THE UN IV ER SIT Y OF MI C I GAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Final Report - Part 2 ESR RESULTS OF ZnSe AND OTHER HEXAGONAL AII BVI COMPOUNDS Go Hossein Azarbayejani ORA Project 04385 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. G-15912 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR June 1963

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TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vii ABSTRACT 1 CHAPTER I. INTRODUCTION II. RESULTS 3 A. Experimental 3 B. Theoretical 6 C. Determination of g, A, D, a, and F 21 III. DISCUSSION 23 REFERENCES 30 iii

i I I i i i i i I 1,

LIST OF TABLES Table Page I. Absorption Peak Positions (in gauss) of the Lines Designated in Figs. 1 and 2. 6 II. Mn++ ESR Parameters in Hexagonal AII BVI Compounds. 23 III. Crystalline Parameters of AII BVI Compounds. 25 V~~~~~~2

LIST OF FIGURES Figure Page la-4a. ESR Spectra of Mn+ in the Plane (1010) of ZnSe. Q is the 4 angle between c-axis and H. 4 lb-4b. ESR Spectra of Mn++ in the Plane (1010) of ZnSe. 0 is the 5 angle between c-axis and H. (M = -1/2 -+ +1/2 transitions; signal level is reduced by a factor of 3.) 5 5. Relative positions of the magnetic (H Polar Axis) and Crystalline coordinate systems. 8 6. Change of the axis of quantization (z-axis) from [100] to [111] direction and the result of C2 operation on xyz coordinate system. 10 7. Ideal hexagonal packing. 26 8. Schematic representation of surrounding ligands around the Mn++ ion in ZnSe:Mn. 28 vii

ABSTRACT ESR experiments on ZnSe:Mn have been carried out at room temperature with X-, Ku-, and K-band microwave spectrometers. Parameters g and A are isotropic, with values of 2.0055 + 5 and -61.2 +.5 x 10-4 cm-1 respectively. The following values were obtained for parameters D, a, and F: D = 425.1 x 10-4 cm-l; a = 17.66 x 10-4 cm-1; and F = 6.61 x 10-4 cm-1. I. INTRODUCTION It was pointed out previously1'2 that the Watanabe3 theory regarding the cubic crystalline factor a holds satisfactorily in the case of oxides of Mg and Ca. In the case of sulfides, selenides, and tellurides, however, the prediction from this theory is not in agreement with experimental results. The disagreement is due to the covalency nature of the bonds, which is more pronounced in the heavier compounds of the two elements zinc and cadmium. Since knowledge of the sign, magnitude, aind derivation of D from one-electron orbitals aids in the evaluation of a, it is necessary to find theoretical expressions for D first. Therefore, the experimental results obtained by the author and other investigators concerning hexagonal AII BVI compounds are given in Section II. In Section III, a theoretical discussion of the variation of the D parameter among the AII BVI compounds with hexagonal structure is given.

I I I i i i I i i i i T i I i I i i I I i i i I i

II. RESULTS A. EXPERIMENTAL Pieces of ZnSe crystals were examined in an ESR system and the microwave absorption was detected by a phase-sensitive detector. The derivative of the paramagnetic absorption with respect to the dc magnetic field, dx/dH, was obtained through a Varian detection and recorder system. The magnetic field associated with these resonances was measured by a Varian fluxmeter and a Beckman transfer oscillator. The resonance spectra obtained through the graphic recorder are shown in Figs. 1-4. The magnetic field was rotated in a (101 0) plane. Figures la and lb give the spectrum at Q = O0 where the dc magnetic field is parallel to the c-axis of the crystal. Figures 2a and 2b give the spectrum at 0 = 25~ to the caxis; Figs. 3a and 3b at 0 = 55~; and Figs. 4a and 4b at 0 = 90~. A comparison of these spectra indicates that the crystal is at best a twin, which is obtained by dendritic growth of ZnSe. That such a phenomenon occurs in crystals grown from melt has been observed in the growth 4 of germanium from melt. Due to the presence of these imperfections, the spectra are more complicated in this ZnSe than in a pure hexagonal crystal such as ZnO. Cuceaneau5 reports the simultaneous presence of hexagonal and cubic structure in ZnSe. Bube, in his detailed investigation of the optical properties of ZnSe, also reports the presence of both cubic and hexagonal structures. According to these investigations,

