Magneto-optical investigations of diluted Cd1 ... - APS Link Manager

PHYSICAL REVIEW B

VOLUME 56, NUMBER 4

15 JULY 1997-II

Magneto-optical investigations of diluted Cd12x Mnx S magnetic semiconductors in the B-exciton region Yu. G. Semenov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine, Prospekt Nauki 45, Kiev 252028, Ukraine

V. G. Abramishvili, A. V. Komarov, and S. M. Ryabchenko Institute of Physics, National Academy of Sciences of Ukraine, Prospekt Nauki 46, Kiev 252028, Ukraine ~Received 16 January 1997! The measurements of magnetoreflection spectra in the B-exciton region of diluted magnetic semiconductors ~DMS’s! Cd12x Mnx S are carried out for the magnetic-field directions both along and across the crystal c axis. It is pointed out that giant spin splitting ~GSS! of the B exciton does not demonstrate a nonlinear dependence on the magnetic-component fractional content x, contrary to the A-exciton GSS situation. It is supposed that the effects of multiple scattering of valence electron on the magnetic ions, which lead to a sublinearity of the A-exciton GSS x dependence in Cd12x Mnx S, are different for the A and B valence-electron subbands. In particular, due to a smaller effective mass of valence electrons in the B subband, the multiple scattering of this electron on Mn21 ions may be not important and the GSS of B excitons can be described by the mean-field approximation ~MFA! with the ‘‘bare’’ exchange constant J h for B-hole interactions with magnetic ions. The approach to the determination of band parameters D 1 ,D 2 ,D 3 and the hole-ion exchange constant J h in hexagonal DMS’s is suggested. This approach is based only on the B-exciton GSS MFA description and data of energy distances between A, B, and C excitons. The method is applied for the treatment of exciton magnetoreflection spectra of Cd12x Mnx S crystals with different x. It is established that J h 521.7 eV and is independent of x. This value J h turns out to be several times smaller than that obtained from MFA for A-exciton GSS. However, this value is in good agreement with J h 521.4 eV, the value obtained earlier from the A-exciton GSS treatment beyond the MFA. In the limiting case of small x, we obtained the band parameter values D 1 517.262 meV, D 2 525.660.7 meV, and D 3 521.760.3 meV. They differ from those known in the literature for pure CdS. @S0163-1829~97!01928-0#

INTRODUCTION

The investigations of giant spin splitting ~GSS! of exciton reflection spectra in Cd12x Mnx S crystals show1–5 a number of peculiarities as compared to other diluted magnetic semiconductors ~DMS’s! of A 212x Mnx B 6 type. The most widely discussed peculiarity is the sublinear concentration dependence of A-exciton GSS, which was interpreted in Refs. 1–3 to be a result of the effective dependence of the hole-band exchange constant J h 5J h (x) on the magnetic-component fractional content x. The measurements in Refs. 1–3 were performed on crystals with such small Mn21 concentrations that the interaction between magnetic ions was negligible and could not be the reason for the dependence of J h (x). An explanation based on the mean-field approximation ~MFA!, acting on carrier spins has been suggested in Refs. 1–3. In that case the J h value reached 3–4 eV, several times larger than its value in the other DMS’s. This explanation, however, cannot be regarded as a satisfactory one because the resulting J h value turn out to be greater than the CdS valence-band width. Due to this situation, attempts to explain GSS peculiarities of Cd12x Mnx S A excitons beyond the MFA were made in Refs. 6–8. The result of the investigations was a qualitative understanding of observable peculiarities as a manifestation of multiple hole scattering on the magnetic ions. This scattering influences GSS; it is also important due to the 0163-1829/97/56~4!/1868~8!/$10.00

