raman effect in zno

8/3/2019 Raman Effect in Zno

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PH YSICA L RE VIEW B VO LUME 16, N UMBER 8Resonant Raman scattering in ZnO

15 OCTOBER 1977

J.M. Calleja* and Manuel CardonaMax-Planck-Institut fur Festkorperforschung, 7000 Stuttgart 80, Federal Republic ofGermany

(Received 19 May 1977)The resonance of the Raman scattering by EAlp' Eil and E phonons, and several second-order

features, has been studied for ZnO for photon energies between 1.6 and 3 eV. The results are interpretedwith a dielectric theory based on the first and second derivatives of the dielectric constant. By combining ourresults with absolute scattering cross sections previously determined by Arguello et al. at 2.41 eV absolutevalues of the deformation potentials of the band edge can be determined. The difference in strength betweenthe longitudinal and the transverse modes provides the signs of these deformation potentials. Theantiresonance around 1.6 eV suggested by the earlier work of Callender et al. and attributed to acancellation of the deformation potential and electro-optical contributions to the Raman tensor is confirmed.The deformation potentials of the A, phonons at the band edge have been obtained from a pseudopotentialcalculation. While the sign of these deformation potentials agrees with the experimental determination, theirmagnitudes do not agree. This fact is attributed to difficulties with the pseudopotential of the 0' ion. Anestimate of the deformation potentials from the dependence of the band edges on uniaxial stress is also made.

I. INTRODUCTIONZnO, the mineral zincite, crystallizes in thewurtzite structure (space group C~6) which pos-sesses four atoms per primitive cell.' The dis-persion relations of phonons (and also of elec-trons) along the 6 direction (hexagonal axis) canbe approximately derived by folding those of thecorresponding zinc-blende crystal (two atoms perprimitive cell) along the [111]direction. ' Thus

one finds at k=0 (I') two sets of infrared- andRaman-active modes, A, (singlet) and E, (doublet),which correspond to the optical modes of zinc-blende. '4 Besides, as a result of the folding,additional infrared-inactive modes appear at I',as shown in Fig. 1. They are two sets of ERaman-active doublet modes and two Raman-in-active singlets (B}.Raman AEand E, modeshave been recently observed by Arguello et al.'and identified according to their polarization selec-tion rules. The infrared-active E, and A, modescan be seen to split into E,~ and A (L, longitudi-nal; T, transverse) for propagation perpendicularto the c axis and Eiy A]g for parallel propagationas a result of the polarization which accompanieslongitudinal infrared- active excitations. Forpropagation at an angle other than 0 or 90' to thec axis mixed modes are obtained. Neutron-scat-tering data, confined mainly to the low-frequencybranches of the dispersion curves, have recentlycomplemented our knowledge of the lattice dynam-ics of ZnO, ' and provided a basis for param-etrized calculations such as that shown in Fig. 1.In spite of the interest in ZnO and of a con-siderable number of calculations, "the knowledge

of its electronic bands is rather imperfect except

for the details of the lowest band edge at k =0(which is usually used as an experimental fittingparameter for calculations! ). The Korringa-Kohn-Rostoker (KKR} calculations' yield the wrong posi-tion for the 3d core levels of the Zn while theempirical pseudopotential work' is not able to re-produce the details of the optical-absorption spec-trum, a fact believed due to difficulties in settingup a local pseudopotential for 0' .However, the experimental picture concerningthe lowest direct edge (actually excitons) seemsclear. At room temperature there is an edge at3.32 eV polarized perpendicular to the c axis,(A, B}with a parallel polarized counterpart (C}at 3.36 eV.' The (A, B) exciton exhibits a verysmall negative spin-orbit splitting' which onlybecomes observable at low temperatures. The A,B, and C excitons correspond to the so-called E,edge. Note that the E, edge in ZnO is smaller thanthat of Zns (3.7 eV)," a fact which has thus farescaped theoretical understanding. The Raman-scattering efficiencies are expected to resonatewhen the laser frequency approaches E,. Theseresonances have been investigated for a numberof zinc-blende-type semiconductors. " They canbe used to study the details of the scatteringmechanism and to determine electron-phononcoupling constants or deformation potentials.Work of this sort for ZnO has been rather limitedbecause of the awkwardness, from the point ofview of laser technology, of the region in whichE, occurs (3.3 eV=3750 A). A study'~ at 4.2' Kof the Frohlich-interaction-induced 2E,~ scatter-ing, using a pulsed dye laser in the region from3.28 to 3.36 eV, has been published but unfor-tunately this type of scattering is difficult to in-

16 3753


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J. M. CAL LE JA AND MAN UEL CARDONA 16600

-A)L600

400E

200

EgEirAir8

FIG. 1. Phonon disper-sion relations for ZnQ atroom temperature takenfrom Ref. 4. The points in-dicate the phonon frequen-cies determined by Ramanscattering (Ref. 21); thedashed bars represent atentative assignment of thesecond-order structuresshown in Fig. 2(a).