+ I 4 L < XIrI F+III Ilr - r l::i::::7I:1:1-::I:::::t:::iI-I.I'.........:......: 1:.. i i~i..i......!. _/....!..1111 —J..:~ J..;.ii~~~~~~~~~~~!~ (~~..../~~~~..-~.I —'=. I............ I Ll -....'...........................'1 1 -r TI Jr ii.I;jJ Ii W~ _i_.~......: o 0i..-l~~ -..!..s~~~I....I....I:...l.:.~!1. ~ 1, i~, 0i~ I~ so 2 - I I I 3. 1.... -_'! i J.. lrl:::|:: I: i...... d I I III Ir'' 1'' 11 1 | I::::::::::: II, ~~1.:~ 1''i J I J+,...,............'.'''''''' f i~i!~!~, f cli i... I I:' [; _.._..._._.._ 1:.4

-=0 Fig. I-b 0=25 Fig. 2-b 8=55 Fig. 3-b 8=90 —' i";: — - ---— 1 -. —-.- —'. —---- - - --. y 6 - 7 ~-I"- iA...... C Fig.4-b Figs. lb-4b. ESR Spectra of Mn++ in the Plane (1010) of ZnSe. 0 is the angle between c-axis and H. (M = -1/2 + +1/2 transitions; signal level is reduced by a factor of 3.)

it is possible that the complexity of the central portion of the spectra shown in Figs. 1-4 is caused by the presence of microcrystallites of cubic structure. Comparing the spectra in Figs. la and 3a, one finds at least one spectrum behaving as though there were a hexagonal crystal, one of whose (101 0) planes is horizontal. The measurements were carried out with respect to this set of spectra and the results for the two orientations @ = 0 and 0 = 90~ are given in Table I. TABLE I ABSORPTION PEAK POSITIONS OF THE LINES DESIGNATED IN FIGS. 1 AND 4 (in gauss) 9 position a Y A High Side 6920.70 7861.58 8745.66 9639.99 10577.08 8591.50 Low Side 6587.51 7531.67 8420.58 9315.24 10255.68 - Diff = A 333.19 329.96 325.08 324.75 321.22 - Average = a 6754.10 7696.60 8583.12 9477.61 10426.29 - High Side - - 8745.22 - 9691.35 8590.41 Low Side 7549.25 - 8419.61 - 9347.71 - A - - 325.59 - 343.97 - a - - 8582.40 - 9519.69 - B. THEORETICAL To analyze the data given in Table I the usual procedure 3 is to express the crystalline electric field as a function of the spin operators. This is done simply by transforming the: terms of the crystalline field from the crystalline coordinate system to the magnetic coordinate system

and then substituting the spherical harmonics in the new magnetic system by Racah's TT tensors. These tensors are functions of spin operators, and tile coefficients of transformation from the crystalline to the magnetic coordinate system are functions of the Eulerian angles a, A, and y through which this transfornmation takes place. By following this procedure one can obtain expressions which are easy to compare with experimental data. To illustrate this procedure consider Fig. 5. The MA+ ion, which is surrounded by four Se ions, is at the origin. The three Se ions, 2, 3, and 4 are equidistant from the Mn ion when the Se ion 1 is slightly further from it (R1>R2). At the Mn ion site this produces a local field of the trigonal symmetry with OA, the axis of threefold symmetry. We take as the polar axis and express the crystalline field terms in such a manner that this axis represents the z-axis of the crystalline coordinate system. The crystalline field V'(r,O,4) produced at Mn++ site by the neighboring ions can be expressed as a linear combination of the spherical harmonics provided V2V' = 0: V(r,,4) = jA Y (G,+). (1) ~m For the case of d-electrons of the Mn++ ion (the trigonal axis being the polar axis),, the effective field V, when it meets the requirements of crystal symmetry,* can be obtained from Eq. (1) by retaining only those *These requirements are that the fields by (1) real, (.2) invariant under rotation + 2</3 about the c-axis, and (3) invariant under rotation </3 followed by reflection in a plan normal to the c-axis.