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large effective mass of A valence electrons. A quantitative description was carried out in the aforementioned works on the basis of a simplified band structure in the isotropic effective-mass approximation m h 51.3m 0 for the A subband of CdS crystal valence electrons. The ‘‘bare’’ exchange constant value J h 521.4 eV, obtained in the latter approach, is in agreement with the data for other A 212x Mnx B 6 and seems to be reasonable. At the same time much less attention has been paid to magneto-optical investigations of the B band in Cd12x Mnx S. One of the reasons was the absence of vivid effects, similar to those in A-band investigations. Moreover, the substantial splitting of B excitons was not observed in s 6 polarizations for the magnetic field Hi c ~c is the hexagonal axis of the crystal!. The latter circumstance, in fact, was explained in Ref. 9. The explanation was reduced to the reconsideration of the band parameters D 1 ,D 2 ,D 3 well known in the literature; the difference D 1 2D 2 turned out to have a different sign. The authors of Ref. 9 were not able to give a reliable value of the exchange constant J h along with separate band-parameter D 1 ,D 2 ,D 3 values due to the impossibility of a joint description of A- and B-exciton GSS in the MFA framework. In the present paper we pay attention to the absence of a visible splitting of the B-exciton spectrum in s polarization for all magnetic fields and samples under investigation in a wide range of x values. This points to the equality between 1868

© 1997 The American Physical Society

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MAGNETO-OPTICAL INVESTIGATIONS OF DILUTED . . .

the GSS of the B valence subband and conduction band at D 1 2D 2 ,0. However, the conduction-band GSS is proportional to x ~for small x, when the effects of the Mn-Mn interaction are negligible!, so that J e does not depend on x and therefore the B-band GSS should not have a nonlinear dependence on x either. To explain this circumstance we suppose that the effects of the multiple scattering of valence electrons on the magnetic ions, which lead to a modification of A-exciton GSS, are different for the valence electrons of B and C subbands of the same crystal in comparison to that for the A subband. In particular, the effective masses of B valence electrons * 50.67m 0 are substantially less than m* B i 50.42m 0 and m B' 53.03m and have the same order of magnitude as those m* i 0 A for other DMS’s. The multiple scattering of this electron on Mn21 ions may not be so important due to the smaller valence-electron effective mass in the B subband than that for the A subband. The latter circumstance gives us the possibility of a B-exciton GSS MFA description ~except for A and B excitons mixing by exchange-field effects!. In this case the GSS of B excitons will be described by the bare exchange constant J h for B-hole interactions with magnetic ions. This approximation in turn is the basis for our approach to determine D 1 , D 2 , D 3 , and J h . This is done both from spin splitting of the B exciton, measured at magnetic-field directions along (Hi c) and across (H'c) the crystal hexagonal axis c, and from data on the A-B and A-C energy intervals between A-, B-, and C-exciton spectrum components, measured at zero magnetic field. The application of the developed formalism to the case of Cd12x Mnx S gives an independent confirmation of the value of the bare exchange constant J h obtained in Refs. 6–8. It also permits us to obtain more accurate values of the band parameters D 1 ,D 2 ,D 3 of CdS (Cd12x Mnx S) crystal. OUTLINE OF THE METHOD

Let us consider first the energies of A, B, and C subbands of DMS Cd12x Mnx S valence electrons in the absence of a magnetic field. The group of A, B, and C levels of a nonmagnetic semiconductor with wurzite structure in the basis of valence-electron wave functions u X & , u Y & , and u Z & is described by the Hamiltonian H v 5D 1 L 2Z 12D 2 L Z S Z 12D 3 ~ L X S X 1L Y S Y ! ,

~1!

where L and S are the angular momentum (L51) and spin operators, D 1 is the crystal-field parameter, and D 2 and D 3 are anisotropy spin-orbit interaction constants. Known solutions of the Hamiltonian ~1!, corresponding to the A, B, and C states of the valence electrons are conveniently represented in the form u A 6 & 5 u 61 & u 6 & , u B 6 & 5 A~ 11 j 0 ! /2u 61 & u 7 & 2 A~ 12 j 0 ! /2u 0 & u 6 & , u C & 56 A~ 12 j 0 ! /2u 61 & u 7 & 7 A~ 11 j 0 ! /2u 0 & u 6 & ,

~2!

j 0 5 ~ D 1 2D 2 ! / v 0 ,

v 0 5 A~ D 1 2D 2 ! 2 18D 23 ,

~3!

u 61 & 56(X6iY )/&, u 0 & 52Z, and u 6 & are eigenfunctions of the L Z and S Z operators. Note that the Hamiltonian ~1! is derived from symmetry considerations only10 and thus remains valid for DMS’s also in the case of valence-electron scattering by spin fluctuations.11 In the latter case, however, D 1 ,D 2 ,D 3 may depend on impurity-center characteristics. In particular, they depend on Mn concentration. So the energies calculated with the help of the eigenfunctions ~2!, are determined as10

E A 5D 1 1D 2 ,

~4!