0I T K 04

terpret quantitatively. ' A study of the E,~, E,A, and E, resonances in the energy range 1.92-2.57 eV has appeared more recently. " While thisrange is still quite far from the Eo edge of ZnOone can already identify in it resonance effects.In fact, by extrapolation of the observed E reso-nance the data of these authors suggest the exis-tence of an antiresonance in the scattering by E,~phonons at about 1.6 eV. They attribute this be-havior to a destructive interference between de-formation potential and interband Frohlich inter-action (i.e., electra-optic effect) scattering.In this paper we investigate resonance Ramanscattering of all modes mentioned above for ZnOplus a few second-order structures at room tem-perature in the 1.6-3 eV region which, approachessufficiently. the Eo edge to clearly distinguishresonance effects. On the low-photon-energy sideour experimental region reaches the energy of thesuspected E antiresonance and we are thus ableto confirm its existence. In Sec. II we discuss theexperimental details. In Sec. III we present the ex-perimental data and we fit them with expressionsbased on derivatives of the dielectric constant.By normalizing to the values of absolute scatter-ing efficiencies given in Ref. 3 for 2.41 eV it ispossible to present our data as absolute effi-ciencies. In Sec. IV we discuss the results interms of deformation potentials and electroopticcoefficients. We present the results of a pseudo-potential calculation of the deformation poten-tials of the A, modes for the E, gap of ZnO.While this calculation predicts the sign of thedeformation potentials extracted from our Raman

' data (by comparison to the sign of the electro-optic coefficients") the calculated magnitudes donot agree with experiment.We also make an estimate of the 4.E and E,deformation potentials from the observed depen-

dence of the Eo gap on uniaxial stress along sev-eral directions. "II. EXPERIMENTAL DETAILS

(aA, (z)- a

(0Ei(Y) 04

(0E,(X)- 0

0 0)0 c. ofO c)0 00 0

D O~E&-, 0 d 0

(0 doE,"'- doo

The measurements were performed at roomtemperature on a 2 x 2 && 10 mm ZnO single cry.stalwith all faces polished and the long dimensionalong the c axis. We tried to etch the crystal withphosphoric acid, but such procedure, as well asthe use of cleaved surfaces, resulted, for fre-quencies above the gap, in a strong luminescencewhich largely obliterated our Raman spectra.Since the measurements discussed here are con-fined to a region in which the crystal is trans-parent we do not think etching is necessary. Thex and y axes were perpendicular to two long faces;a distinction between x and y is not necessary forour work. The symmetries of Raman active pho-nons and the corresponding Raman tensors for theC,point group are'


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RESONANT RAMAN SCATTERING IN ZnO 3755'7he A, and E, phonons are infrared active andhence they split into transverse and longitudinalmodes (A, ~,A, r E,~,E,r}. There are thereforetwo independent sets of Raman tensor componentsa, b, and c (a~, ar, b~, br, c~, cr). The scatteringgeometries used in our experiments, the corre-sponding phonons, and the scattering efficienciesS in terms of the parameters of Eq. (1) are (instandard notation)

"'12os

p380(A&v&

541

1160(a)

x(zz)X-A, r; S=b2r1 2 1 2x(zx) y -E,E,I., S= -,c,cz

x(yy)x-A, r, E2; S=a2r, d' .(2)

III. RESULTSTypical spectra obtained with 4880 A (2.54 eV)excitation wavelength are shown in Figs. 2(a)-2(c)for the three configurations of Eq. (2), respective-ly. These spectra agree basically with those re-ported earlier. 3'2' The arrows in Figs. 2(a)-2(c)indicate the position of the phonons and two-phononstructures whose resonance is shown in Figs. 3-5.In addition to the phonons shown in Fig. 2 we ob-

served for 4067-A laser excitation the E phononin the forbidden configuration x(zz)x. This is astandard occurrence for ir-active phonons which isbelieved to be due to intraband Frohlich interac-tion. ' Similarly, we detected for this wavelengtha weak A, z phonon at 574 cm ' in the z(xx)z al-lowed configuration. We were not able to see thisphonon at longer wavelengths.We have also performed some measurementswith the 3638 A (3.41 eV, Ar'), 3507 A (3.53 eV,Kr'), and 3250 A (3.82 eV, HeCd) laser lines. In