Fig. 5. Relative positions of the magnetic (H polar axis) and crystalline coordinate systems. terms of even 2 up to 2 = 4, with ImI_< Land mbeing 0 or + 3. Therefore V(rG,4) = A Y + A Y + A Y + A Y + A Y' (2) 00 00 20 20 40 40 43 43 4-.3 4-3 where A's are independent of 0 and 4 and are numerical constants multiQm plied by r2. Since the term Aoo00 YOO shifts all of the spin levels by the same amount, it does not produce any change in the relative energies of the levels. The coefficients A are related to each other. To find 4~3 this relation, use will be made of symmetry operations on the xyz coordin

ate system. Before doing so we write Eq. (2) as a sum of two axial and cubic terms V and V a c V(rQ,4) —V(xyz) = V a(xyz) + v (xyz), (3) V (xyz) = A Y +A Y + A' Y, (4) a oo00 00 20 20 40 40 v (xyz) = A" Y +A Y +A Y (5) VC~xYZ = 40 40 43 43 4-3 4-3 where7 Y40(xyz) (=35Z4 30Z2r2+3r4)/8r4, (6) Y (xyz) =; (y Y43(xyz) z5 (x+iy)3/r4 (7) Substituting for Y40 and Y4~3 in Eq. (5) and operating with C~ (face diagonal = yy'), we find: C.Vc(xyz) = Vc(-x,y,-z). (8) Since Ck is a symmetry axis of the cube (Fig. 6), Vc(xyz) = Vc(-xy,-z), resulting in A43 = - A4_3 (9) or Vc(xyz) = A4Y40 + A43(Y43-Y4-3). (10) There is also a numerical relation between A40 and A43, which can be

Ca y(C2) mI Fig. 6. Change of the axis of quantization (z-axis) from [100i to [111] direction and the result of C2 operation on xyz. obtained by operating on Vc(xyz) with C2 axis of symmetry (Fig. 6). This operation brings the xyz system into the x'y'z' system. A close inspection of Fig. 6 reveals that this operation is equivalent to the initially orthogonal rotations = /2, cos'(-l/3), and 4 = 3i/2, around the z, the a, and the z' axes respectively. Therefore 10

x, /cos* sin* oC 0 0 o s/*o sin$ 0 x -sin* cos 0 0J cosO sing (-sinS cos$ O cosxI sin [ Y 0* z 0 1 -sing cos/ \ 0 1 = ( 0 (0 -1/ 247 1 o (11) o 1/ \0 237 -1/3 / 1 /-1/3 0 2o/ x \22 3 0 12/3 /x) Since C2 is a symmetry axis, as in the case of C~ we have C2Vc = Vc. Substituting for Ye's in V (x,y,z) from Eqs. (6-7) and employing the transformation represented by Eq. (11), one obtains: r-4A40 8 & L 5Z4 3 2 3+r4 3- (A43/A 40) 1z(x+ly) 3+z(X-ly) 3] ~-4x 2\ 4~ 4 2\ 40352' I — - 3- r + 3r4 (12),,:~- ~2x 2 x - (A43/A40) 16 +, _ iy +e + + -iy). Equation (12) holds if the coefficients of each power of x, y, or z are the same on both sides. We will take the coefficients of z: 1 3-0,~5~-3o+1) (5 __ r 2/22. 5303 - +5) - (A43/A40 - A43/AJo) (13) 410/7. Thus, the cubic part of the crystalline field V (x,y,z) is c+ = A40[Y40 77 (43-Y4-3)]. (14) The Vc belongs to the field produced on those Mn+ ions which are surrounded by tetrahedrons of ligands represented by,~, (Fig. 3). 11