E A 2E B,C 5 21 $ D 1 13D 2 7 A~ D 1 2D 2 ! 2 18D 23 % .

~5!

It is convenient to introduce the dimensionless ratio P5 ~ E A 2E B ! / ~ E A 2E C ! ,

~6!

which is expressed through the dimensionless parameters

d5

D 1 2D 2 , D 1 1D 2

z5

D 3 2D 2 ~ 11 d ! D 2

~7!

.

Here z determines the spin-orbit interaction anisotropy. One can find, with the help of Eqs. ~4!–~7!,

j 05

P5

d

Ad

2

12 ~ 12 d ! 2 @ 11 z ~ 11 d !# 2

,

22 d 2 Ad 2 12 ~ 12 d ! 2 @ 11 z ~ 11 d !# 2 22 d 1 Ad 2 12 ~ 12 d ! 2 @ 11 z ~ 11 d !# 2

~8! .

Two additional equations besides Eqs. ~8! are necessary for the unambiguous determination of the three band parameters and exchange constant from the experimental results. They can be obtained from the description of the exciton spectrum magnetic splitting. The exchange interaction of electrons with the system of magnetic ions is described by the Hamiltonian H ex5

(i

~9!

j e ~ h ! ~ r2Ri ! Sli Se ~ h ! ,

which should be averaged with magnetic ions ~with spin Sl and coordinate Ri ! and electron ~hole! wave functions. j e(h) (r2Ri ) is the exchange interaction operator, often represented in the form a d (r2Ri ) for conduction electrons and b d (r2Ri ) for valence electrons. The values N 0 a 5J e and N 0 b 5J h are the exchange constants, N 0 is the cation concentration in the crystal, and d (r2Ri ) is the delta function. When the MFA is valid, we can represent the main effect of this Hamiltonian as the action of the exchange field Ge(h) on the electron ~hole! spin Se(h) . The corresponding effective Hamiltonian has the Zeeman form H ex5Ge ~ h ! Se ~ h ! 5J e ~ h ! x ^ Sl & Se ~ h ! ,

~10!

where the ^ Sl & component

6

where

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^ S la& 5

1 Tr Nm

H ( S DJ Y H S DJ Nm

j51

S aj exp 2

Hm k BT

Tr exp 2

Hm k BT

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SEMENOV, ABRAMISHVILI, KOMAROV, AND RYABCHENKO

is the thermodynamic average of the magnetic ion spin moment components, induced by an external magnetic field H. The spin moment interaction with H, the crystal-field influence on the spins, and the spin-spin ~first of all the ion-ion exchange! interaction are described by the Hamiltonian H m . N m is the total number of magnetic ions in the crystal and a 5X,Y ,Z. Usually ^ Sl & is determined by a crystal magnetization measurement. For the calculation of GSS effects on valence electrons ~holes! in different subbands A, B, and C, the Hamiltonian ~10! should be added to Eq. ~1!. For the case of comparable or large GSS in comparison to D 1 ,D 2 ,D 3 , the exchange field action can mix the subband states ~2!. If the effects of multiple scattering are important, Eq. ~10! becomes incorrect. This should take place if the ratio J e(h) /W e(h) , where W e(h) is the width of the corresponding band, is not small. Generally speaking, in this case the solution of the problem involves the diagonalization of the exchange Hamiltonian in the basis of all energy bands. If, however, the influence of the interband mixing is small, it can be made for each of the bands separately. The approach proposed here is reduced, indeed, to a separate treatment of different valence subbands with the neglect of their mixing by an exchange interaction. There is also another way for such an approach. Even if the MFA is not valid, we can formally introduce the ex* !, but the change fields too ~we shall denote them by Ge(h) * will be nonlinear functions of the magneticvalues of Ge(h) component fractional content x and will depend on the multiple-scattering characteristics. For instance, such a description was used in Refs. 1–3 for the analysis of A-exciton GSS data. In general, Gh* may be different for different hole subbands A, B, and C if the multiplescattering probabilities in these subbands are different. So they may be invalid for the description of subband mixing. * value may be represented by a relation similar to The Ge(h) * (x). Eq. ~10!, but with a modified exchange constant J e(h) Just those values were determined for the A exciton in Refs. 1–3 and clearly they are not suitable for the description of * (x)ÞJ e(h) , where J e(h) is B-exciton GSS. In general, J e(h) the true or bare exchange constant. If the multiple-scattering * →Ge(h) and J e(h) * (x)→J e(h) for the effects are small, Ge(h) corresponding band, i.e., a formally introduced MFA-like description will tend to correct the MFA. In the case of multiple-scattering probability differences for valence electrons of A, B, and C subbands ~due to a large difference in effective masses!, the formally introduced ‘‘effective fields’’ acting on A, B, and C valence-electron spins and modified exchange constants will be different. The B and C subband electrons are subjected to the exchange field with the bare exchange constant because the MFA is actually suitable for them. In contrast, the A subband electron ‘‘feels’’ a modified x-dependent exchange constant because of the mentioned multiple-scattering processes. So we cannot describe the A and B subband GSS by the same J h or J* h (x) in the exchange-field representation as well as the mixing of the A subband with B and C by the exchange interaction. The exchange interaction of valence electrons in the B subband (h) as well as conduction electrons (e) with mag-