The purpose of the present work is to determinethe dependence of a~, b~, cr, ~~, and d on ex-citing wavelength and to interpret this dependencetheoretically. We found no convenient configurationyielding the parameters a~ or b~ (see Refs. 3 and15).The spectra were excited with all the lines ofSpectra Physics models 164, 165, and 185 lasers(Ar', Kr', and He-Cd, respectively} and analyzedby means of a 4-m SPEX double monochromatorwith Jobin-Yvon holographic gratings (1800 lines/mm) and a RCA C-31034A photomultiplier. Thedata were then corrected in the usual way (by com-parison with the scattering from a CaF, crystal)for the throughput function of the spectrometer andthe ~~ factor so as to obtain data which can be re-lated to the square of the transition susceptibilityor Raman tensor. " Corrections for the absorptionand the reflectivity" "were only needed for thedata at 4067 A (3.05 eV); they changed the scat-tering efficiencies by only 20/p.

101(E2)I a I8 437(E2)

II

I i I s I

(b)

ItllXUJZ

I' nI

I' ~

st I

ipl2~!p4os&s, r) I i I i I(c)

se4 (F&

all these cases we observed, like in previousworks, "'"up to six overtones of the E,~ phonon ofintensity decreasing with increasing order and in-dependent of polarization configuration. This lastfact may be due to the poor quality of the polishedsurfaces and the strong absorption coefficient inthis region (&10' cm '). However, we saw in thisregion no traces of the A mode or its overtonesin spite of having seen it for 4067-A excitation. .In Fig. 3 spectra similar to those of Fig. 2(a)are presented for several exciting wavelengths.It is worth noting that the 2E,~ peak is resonantlyenhanced with respect to the rest of the spectrumalready at photon energies -1 eV below E,. Moredetails about the 2E resonance can be found inRefs. 13 and 23.The integrated scatterering efficiencies ob-served for the two E, phonons [d' coefficients of

I I I I I I I I I I I I200 400 600 800 1000 1200FREQUENCY SHIFT tcm-1)

F&G. 2. Raman spectra recorded at room temperaturewith 4880-A excitation wavelength for the xgz)x (a),x(yy)x (b), and x(zx)y (c) configurations. The arrowsindicate the position of the one-phonon and two-phononstructures whose resonance is presented in Fig.s 4.In the shadowed regions of (b) the Raman intensity hasbeen divided by 2.


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8756 J. M. CALLEJA AND MANUEL CARDONA 16&579k------ 466OA47l A

IlAUJIK

FIG. 3. Raman spectrarecorded for several wave-lengths in x(zz)X configura-tion. Note the nearly con-stant A ~&height and thestrong resonance of the2LO region.

WoI

600 800FREQUENCY SMIFT (cm~)

I)200

Eq. (2)j and for the A phonon (a2r coefficient) areplotted in Fig. 4 as a function of photon energy.Absolute values of these scattering coefficients(in units of cm 'sr ') are given; they were ob-

tained by normalizing our data to the absolute scat-tering cross sections measured by Arguello et'al. 'at 2.41 eV for the a~ coefficient of the A phonon.Note, however, that the ~4 factor has been re-ill 107" to-8

atLU

EA63i.s I2.0 I I2.4a tev)

~QLJ

I2.8

LUlhu) 10 9VlE3g-5

ILJIFIG. 4. Absolute Raman efficiency vs exciting fre-quency for theA, z(xx) (O), E2 (437 cm ) g,), and E2

(101 cm ') (0) phonons obtained in x(yy)X configuration.The full symbols (o, k, r) represent data taken fromRef. 15 ~ The results have been normalized to the abso-lute values of the Raman cross section for co=2.41 eVgiven in Ref. 3. Note that the co dependence has beenremoved and thus the absolute values of the cross sec-tion in the sense of Eq. (8) are obtained by multiplyingthe strengths shown in the figure by (u/2. 41)4. Thisalso holds for Figs. 5 and 6. The dashed and solid linesare the least-squares fit to our data, with Eqs. (3) and(4), respectively. The corresponding values of the A,B, A ' and B' coefficients are listed in Table I.