We must now find the matrix elements of these V and V between the a cmagnetic states of the Mn++ ion. This can be done by transforming Va and V into a magnetic coordinate system where H, the dc magnetic field, is considered the z-axiz. The transformation of Va and Vc~ involves the transformation of the spherical harmonics Ym. These spherical harmonics transform like the monomials m T /~(+m).!(A-m)!J where 5 and r, under a rotation of the coordinate system through the Eulerian angles a, A, and y, transform into i' and i' as follows*t (* a* (, (15) where a = e i/2 cos e/2 b = -e/2 si iy/2 6 The coefficients of transformation of Y are ~m m'm (Q-m'-A): ( +m-k) t (!A+m'-m) ) im'o 22+m-m'-2 sin -m+m'+2 (16) (x) e cos 1sin (x) Fimy Considering the axial nature of the spectra (Figs. la-4a), we obtain D(2) moo for m' = -2,-1,0,1, and 2 for Yam terms, whereas for Y4m terms we find only the terms corresponding to D(4) and D(4) It is necessary to note 00 03 *Do not confuse 5, i, 5' and r' with the Act and ~'~'5' above. 12

that the angle y in Wigner's notation is similar to + of the Goldstein + rt/2. For the case where the magnetic field is in one of the planes parallel to the c-axis we can take a = o. Therefore we find only Dm m(o,),y), as follows: (2) 2.,Af 2=m )! (2+m, 4-m'-2)(m\ D2)(o,Z,) = Z (-1) cos m'o A (2-m' - )' 2)!(2+m') 2 (17) (x) sinm +2\(). (17) Therefore (o,2,C) o 2 I D(2)(oo) = 4 Ccos cos sin + ~ sin = os) - sin c ( sin cos 1 2 = -(3 cos-l), 2 (2) ____ ___ ____ __J_ _ 3-22\ 1+2?\ D(O,~,Y) Z (-li 1-)::!l(t cos 2 sin 2 10 _ l 2-)'2 2 O= 6C o sin - cos sin (19) NT6 = 4 sin P cos P, and (2) _ _ __4 2-2?\1\ 2+ D20 P'Y) C (-1) cos sin ) (20) ~~~~(R~~~~~~~~~~~)~(o) Since D (o,P,y) coefficients are real, it is obvious that m' m D(~) ( o,) = (-1)m' D()(o,f,y), (21) -m'm m'm 13

giving D10(o,2yY) = -6 sin D cos P (22) and (2) 1 2(2 D(20(o,By) = - sin P (23) For transforming the remaining terms, it is necessary to find D(oo and 00 (4) 03 (4) (42 D(4o(o,,:) = Z (-1) -2____2(\)2 cos 2 sin - [(_4-\),]q(:)2o 2 2 8: (4!)2 6 2 f M!42 4 sin4 C cos - 4 Cos -sin- + cos - sin P 2 (3!)2 2 2 (2X)4 2 2 (4!)2 2 6, 8 =o- (-) cos - sin- + sin (3:)2 2 2 2 (24) = (cos - - sin - 3 cos(-cos ) (24) \~os" 2 2 + 3(1-2 cos2%+cos%4) -= [8 cos4~+24 cos4p-24cos2p ~8 - B~ + 3-6cos2 +3 cos+3 os4] = [35 os4-30 cos2+3], and (4) 2 _ _1-2_\ -5 (i57 03 D3 (4-A)J!(7-A) )A!(-3 (3) cos F 1 Eji1P =-4! 47 cos:' sin 3ei3 - cos - sin5 ei37 (25) l43! 2 2 4:3! 2 2 3 i7 3.-. e sin B cos B. 8.3! It is evident that 14