56

netic ions can be represented as the action of the mean ~or exchange! -field Ge(h) on electron spin. In the B subband eigenvector basis ~2!, the Hamiltonian ~10! still has a Zeeman form, but with an anisotropic g tensor, H ex~ B ! 5g B i G hZ S BZ 1g B' ~ G hX S BX 1G hY S BY ! ,

~11!

where g B i 52 j 0 , g B' 5 ~ 12 j 0 ! /2,

~12!

and S B is operator of fictitious spin 21 , introduced on u B 1 & and u B 2 & states. Equation ~11! describes the B subband valence-electron GSS DE (B) in a linear exchange-field G h approach and is suitable for small G h values. At larger G h values, the B subband GSS may be influenced by mixing of A and C subband states by the Hamiltonian ~9!. Since the MFA is invalid for the A subband, we shall suppose that experimentally observed dependences DE (B) (H) permit one to extract its linear part, described by the Hamiltonian ~11!. Thus we obtain two more independent equations for the ratios of the B subband GSS DE (B) at Gh i c and DE (B) at Gh'c to the conduction-band GSS DE (e) , that have no anisotropy: Q5DE ~ B ! u Hi c /DE ~ e ! 52 h j 0 ,

~13!

R5DE ~ B ! u H'c /DE ~ e ! 5 u Q/ j 0 u ~ 12 j 0 ! /2,

~14!

where h 5J h /J e and for convenience we substitute the expression for uhu from Eq. ~13! into Eq. ~14!. It should be noted that the sign of Q is an important experimental parameter. It depends on the mutual position of split energy levels with spin projections 6 21 for both conductivity and valence electrons of the B subband. The value of the splitting between the s 2 and s 1 components of the B exciton at Hi c will be equal to u DE (B) u Hi cu 1 u DE (e) u if Q.0, h ,0 and to u DE (B) u Hi cu 2 u DE (e) u if Q,0, h ,0. For the value of the two allowed p components, splitting the B exciton at Hi c the situation should be opposite. The sign of Q may be determined from the analysis of intensity variations of different B-exciton spectrum split components at H'c for different linear polarizations of light ~Ei c, Ei H, or H'E'c, where E is the E vector of light wave! at an H-field increase. The simpler way to determine the sign of Q is from GSS data for both A and B excitons at Hi c. In this case we should take into account that multiple scattering changes the value of the A valence subband electron GSS, but not its sign. The sign of h is known independently from the data of both A exciton GSS and spin-flip Raman scattering on shallow donors ~h ,0 for Cd12x Mnx S!. Note that the signs of Q and h determine, in accordance with Eq. ~13!, the sign of j 0 @i.e., the sign of D 1 2D 2 according to Eq. ~3!#. Thus it follows from Eqs. ~7!, ~8!, ~13!, and ~14! that for a complete and unambiguous determination of the parameters j 0 , d , z , h it is sufficient to know the experimentally determined values of P, Q, and R. The knowledge


56

of the parameters d,z,h and E A ,E B ,E C ,J e permit one to calculate all the band parameters in the Hamiltonian ~1! and the exchange constant J h :

j 0 51/~ 112R/Q ! ,

~17!