1.6 2.4Q (eV)FIG. 5. Absolute Raman efficiency vs exciting wave-length for Eq +(O) and E&1 (6) phonons in x(zx)y configura-tion and for A &&(zz) (CI) in x(zz)7 configuration. (Notethat the A f p(zz) data are shifted two decades downwardsfor display purposes. ) The dashed and solid lines areagain the fits of Eqs. (3) and (4), respectively. The fullsymbols (o, i, r) are data taken from Ref. '15. Thetriangle marged with an arrow is an upper-limit esti-mate of the E&~ Raman cross section at 1.65 eV.


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16 RESONANT RAMAN SCATTERING IN ZnOour data except for a slightly smaller slope.The dashed lines in Fig. 4 represent a least-squares fit to the experimental scattering effi-ciencies Iwith the expression

VllAVlO

Q 7 I=A Bwhich is based on the dielectric theory (see Sec.IV). The frequency derivative of the dielectricconstant (in eV ') has been obtained by differen-tiating the data of Refs. 24 and 25. The constantB represents a nonresonant contribution to theRaman tensor. The values of A and B requiredfor the fits are given in Table I. We have alsoperformed a fit to the data of Fig. 4 (solid line)with the expression

UJ n10@

1.6 28 2.4A (eV)

I=A'[ -g(x)+B']'with

g(x)=x '[2(lx)' '(1+x) ' 'jand

(4)

FIG. 6. Absolute Raman efficiency vs exciting wave-length for the second-order structures shown in Fig. 2(a)in x(zg)X configuration. The full triangle has been takenfrom Ref. 23. The dashed and solid lines are the least-squares fit tothedata with Eqs. (6) and (7), respectively.The corresponding C, C', D and D' coefficients arelisted in Table I. The ordinate scales of the differentcurves have been shifted for display purposes.moved from the scattering efficiency of Figs. 4-6.The experimental points of Ref. 15, also re-normalized in this manner, have been includedin Fig. 4. They show substantial agreement with

X= (d (do y~, being the frequency of the E, gap. The functiong(x) is one of a family of functions used in the theo-ry of dielectric properties of three-dimensionalenergy bands extending to infinity. '~" It repre-sents the derivative of the dielectric constant withrespect to the energy gap &, under the assumptionof a density of states mass independent of ~,."This function is used often to represent the reso-nance of the first-order Raman tensor near three-dimensional critical points. " It is easy to showwithin the model of parabolic bands that for &close to but below (d, the following equation holds:

TABLE I. Coefficients resulting f'rom the least-squares fit to the experimental Raman crosssections with Eqs. (3), (4), (6), and (7). A and A' have been normalized so as to obtain at2.41 eV the absolute cross sections given in Ref. 3. Units are 10 eV cm sr for A, 10cm ~sr for A' and C', eV for B, 10 eV cm ~sr for C, and eV for D. B' and D' aredimensionless. For the purpose of comparison with B we have tabulated B'+ 7 instead of B'[see Eqs. (4) and (5)].First orderCz2

2Cz2Cgd (437 cm )d (101cm )

1.860.021.451.252.940.07

0.6612.020.280.080.904.46

A'3.850.032.033.036.960.09

B'+ 340.349.310.24-0.150.494.05

Second order C' x102 D'+ 1208 cm332 cm541 cm '1160 cm ~

0.542.152.507.770.31-0.24-0.50-0.22

0.39-1.701.906.3014.176.803.686.63


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3758 J. M. CALLE JA AND MANUEL CARDONA 16d& =-K[@(x)+s]. (5)

While ZnQ has strong excitons at E the parabolicbands expression [Eq. (5)] should be valid in ourexperimental region (x& 0.9) as can be seen byexamining Eqs. (58) and (60) of Ref. 26. The valueof SP evaluated from the mass parameters of ZnQis 0.6,"in reasonable agreement with the ratio ofA and A' shown in Table I.The resonance in the scattering efficiencies ofthe E,z (c2~), E (c'r), and A, r (b'r) phonons isshown in Fig. 5. The absolute cross sections weredetermined by fitting to the measurements of Ref.3 at 2.41 eV. Fits with Eqs. (3) and (4) are alsorepresented in this figure, which includes the dataof Ref. 15, by dashed and solid lines, respectively.The efficiencies of E at 1.65 and 3.05 eV havenot been included in our fit since they are affectedby considerable uncertainty. In fact, the E pointat 1.65 eV, indicated by an arrow, is only an upperlimit for the Raman cross section, since we couldnot detect any signal in that case. Qur data forE clearly indicate the existence of an antireso-nance at -1.6 eV, corresponding to a cancellationof the resonant and nonresonant terms in Eqs. (3)and (4). The A, r mode hardly shows any resonanceat all; the corresponding values of the resonatingterms are zero within the experimental uncertainty.In Fig. 6 we present the resonance behavior ofthe main second-order structures of Fig. 2, the2E structure at 1160 cm ' and the weaker struc-tures at 541, 332, and 208 cm '. Since these fea-tures seem to resonate more strongly than theirfirst-order counterparts (except for the behaviorof the E mode in Fig. 4 which can only be ex-plained as an antiresonance), we have chosen tofit them with the second derivatives of the dielec-tric constant