(4) + -7. e sin P cos A. (26) 0-3 8. Considering the Eqs. (3-5) and substituting for Y20, Y40o and Y4~3, one finds for the transformed potential in the magnetic coordinate system 2 V( +' A0 7D(2)(o,~,r) Y2m,(2, $) + A40Do(4(o,B,y) Y40(@9 $') + Ao[Doo) (o,,7) (27) + D(4) (4)(o,B,) (Y43(',P') -Y4- 3(e','))]. 7 03 The electric field represented by Eq. (27) does not predict any splitting of the magnetic levels of Mn+ ions; but a splitting of the levels does exist (see Fig. la-4a), and Eq. (27) must be modified to include the spin 9 operators. This has been done by Bleaney and Stevens by a method similar to the one they used for substitution of Y~+m(9',$') with L+ operators. 10 An alternative procedure is suggested by Kikuchi and Matarrese: instead of writing the Ym (@',4') as a function of x, y, and z [see Eqs. (6-7)] and then substituting x, y, znd z with appropriate linear combinations of 2 s5, Sy, and sz as shown elsewhere, one can substitute those Y functions 11 with Racah's T tensors. These tensors are generated by the process -1/2 Tkl = [k*2-(-.l). [S-, Tkq. (28) To obtain all terms of Eq. (27) as functions of the spin operators, we have to define T22 and T44. According to Kikuchi and Matarrese they can be taken as: 15

T2~2 - 61/2s+, T4+4 = (16) 4J70 S+. (29) Substituting from Eqs. (28-29) for the Y~'s of Eq. (27), and substituting from Eqs. (18-26) for the D(~)(a,y) of Eq. (16), one obtains VI (r,9,$,i 7) - AooYoo = TA20 [(3 cos2-l)T2o+9l sin P cos f (x) (T21-Ta2- +,2 sin2 (T22+T2_2) + - A4o (x) (35 cos4@-30 cos2 @+3)T40 + 8ltA40 (x) [(35 cos4P-30 cos2P+3)T4~+8 i-0 3 sin3k 7 (x) cos f (e3iYT43-e-3iYT4-_3) (30) where the it parameters are numerical constants arising from the substitution of Tm'.s for Ym's. The energy associated with each magnetic level M, m of the Mn++ ion to the second-order perturbation can be found from the Hamiltonian'c-= gPS_'H + AI-S' + V' - AooYoo - gN\IH'I (31) and is f h E (,') = E + EMm f(O) f(l) f(2) = Mm + EMm + EMm (32) h(l) h(2) + EMro + EMm where f and h denote the fine and hyperfine structure and M and m the electron and nuclear spin quantum numbers respectively. For transitions due to AM = + 1 and Am = O corresponding to [EMm(,ry)-EM lm(,Y) ]/gS, we 16

will find*: 1 D2 H M = + _ - + m = Ho+ DF1 + 2pa + F - 32-F2 H H + +2m,,7+71 = H0 +{4DFi + 2p'a + D2 A2 _r 6 A F 2 3 + H- F - Am - m+ 4m + C; H+ ~1 H0 f H +pa 5 q} H0 H.2Ho 3D2 _ -Hf - A2m 231- 2Ho - m + 4m + El _ 2 HMM H ==H+ 2p (6 2 2 2 24 3 Ho (34) here D = tA20 is the so-called axial fieldm splitting of second d2megree 4 Ho 2Ho4 + C2; 3 Hrelated to parameter associated with - 2 24 + D2 -F - Am..... ma + 2m + ca; Ho 4 Ho 2Ho and 16D 2 2D2 M =H+ -v -mF2 - F3 + E:3y (36) - 2 o Ho Ho (I = 2); F is the axial field splitting parameter of fourth degree (~ = 4) related to A40 [Eq. (30)]; a is the cubic field parameter associated with Alo; F1 corresponds to the first-order perturbation and F2 and F3 to the *Do not confuse,y with f in g$, which is Bohr magneton. 17

second-order perturbation; and p, p', and g are related to i'A' and HO r"AH". These parameters can be obtained from Eq. (30) as follows: HO F1 = 1(3 cos2B-l), (37) 2 2 F2 = sin a cos P, (38) F3 = sin4, (39) g = 35 cosr -3i cos2 +3, (40) pp = - -+ sin3P cos 3y (41) 12 12 The coefficients EcP E2, and cs are derived elsewhere and have the values a2 a2fH(l~7D}./Ho (42) c~ = a2~-5-$(1-75)j/Ho~ = -a2 (5(3+1784-62542)/48)/H0, (43) 3 = a2(lo(7-25d)/3 /Ho, (44) where 22 222 22 = 2m2 + mn + n2, (45) A, m, n being the direction cosines of H referred to Srl axes of the cubic field. [See Fig. 5 and recall that a,,y here are not necessarily the same as those in D (ay)]. Comparison of Figs. 1-4 reveals that the magnetic field lies in the (1010) plane of the hexagon; thus y = c/6 and y'= 77c/6 yields p = p' = -q/12. For this special case one has (putting 18