D 25

DE AB j 0 ~ 12 d ! , 2 j 02 d j 02 d

~18!

2 j 02 d j 02 d

a

,

~19!

Q , j0

~20!

J h5 h J e ,

~21!

h 52

0.013 0.0056 0.0013 0.0002

~16!

DE AB j 0 ~ 11 d ! , 2 j 02 d j 02 d

DE AB d A~ 12 j 20 ! /2

E A ~eV!

x a

D 15

D 35

TABLE I. Measured positions of A-, B-, and C-exciton bands in zero magnetic field for the three crystals investigated and the ratios P5(E A 2E B )/(E A 2E C ) calculated from these data.

~15!

2j0 d5 , ~ P11 ! j 02 ~ P21 !

where DE AB is the experimental value of E A 2E B . In this approximation we obtain the splitting of the B exciton at H'c on the four allowed lines with energies

~22!

where G h' corresponds to the value of G h for H'c ~in the usual case Gh i H and there are no reasons for G h anisotropy! and E B0 is the B exciton energy in zero H field. It is convenient to represent the energies as one pair of lines with ‘‘additive’’ splitting values E 1,25E B0 6 ~ u DE ~ B ! u H'cu 1 u DE ~ e ! u ! /2,

~22a!

and another pair of lines with ‘‘difference’’ splitting values E 3,45E B0 6 ~ u DE ~ B ! u H'cu 2 u DE ~ e ! u ! /2.

a

2.543 2.5435 2.550 2.554

E B ~eV! a

2.575 2.568 2.570 2.570

E C ~eV! a

2.650 2.640 2.635 2.632

P 0.30 0.254 0.235 0.205

Data from Ref. 4.

of G h . The case of D 1 .D 2 is similar to that of D 1 ,D 2 , but specific values of the transition probabilities and their changes with G h are a bit different. The effects of A, B, and C valence-subband mixing by the exchange field will influence the B-exciton line E 1 – 4 positions also. First of all, it will be manifested in the joint shift of all lines with the magnetic-field increase. The changing of energy distances between different components of the B-exciton GSS will be at the same time essentially weaker. Note that values of DE (B) u Hi c , DE (B) u H'c , and DE (e) in Eqs. ~13! and ~14! are determined as those induced by the exchange field only. Actually, there are contributions to the splitting from the direct action of H on the hole and electron spins. They should be determined and excluded from experimental results before the calculation of R and Q. EXPERIMENT AND DISCUSSION OF THE RESULTS

E 1,2,3,45E B0 6 ~ G e 6g B' G h' ! /2 5E B0 6 ~ DE ~ e ! 6DE ~ B ! u H'c! /2,

1871

~22b!

The transition probabilities to the exciton states with energies E 1 – 4 can be better considered with respect to the effects of A, B, and C valence-subband mixing by the exchange field. Such considerations have been made in Ref. 12, but we reconsider it because some results in Ref. 12 do not coincide with the symmetry predictions. The consideration for the case of D 1 ,D 2 shows that lines E 1 and E 2 should be active at Ei ci Z and Ei Y polarizations if Hi X ~the values of the transition probabilities depend on G h !. The E 1 line is strong at Z polarization and initially weak at the Y one, but with an increase of G h the Y transition increases. The E 2 line is strong at Z polarization too; at Y polarization it is initially weak and additionally weakens as G h increase during the initial stage. Lines E 3 and E 4 should be active at Ei X (Ei H) polarization only; the upper line (E 3 ) should be initially weak and decreases drastically with an increase of G h . Line E 4 is initially weak too, but increases essentially with an increase

