component is plotted here. Above 100'K theelectric-field shift could not be seen, and so the curvedoes not continue to the highest temperatures ere ex-amined.

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VOLUME 19,NUMBER 20 PHYSICAL REVIE%' LETTERS 13NovEMBER 1967coupled with the large scattering efficiencyinduced by the field, makes the soft mode aprime candidate for an electric-field-tunableRaman laser. In addition, the natural infraredactivity of the mode would provide radiationin the 10- to 100-cm ' region of the spectrum.The knowledge of the soft-mode frequency andits variation with temperature and electric fieldshould prove useful in other experiments. Oneexample has already been given here by relat-ing the variation in the phonon frequency tothe dielectric properties of SrTiO, . Many oth-er physical parameters, such as thermal con-ductivity, depend upon the phonons for theirmicroscopic origin. A movable phonon modeis a very useful probe for testing theories ofoptical, thermal, and mechanical propertiesof solids.We are grateful to A. S. Barker for a sam-

ple of SrTiO to D. H. Olson and H. L. Car-ter for technical assistance, to A. Albert forcrystal polishing, and to T.W. Armstrong,Jr., for electrode evaporation. We are alsograteful to S. H. Wemple for a copy of D. Itsch-ner's dissertation, and for helpful discussions.

~W. Cochran, Advan. Phys. 9, 387 (1960).2P. A. Fleury and J. M. Worlock, Phys. Rev. Letters18, 665 (1967).J.M. Worlock and P. A. Fleury, Bull. Am. Phys.Soc. 12, 662 (1967).4H, . F. Schaufele, M. J.Weber, and B.D. Silverman,

Phys. Letters 25A, 47 (1967).5H. E. Weaver, J. Phys. Chem. Solids ll, 274 (1959).6R. A. Cowley, Phys. Rev. 134, AS81 (1964).W. G. Spitzer, R. C. Miller, D. A. Kleinman, andL. E. Howarth, Phys. Rev. 126, 1710 (1962); A. S.Barker and M. Tinkham, Phys. Rev. 125, 1527 (1962);A. S. Barker, Phys. Rev. 145, 391 (1966).P. W. Anderson, in Fizika Dielektrikov, edited byG. I. Shansvi (Academy of Sciences of the USSR, Mos-cow, 1960).~R. H. Lyddane, H,. G. Sachs, and E. Teller, Phys.Rev. 59, 673 (1941).A. F. Devonshire, Advan. Phys. 3, 85 (1954).~~G. Rupprecht, R. O. Bell, and B.D. Silverman,Phys. Rev. 123, 97 (1961).~2D. Itschner, Promotionsarbeit, EidgenossischenTechnischen Hochschule in Zurich, 1965 (unpublished).~3P. S. Narayanan and R. Vedam, Z. Physik 163, 158(1961); R. F. Schaufele and M. J.Weber, J. Chem.Phys. 46, 2859 (1967); D. C. O'Shea, R. V. Kolluri,and H. Z. Cummins, Solid State Commun. 5, 241 (1967);L.Rimai and J.L. Parsons, Solid State Commun. 5, 387(1967);W. G. Nilsen and J. G. Skinner, to be published.

ANOMALOUS TOTAL-ENERGY DISTRIBUTION FOR A MOLYBDENUM FIELD EMITTERL. W. Swanson and L. C. Crouser

Field Emission Corporation, McMinnville, Oregon 97128(Received 11October 1967)The possibility of using field-electron micro-scopy to elucidate special features of bulk elec-tronic band structure for metals has been pro-posed in previous theoretical'~' and experimen-tal findings. &~ In a previous publication ananomalous total-energy distribution (TED) fromthe (100) direction of tungsten, consisting ofan enhanced emission (see Fig. 1) approximate-

ly 0.35 eV below the Fermi level EF, was re-ported. Upon examination of theoretically andexperimentally suggested band structures andFermi surface shapes along the (100) of tung-sten one could correlate the TED results withspecial bulk electronic features. , Although mo-lybdenum is nearly identical to tungsten bothgeometrically and electronically speaking, somenotable differences along the (100) directionband structure, which might be manifested inTED rneasurernents, motivated our undertak-

ing of a similar investigation of Mo.The expected electronic band structure fortungsten along the (100) direction (the FH di-rection in k space) is reproduced in Fig. 2 froma previous publication by Mattheiss. ' Introduc-tion of spin-orbit interactions (proportionalto 5d) results in a splitting of the b,, degen-eracy and removal of the 6, crossing. Mattheisspoints out that a value of 5d = 0.03 Ry (0.4 eV)can account for the disappearance of the elec-tron lenses along the (100) axes in tungsten,a result obtained by Sparlin and Marcus' fromde Haas-van Alphen measurements. Thus,if the upper 6, band does not dip below the Fer-mi level EF [see Figs. 2(e) or 2(d)] one expectsTED measurements of field-emitted electronsnear EF to be normal (i.e., obey the Sommer-feld model for field emission); however, atapproximately 0.35 eV below EF, emission

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