= O) H IM =+ < 2 3 = Ho + i{2D(32cos o-1) 6a-F (x) (35 cos49-30 cos2@+3) - 32D2 cos2@ sin2g (46) Ho D2 A2 (35 2 + sin4 - Am - - - m 4m + El, H M4 = + 2 > ~ 2m = O,r = = Ho $ D(3 cos2-l) + -(a-F) L ~2 0 L 422 (x 55 os- 46 cos2+ H CO2 sin2g - sin44 - mA ~~~~6 J ~~H0 4 H0~ ~ (47) 2H0 -3 m2 + 2m)+ 2, 2Ho 4 16D212D2 4 H[LM = - - -,m,~ = Or =] = H0 + 16D2 sin2Q cos2Q sin4 2~ 2 11, 6 H0 Ho / \ (48) mA A 7 - m2 + E3 2 \4 For 0 = 0, H M =3 - 5- = m -H0 = -4D + - (a-F) L 2 2 3 A2Ho~~ (4 ~ ~+ 4? +(49) -Am- A2 5 2 + 4m + ~E H[M 12 0, Ho 2D (a-F) ~2 21 3 / 19~~~~~~(50) -Am A2 I m2 + 2m + E2)

H M = - 1,0 = O,m - Ho = 2D + (a-F) 2 LAAm 2 (52) -Am- - - m2 2m + c2, 2Ho 4 and [H 2 2, om] - (a-F) A2 2 A2( (ss) - Am- 2H 4 m2 4m + e1. In all the AII BVI compounds of sphalerite and Wurtzite structures studied so far, A has been found to be negative. Therefore the low side and high side fine-structure components such as LS. HS... LS and HS..... (see Figs. la-4a) are related to m = -5/2 and m = +5/2 respectively. It is also wellknown that the relative intensities of the resonances corresponding to M = ~5/2 4-> ~:+3/2, +3/2 <-> +1/2 and -1/2 + 1/2 transitions vary as 5:8:9. Thus the transitions a...... 6 and 61.....'6 (see Figs. la-4a) represent the M = +5/2 <-> + 3/2 transitions. Consider ing Eqs. (49) and (51), one has A (+-5,-)= H (+ 25= + -H + - m = + H + > + =-) - 5A + 10A2/Ho. Equation (54) indicates that A for -5/2 + -3/2 transitions should be greater than A for 3/2 -* 5/2 transitions. Comparing the data in Table I and Eq. (54), we find that a-, -, y-, 6-, and A- absorptions in Fig. la correspond to M = -5/2 + -3/2, -3/2 - -1/2, -1/2 - 1/2, 1/2 + 3/2, and 20

3/2 + 5/2 transition respectively, and that the individual lines such as al..... in each group correspond to nuclear magnetic quantum numbers ranging from m = -5/2 for al to m = +5/2 for oc. C. DETERMINATION OF g, A, D, a, and F To compute g [Eq. (30)], a strong set of six lines corresponding to y transitions (Fig. la) has been considered. This set is believed to belong to the cubic crystallites inside the crystal. The g factors at 0 = 0 and 0 = 90~ are obtained through hydrazyl with the magnetic probe being located close to the bottom of k-band cylindrical cavity: gll( = 0) = 2.0056 (55) g~( = 90g) = 2.0055 A comparison with a, I, 6, and A transitions reveals that the g corresponding to the hexagonal components is slightly smaller, and that at 0 = 0 we have h g11(o = 0) = 2.0050 (56) The computation of A has been carried'out through the relation 1A1 = 1( 1 (Y6 - Y1) with the following results: A = - 65.09 gauss or - 60.86x10-4 cm-m c(57) All1 = - 65.02 gauss or - 60.78x10-4 cm1, 21