The light reflection spectra in the region of A, B, and C excitons of the Cd12x Mnx S crystals with Mn fractional contents x 1 50.0002, x 2 50.0013, and x 3 50.0056 were obtained at a temperature T52 K in the magnetic fields H ,3.5 T for Hi c and H'c. The x values were measured by the x-ray microanalysis method and were checked additionally by the comparative integral intensity of Mn21 ion electron paramagnetic resonance spectra. Crystals with x 5x 2 ,x 3 were grown by the Bridgemen method from a melt and crystals with x5x 1 were grown from the gas phase. The optical measurements were carried out by double-grating spectrophotometer with a linear dispersion of 5 Å/mm. The spectra were recorded at different fixed values 0,H ,3.5 T. The positions of the exciton reflection line inflection points were used for plotting the magnetic-field dependences of excitonic band splitting energies. The measurements for x5x 1 were carried out for Hi c only. The reflection lines in the C-exciton region were detected for all crystals under investigation, which permitted us to find E A 2E B and E A 2E C values for them ~Table I!. At the same time we were not able to observe the magnetic-field effects of the C exciton on the linewidth background. The reflection spectra in the B-exciton region show the absence of visible splitting in s 1 and s 2 circular polarizations at Hi c for all samples under investigation. The s 1 and s 2 circular-polarized spectra are observed separately and this result cannot be explained by the insufficient resolution of s 1 and s 2 lines. At the same time we could observe only two allowed p lines together and they were unresolved in the experiment for each sample investigated. But one wide p line that we observed was shifted slightly with an increase in

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the H field. This indicates, from our point of view, the splitting of two unresolved p lines that have different widths in the H field, similar to those observed separately at s 1 and s 2 polarizations. Perhaps the other reason for a p line shift is the mixing of B and C valence subbands by the exchange field ~the A and B subbands are not mixed by Gh i c!. However, in this case the same shift should be present for lines of s 6 polarizations also, but it is absent. Moreover, the quantitative estimations of the possible shift from B to C mixing do not coincide with experiment. Thus the absence of a splitting of B-exciton s 1 and s 2 lines at Hi c identifies DE (B) u Hi c with the conduction-band GSS and establishes the ratio Q521 in Eq. ~13!. At H'c the spectra were measured, first of all, for the crystal with x5x 3 in three polarizations Ei X, Ei Y , and Ei Z, where the coordinate axes X and Z were directed along H and c, respectively. In this case, either one of the components corresponding to transitions from one of the splits in magnetic-field B-subband states to the appropriate conductivity-band split states ~at Ei X and Ei Y polarizations! or two components together ~at Ei Z polarization! should be observed. At H50 the energies of the B excitons for Ei c and E'c polarizations were slightly different, which is probably connected with the intraexciton’s electron-hole exchange interaction. The interaction and the difference are not so large and we did not take it into account in the treatment of the experimental results. For Ei Z we observed the B-exciton splitting in the H field on the two lines with similar intensities. Only one line was observed for each polarization Ei X and Ei Y in the entire range of magnetic fields used. Each of these lines was shifted up or down in energy in comparison to the center of gravity position of the Ei c spectrum for the same H-field values. Therefore, each of them is one of the B-exciton splitting components. The observed line positions versus the H field and versus G e for different light polarizations are shown in Figs. 1~a! and 1~b!, respectively. The G e (H) values were calculated in the usual manner @see Eq. ~10!# with J e 50.23 eV ~Ref. 1! and with ^ S l & as the Brillouin function: ^ S l & 52SB S (y), with y5g Mnb HS/kT and S5 25 . Due to a small x value for the samples investigated we do not use modified x, S, or T values in ^ S l & . As can be seen from Fig. 1~b!, the B-exciton splitting is the linear function of G e with good accuracy, although the joint shift of all B-exciton components to high energies with an increase of G e is present. According to theoretical predictions, we suppose that two components of the Ei ci Z spectrum @lines 1 and 4 in Figs. 1~a! and 1~b!# correspond to the E 1,2 lines of Eq. ~22a! and the components of the Ei Y and Ei X spectra @lines 2 and 3 in Figs. 1~a! and 1~b!# are the E 1 and E 4 lines of Eqs. ~22a! and ~22b!. The difference of the positions of line 1 and 2 in Figs. 1~a! and 1~b! is not understandable if both of these lines are due to the E 1 transition at different allowed polarizations and if the difference of the B-exciton positions at Ei c and E'c polarizations for H50 is due to the intraexciton’s electronhole exchange interaction only. We will not analyze the na-

56

FIG. 1. Exciton line positions at H'c for the sample with x 5x 3 50.0056 at different polarizations of light versus ~a! the H field and ~b! G e . Full and open circles stand for high- and lowenergy components of the B exciton at polarization Ei c; triangles up ~open! and down ~full! stand for the B exciton at Ei X and Ei Y , respectively; full and open squares stand for A exciton lines at Ei X and Xi Y polarizations. Lines 1–4 for the B exciton and 5 and 6 for the A exciton are plotted in ~a! for the better view and ~b! as a linear regression fitting.