or, corresponding to Eq. (4),f=C'[(1x) +B'] (7)We have left in Eq. (7) only the most dispersivepart of the derivative of g(x). The parameters C,C', D, and D' required for the fits of Fig. 6 arealso given in Table I.

IV. DISCUSSIONWe have already mentioned that Eqs. (3), (4),(6), and (7), which have been used to fit the exper-imental resonance data, are based on the dielec-tric theory of Raman scattering. This theory con-tains several assumptions:(a) The phonon frequency 0 is always much

smaller than &ov, (e is the laser frequency, ~,that of the E, gap) so that the phonon can be treatedas a quasistatic perturbation. Under this assump-tion it is easy to see that the efficiency for Stokesscattering per unit length and unit solid angle isgiven, in units of sr &bohr x by

BXgg, BXyp Bp BXgp BpBQ& Bcdp BQ; BOO BQ& (9)

The replacement of dye, /d(u, by dye~/d&u in Eq.(9) strictly speaking requires the addition of a lessdispersive function which is omitted since it canbe lumped into the constants B and B'.The dispersionless background B (or equivalent-ly B') is dominant for the A, r mode in (zz) config-uration [5'r coefficient of Eq. (2)]; in this case theresonant term is- nearly zero within the experi-mental error. The background is also large forthe E, mode although in this case a resonant con-tribution clearly exists.We now proceed to give explicit expressions forthe A's of the various Stokes modes in terms of

op((d) = "~ [ (n, + 1 ] ug (n, ) ( ';Cp BQ]where j and k are the directions of the incidentand scattering fields, X the susceptibility tensor,Q, the atomic displacements associated with theith phonon, n its Bose occupation number, andc, the speed of light in vacuum. Equation (8) iswritten in atomic units (i.e. , e = I=m = 1, c,= 137).(b) We assume that ay/du, has two contribu-tions, a strongly dispersive one arising from thelowest gap E, and a contribution from all highergaps, represented by B and B', which is nearlyconstant below E,. The,E, contribution can also besplit into two components. The most strongly dis-persive of these components originates from amodulation of the E, gap (or any of its three com-ponents A,B,C) by the phonon displacement u, .This type of contribution exists for the A (thisphonon modulates the conduction and all the va-lence bands which have E, symmetry). The othertype of contribution is usually less dispersive verynear the gap. It corresponds to changes in thetransition matrix elements due to interband mixingof wave functions. To this type belongs the effectof the E phonon which couples the A, B valencebands with the C band. Both types of effects whichoccur in ZnQ cannot be distinguished if the threevalence bands are degenerate, i.e. , if ~= 0, orequivalently if &~&p& for the experimentalregion of interest. This condition is well fulfilledin our case (~, -&o a 0.3 eV, (uc=0.036 eV) andtherefore we only have one type of Ep contributionwhich can be written in terms of derivatives of Xwith respect to energy gaps:


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16 RESONANT RAMAN SCATTERING IN ZnO 3759