and Ahl = - 65.44 gauss or - 61.18x10-4 cm-1. Considering Eqs. (49-53) and Table I, one finds: - 8D+ 8(a-F) = 3672.19 gauss 3 (58) 4D - 10(a-F) = 1781.01 gauss, 3 giving D = - 455.03 gauss or - 425.1x10-4 cm-1. (59) Equations (58) give: a-F = 11.8 gauss or 11.05x10-4 cm-1. It is found13 that a = 17.66x10-4 cm-1; (60) therefore F = + 6.61xl0-4 cm-l. (61) 22

III. DISCUSSION In Table II the parameters g, A, D, a, and F corresponding to the hexagonal AII BVI compounds ZnO, ZnS, ZnSe, CdS, and CdSe are given. TABLE II Mn++ ESR PARAMETERS IN HEXAGONAL AII BVI COMPOUNDS Crystal g AxlO4cm-1 DxlO4cm' axlO4cm' FxlO4cm1 T(OK) Ref. ZnO 2.0016+6 -76.0+.4 -216.9+2.2 +6~1.5* - 77 a ZnS 2.0016+1 -651 -105+2 -7.61** b ZnSe 2.0050~+5 -61.2+5 -425.1+1 +17.66 +6.61 300 c CdS 2.0029+6 -65.3+4 +8.2+2.2 +4.2~1.5* 300 a CdSe 2.0042~10 -62.7+. 5 +15.2~.5 +14.3+1 -2+1 77~ d *In ZnO and CdS these a factors should be considered a-F **F here is actually a-F References: a. P. Dorain, Phys. Rev. 112, 1058 (1958). b. Keiler et al., Phys. Rev. 110, 850 (1958). c. This paper. d. Reuben Title, Phys. Rev. 130, 17 (1963). It is very interesting that D is negative for Zn compounds but is positive for Cd compounds. There are two different mechanisms contributing to the D parameter. One involves the spin orbit interaction to second order and the axial field of a single d-electron and the cubic field to fourth order. The other mechanism mixes some 3d44s configuration with 3d5 configuration. These mechanisms give rise to the expression 23

< r2> + < r2 2 D = D1 + D2 = 3 (62) where al and a2 are constants depending on i, the spin orbit coupling factor, and the difference of Mn++ orbital energy levels. According to Watanabe,l4 D1 is more pronounced than D2; therefore we can assume that in an ionic picture, the D parameter in different compounds varies as R3 (R is inter-ionic distance) provided < r2 > remains the same. In Table III the values of R1 and R2 (Fig. 5) are given. A comparison of D values for Mn++ in ZnO, ZnS, and ZnSe (Table II) with the theoretical values obtained from Eq. (62) and Table III reveals that ZnO= 1.7, nO = 2 (63) DZnS/T kDZnS E and (DZnO DZnO> DznseIT - 2 Dnse E-.5 (64) DZnEnSE where E and T stand for experimental and theoretical respectively. It is evident that though an ionic model for ZnO and ZnS gives a satisfactory result [Eq. (63)], it fails to account for the large value of D in ZnSe. This discrepancy is also present in CdS and CdSe. Therefore it is necessary to take the effect of covalency into account. The factor D can be expressed as a function of degrees \i of covalency and overlapping intez15 grals Si as follows:5 D = - 5.05 A cml, (65) 24

where A = (S'2S2) - ('2_\2) (66) Si J= j do OdT' (67) hi = ($i-N4npo) /N3do' (68) TABLE III CRYSTALLINE PARAMETERS OF A B COMPOUNDS L(R-x) = R(A) j II VI Crystal ao(A~) C(A~) c/ao R1(A ) R2(A~) Ref. ZnO 3.2426 5.1948 1. 603 1.95 1.98 a Zn[S 3.811 6.234 1.636 2.33 2.33 a ZnSe 3.98 6.55 1.645 2.45 - b CdS 4.131 6.691 1.619 2.51 2.53 a CdSe 4.30 7.02 1.632 2.63 2.64 a References: a. Wyckoff's Crystalline Structure. b. Crystallography 5, 364 (1960). 3d and p denote the 17ave functions corresponding to the 3rd and the d0o npo np electrons of Mn++ and ligand with J =0. It is obvious that n has the values 2,3,4, and 5 for the ligands 0, S, Se, and Te respectively. Considering Fig. 7, one finds that C - [a221/[a2; 2= 4(a2-a2/4) = a 2/3, L2a 2/31 / or c = 2a, (69) 2~~ ~25 25

Fig. 7. Ideal hexagonal packing. and c= (8/3)1/2 = 1.6327 = 1.63. Comparing Eq. (69) with the c/a ratios obtained for Zn(O,S,Se) in Table III, one finds that in ZnO the tetrahedron of oxygen ions is squashed along the C-axis, whereas in ZnSe it is elongated along the C-axis as shown in Fig. 8. This indicates that in the case of ZnO the relation be

tween the overlapping integrals SI, SII, SIII, and SIV is SIV > S SII or SIII (70) whereas for ZnSe it is SVI < SI, SII or SIII. (71) Assuming S = S = S = S and S = S' and recalling Eqs. (65-66), I II III IV one obtains S2'(ZnO) > S2(ZnO), where S' 2(ZnS) < S2(ZnS) and S' (ZnSe) < S2(ZnSe). Therefore the overlap model (A' = O) gives a correst sign of D for ZnO but an incorrect sign for ZnS and ZnSe. Consequently we must invoke the covalent model (\ O0) for ZnS and ZnSe, so that (S'2-S2) _ (A\2_\2) >. (72) (~t2~2) (M2~2) > 0. (72) Equation (72) indicates that for ZnS and ZnSe, the covalency effect is more pronounced than overlapping, and that A'2 < A2. In the case of CdS and CdSe we expect that A'2 > A2 because the c/a ratio for these compounds 27

SoFig. 8. A schematic representation of the surrounding Se ligands around the Mn++ ion in ZnSe:Mn. (Table III) is less than 1.633. Therefore, according to this model we expect D(CdS, CdSe) = - 5.05[(S'2_S2) - ('2-_A2)] > 0. (73) Experimental results (Table III) confirm this prediction for the D value of Mn+ in CdS and CdSe. Further investigations regarding S and A coefficients and their explicit dependence on the one-electron wave functions are under consideration. Until such relations are established, the only conclusion that can be drawn is that in the selenides of Zn and Cd 28

the covalency effects play a major role, whereas in their oxides the ionicity is more pronounced. 29

REFERENCES 1. C. Kikuchi and G. H. Azarbayejani, J. Phys. Soc. Japan 17,9 Suppl. B-l, 453, (1962). 2. C. Kikuchi et al., The University of Michigan, Report 04275-2-F, Pt. III, (1963). 3. H. Watanabe, Prog. Theor. Phys. (Kyoto) 18, 405 (1957). 4. S. Slicker, J. Appl. Phys. 31, 1165 (1960). 5. Cuceaneau, Crystallography 5,_ 364 (1960). 6. R. H. Bube, Phys. Rev. 110, 1040 (1958). 7. Carl J. Ballhausen, Ligand Field Theory, McGraw-Hill (1962). 8. E. P. Wigner, Group Theory, Univ. of Calif., Los Alamos Scientific Laboratory (1959), p. 166. 9. B. Bleaney and W. H. Stevens, Repts. Prog. Phys. 16, 108 (1953). 10. C. Kikuchi and L. M. Matarrese, J. Chem. Phys. 33, 601 (1960). 11. G. Racah, Phys. Rev. 62, 438 (1942). 12. R. de L. Kronig and C. J. Bouwkamp, Physica 6, 290 (1939). 13. Reuben S. Title, Phys. Rev. 130, 17 (1963). 14. M. H. L. Pryce, Nuovo Cimento 6, Suppl., 843 (1957). 15.Jun Kondo, Proj. Theor. Phys. 23_, 106 (1960). 3o

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