3&3(n(E, r) + 1)c S(u128v2c,oPQ(E, r) S6(E,r) (12)where 8&represents the matrix element of theelectron-phonon interaction between the A and theC valence-band states.For the E, modes an additional complicationarises due to the fact that there are two modes of

electron-phonon, interaction constants or deforma-tion potentials, i.e. , o]' derivatives of the energiesof the various states forming the gap with respectto the phonon displacements u, Using Eqs. (3),(8), and (9), we can write2 (10)

where (u',.)'~' is the zero-point vibration amplitudeof the phonon under consideration and n, the Bose-Einstein factor. For A, phonons we take as u(A, )the change in length of a ZnQ bond parallel to thec axis. It is then customary to introduce the di-mensionless parameter 6(A,),which is the vibrationamplitude divided by the bond length ,c (c and aare the lattice constants). We thus find&3(n(A)+ 1)a' sv, (11)18z'c,pcs(A, r) s6(A, )

where p, is the reduced mass of the Zn and Qatoms. (Note that Eq. (8) has the explicit &o' de-pendence, whereas in Figs. 4-6 and in Eqs. (3)-(7) this dependence has been removed by normal-izing our data to the absolute cross sections givenin Ref. 3 for&=2.41eV. Thus, for using the A's ofTable I in Eqs. (11)-(13), they must be multipliedby (27.2/2. 41)', 27.2 being the factor to convert eVto hartree, and by 5.3 X10 ' in order to convertcm ' to atomic units. ) For the E, mode the vibra-tion is perpendicular to the c axis. We thus useas a normalization for u(E, ) the component of anoblique Zn-Q bond perpendicular to this axis(a/u 3). We obtain

this type. Thus their eigenvectors are not deter-mined by symmetry, but by the solution of the cor-responding dynamical matrix. However, sincethey are well separated in frequency (101 and 437cm ') it is reasonable to assume that the low-fre-quency mode is due exclusively to vibrations ofthe heavy Zn sublattice while that at 437 cm ' in-volves only oxygen vibrations. This fact explainswhy the low-frequency E, mode does not resonateat E,: The vibrations of the zinc sublattice do notmodulate much the A, 8, and C valence-bandstates of ZnQ, composed almost exclusively ofoxygen 2P wave functions. We thus write for theE, (437 cm ') mode the expression

3&3(n(E,) + l)c s~A[E,(437 cm ')]= 32v, c~~&(E ) &6(E ) ~ ( )where 8&represents the matrix element of theelectron-phonon interaction between the A and theB valence bands and M is the, mass of the oxygenatoms. A similar equation could be written forthe E, (101 cm ') mode. The derivatives in Eqs.(11)-(13)shall be referred to as the deformationpotentials d, (A,), d, (E,), and d, (E,), respectively.The values of these deformation potentials obtainedfrom the A's of Table I with Eqs. (11)-(13)arelisted in the first column of Table II. There aretwo deformation potentials for Aone in the (&x)configuration corresponding to a2r of Eq. (2) andanother for the (zz) configuration corresponding toAs already expected, d, [A, (zz)] =0. It is notpossible to determine from our data for transversephonons the sign of do. This sign can be determinedfor the ir-active phonons by analyzing the strengthof the corresponding longitudinal modes. ",In fact,the A's for these modes contain instead of ~d, ~'the square of a linear combination of d, and thecorresponding electro-optic coefficient d, z~. Sincethe signs and magnitudes of these d,- areknown, ""one can in this manner determine the

TABLE II. Deformation potentials for ZnO in eV corresponding to the A fp (+x), A fp(),& ~z, and &2 phonons. The first column contains the values obtained from the A's of Table Iand Eqs. (11)13). The signs have been evaluated (Ref. 15) from the electro-optic 'coeffi-cients given in Ref. 16 for the A~& and E&z phonons. The values obtained from Eq. (14) andthe results of the pseudopotential calculations are listed in the second and third columns, re-spectively.From resonantRaman work From piezo-optical work Pseudopotentialcalculation

A g~(xx)A (~(zz)EirE2 (437 cm )E2 (101 cm )

+0.730.07.660.540.05

+1.84-0.02-2.12+1.2+1.41.39

~ ~ ~


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3760 J. M. CALLKJA AND MANUEL CARDONA 160.2

0.1-d, [A,(xx)]=-(C,+ C,)+ 2"(C,+C,) C,",S33 3S

0.0d [A, (zz)]=(C + C, C,",2Sxs 133 33 (14)

~ -0.1-lK0 -0.2-

-0.3

ySA

-Ol0

G2 {bohr-2}FIG. 7. Pseudopotential form factors v and v (inrydbergs) for ZnO obtained from Ref. 35, vs the square

of the reciprocal-lattice vector (Q).

signs of d, (A,) and d, (E,). The signs so deter-mined" are given with the experimental data inthe first column of Table G. We have made an at-tempt to calculate d, [A, (xx)] and d, [A, (zz)] with thepseudopotential method. ' This calculation is par-ticularly simple since the A, phonons preserve thesymmetry of the crystal: this is not the case forthe E, and E, phonons. The results of this calcu-lation, performed with the pseudopotential formfactors of Fig. 7, are given in the third column ofTable II. The calculation predicts the signs of d,observed experimentally, but fails to reproducethe nearly vanishing d, [A,(zz)]. We believe this isdue to inadequacies with a local pseudopotentialfor the Q' ion. This ion has no p-core states andhence the corresponding pseudopotential cannot besmooth near the core. We have thus made an at-tempt to relate the dp s to the deformation poten-tials C.. . , C6 obtained in Refs. 17 and 28 forZnQ under uniaxial stress. A connection betweenboth types of deformation potentials can be madeby noting that upon application of a uniaxial stressthe unit cell of wurtzite does not deform uniformlybut there are additional relative displacements ofthe various atoms. Hence the effect of the stresscan b6 decomposed into a uniform deformation ofthe unit cell plus various phonons producing therelative displacements. " It has been noted" thatfor zinc-blende-type crystals the effect of a [111)stress on the valence bands at I' is produced ex-clusively by the phonon component. By making thesame assumption for ZnO and assuming also tNatthe bonds do not change length upon uniaxial de-formation, 2' it is possible to relate the C, 's to the

do(E, ) = -3C /2V 2,do(E.) = C5

where Sand S are the elastic compliance con-stants and C~, C" the hydrostatic stress deforma-tion potentials for the (A, B) and C excitons, re-spectively, taken. from data of Ref. 34.The values of the d, 's obtained from Refs. 17and 28 with Eqs. (14) are listed in the second col-umn of Table II. Their signs agree with the ex-perimentally determined ones. The d, [A, (zz)] socalculated is very small, in agreement with theexperimental value. The other dp s so obtainedare larger than the experimental ones. In spite ofthe crudeness of our assumptions this raises thepossibility of a systematic error in the absoluteefficiency measurements of Ref. 3. Much betteragreement with the various deformation potentialcalculations would be obtained if the measured ef-ficiencies were about one order of magnitudesmaller than the true ones. This possibility issupported by the fact that the magnitudes of theelectro-optic coefficients determined from thescattering efficiences in Ref. 3 are also about 2times smaller than the directly measured ones.The antiresonance of the E mode at 1.6 eV,and the corresponding negative value of B, mustbe due to the electro-optic contribution as sug-gested in Ref. 15, since it does not happen for theE phonon. We note that in other wurtzite-typematerials (CdS, ZnS) antiresonances have beenobserved for the transverse modes. '"" It is re-markable that no such antiresonances occur inZnO. We point out that the elastro-optic constantsof CdS exhibit an antiresonance as the gap is ap-proached, with a large contribution of the back-ground. " The same constants also have an anti-resonance in ZnO, but with a much smaller back-ground contribution. This reflects the same trendas the disappearance of antiresonances in the

Ey p and E, modes in going from CdS to ZnO.The second-order resonances of Fig. 6 have beenfitted with the second derivative of the dielectricconstant according to Eqs. (6) and (7). There issome arbitrariness in this choice: The structureat 208 cm ' can certainly be fitted equally well withEq. (3). The 2E resonance, however, canbefit-ted better with Eq. (6) and thus we assume it is theresult of an iterated process involving the Froh-lich interaction. " Two iterated first-order pro-cesses would also have to be involved for the othe~


9/9

16 RESONANT RAMAN SCATTERING IN ZnO 3761second-order structures if the choice of a secondderivative fit proved to be of the essence. A firstderivative fit would imply a process involvingone single- electron-two-phonon vertex. "Finally, by looking at Fig. 1 we can. make atentative assignment for the second-order struc-tures whose resonance is presented in Fig. 6. Thestructure at 208 cm ' corresponds to the frequencyof 2E, (101 cm ') at I': as shown in Fig. 1 the dis-persion relation is very flat around this point.This structure could also contain a contribution ofthe lower TA branch around the M point; the cor-

responding E-space multiplicity would enhancethis contribution. The peak at 332 cm ' should beascribed to two phonons from the K-M-Z around160 cm '. Both structures just mentioned are seenin (xx) and (zz) configurations. The structure

around 541 cm ' is stronger in the (zz) configura-tion and corresponds to phonons from the regionbetween the 1 and M points around 270 cm '. Thepeak at -1160 cm ' is known. to arise from two I,Qzone- center phonons.Note added in proof. We have recently mea-sured (M. H. Grimsditch, private communication)the A(zz) mode of ZnO with respect to diamond(6.1x10 ' cm ' sterr ', M. H. Grimsditch andA. K. Ramdas, Phys. Rev. Bll, 3139 (1975)) andconfirmed the original values of Ref. 3. Hence thediscrepancy between the two first columns of TableII cannot be attributed to an error in the deter-mination of the absolute cross section. We feel theproblem must lie in the oversimplification involvedin attributing piezooptical deformation potentialssolely to their phonon component.

*Supported in part by the Alexander von Humboldt Foun-dation, on leave from the University Autonoma deMadrid.W. L. Roth, in The Physics and Chemistry of II-VICompounds, edited by M. Aven and J. S. Prener (North-Holland, Amsterdam, 1967), p. 122.J. Birman, Phys. Rev. 115, 1493 (1959).3C. A. Arguello, D. L. Rousseau, and S. P. S. Porto,Phys. Rev. 181, 1351 (1969)~A. W. Hewot, Solid State Commun. 8, 187 (1970).R. Loudon, Adv. Phys. 13, 423 (1964).K. Thoma, B.Dorner, G. Duesing, and W. Wegener,Solid State Commun. 15, 1111 (1974).U. Rossler, Phys. Rev. 184, 733 (1969).S. Bloom and I. Ortenburger, Phys. Status Solidi B 58,561 (1973).M. Cardona, Modulation Spectroscopy (Academic, NewYork, 1969), p. 255.' J. O. Dimmock, in II-VI Semiconducting Compounds,edited by D. G. Thomas (Benjamin, New York, 1967),p. 277.See B. Segall, in Ref. 1, p. 67.See, for instance, M. Cardona, Light Scattering inSolids, (Springer-Verlag, Berlin, 1975); andW. Richter, in Springer Tracts in Modem Physics,(Springer, Berlin, 1976), Vol. 78.Y. Oka and T. Kushida, J. Phys. Soc. Jpn. 33, 1372(1972).' R. Zeyher, Phys. Rev. B 9, 4439 (1974)~5R. H. Callender, S. S. Sussman, M. Selders, and R. K.Chang, Phys. Rev. 7, 3788 (1973).'6R. C. Miller and W. A. Nordland, Phys. Rev. 132,4896 (1970).J. E. Rowe, M. Cardona, and F. H. Pollak, Solid StateCommun. 6, 239 (1968).J. L. Freeouf, Phys. Rev. B 7, 3820- (1973).

D. G. Thomas, J. phys. Chem. Sobds 15, 86 (1960).R. E. Dietz, J. J. Hopfield, and D. G. Thomas, J. Appl.Phys. 32, 2282 (1961).'T. C. Damen, S. P. S. Porto, and B. Tell, Phys. Rev.142, 570 (1966).J. Scott, Phys. Rev. B 2, 1209 (1970).T. C. Damen, R. C. C. Leite, and J. Shah, in Pro-ceedings of the Tenth International Conference on thephysics of Semiconductors, Cambridge, Mass. , 1970(unpublished) .E.Mollwo, Z. Phys. 6, 257 (1954).25Y. S. Park and J.R. Schneider, J. A@pl. Phys. 39,3049 (1968).P. Y. Yu and M. Cardona, J. Phys. Chem. Solids 34,29 (1973).VM. Cardona, in Atomic Structure and Properties ofSolids, edited by E. Burstein (Aoademic, New York,1972), p. 514.D. W. Langer, R. N. Euwema, K. Era, and T. Koda,Phys. Rev. B 2, 4005 (1970).

2 R. Martin, Phys. Rev. B 1, 4005 (1970).3 J.M. Ralston, R. L. Wadsack, and R. K. Chang, Phys.Rev. Lett. 25, 814 (1970).S~R. H. Callender, M. Balkanski, and J. L. Birman, inLight Scattering in Solids, edited by M. Balkanski(Flammarion, Paris, 1971), p. 40: J. L. Lewis, R. L.Wadsack, and R. K. Chang, ibid. , p. 41.See for instance, R. L. Schmidt and M. Cardona,Phys. Rev. B 11, 746 (1975).T. B.Bateman, J. Appl. Phys. 33, 3309 (1962).34R. L. Knell and D. W. Langer, Phys. Lett. 21, 370(1966).V. Heine and M. L. Cohen, in Solid State Physics,edited by F. Seitz, D. Turbull, and H. Ehrenreich(Academic, New York, 1970), Vol. 24, p. 1.

raman effect in zno

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