ture of these differences in the present paper because they are close to the errors of the measurements performed. We can determine the R parameter value for crystals with x5x 3 directly from Fig. 1~b! as an averaged ratio of the difference between upper and lower components of B-exciton spectra to G e . In such a way we obtain R54.4 60.4. For a more precise determination of R we plot the H dependences of the splitting between two lines of the Ei c spectrum as well as between the lines observed at Ei Y

56


1873

TABLE II. Cd12x Mnx S band parameter values, calculated from Eqs. ~6! and ~13!–~21! at Q521.060.2, R54.260.2 ~for x 50.0056!, and P from Table I.

FIG. 2. Dependence of the B-exciton lines splitting at H'c for the sample with x5x 3 on ~a! the H field and ~b! G e . Circles stand for the difference between two lines at the polarization Ei c; squares stand for difference between the lines at polarizations Ei Y and Ei X. Curve 1 ~dashed!, the difference E 1 2E 2 @Eq. ~22a!#; ~curve 2 dotted!, the difference E 3 2E 4 @Eq. ~22b!#; curve 3 ~solid!, DE ( B ) u H'c .

and Ei X polarizations. They are shown in Fig. 2~a! along with the E 1 2E 2 and E 3 2E 4 values of Eqs. ~22a! and ~22b!, which were calculated with DE (e) 5G e , DE (B) u H'c5RDE (e) , and R as the fitting parameter. The same dependences versus G e are shown in Fig. 2~b!. The value of DE (B) u H'c5RDE (e) is shown in Figs. 2~a! and 2~b! for the sake of comparison. Note that splitting between the Ei Y and Ei X lines, which are related, by our supposition, to E 1 and E 4 , should be equal to DE (B) u H'c2DE (e) /2. This agrees fairly well with the data of Figs. 2~a! and 2~b!. It can be seen that there is no noticeable deviation of experimental points in Fig. 2~b! from the linear dependence on G e . The best fit was obtained at R54.260.2. The consideration of effects of the H-field direct action on electron and hole spins does not change this result within the experimental error. For the crystals with x5x 2 and x5x 1 we were not able to determine DE (B) u H'c and R due to small GSS effects in these crystals.

x

D 1 ~meV!

D 2 ~meV!

D 3 ~mev!

0.013 0.0056 0.0013 0.0002

27.162 23.062 19.762 17.262

37.260.7 32.760.7 28.560.7 25.660.7

26.260.3 25.260.3 22.860.3 21.760.3

Let us now describe the algorithm of the band parameter determination. First, the problem of the determination of J h ~or the parameter h! and D 1 ,D 2 ,D 3 was solved for the crystal with x5x 3 . This was done with the help of Eqs. ~6! and ~13!–~21!. The value R54.2 obtained above and the values of P50.254 ~see Table I! and Q521 were used for the solution of Eqs. ~15! and ~16!. j 0 and d, which were obtained from Eqs. ~15! and ~16!, were substituted into Eqs. ~17!–~21! for the determination of the exchange constants ratio h 527.4, J h 521.7 eV, and parameters D 1 ,D 2 ,D 3 in the crystal with x5x 3 . The obtained value h 527.4, along with values of P from Table I, was used for the calculation of the band parameters D 1 ,D 2 ,D 3 in the crystals with x5x 1 ,x 2 , where measurements at H'c gave no possibility to determine B subband splitting. We supposed that h ~and J h , respectively! has no x dependence, which follows from the absence of the J e (x) dependence and the absence of s 6 line splittings at Hi c. The results of such an experimental data treatment are in Table II. The dependences of D 1 , D 2 , and D 3 on the Mn fractional content x, obtained for Cd12x Mnx S crystals, are shown in Fig. 3. One can see that they are approximately linear within the accuracy of the values obtained. Both D 1 and D 2 increase with an increase of x. This probably has something to do with the chemistry of the Cd-Mn substitution. We will not discuss this question here.

FIG. 3. Data obtained for the dependence of the parameters D 1 ,D 2 ,D 3 on the fractional content x. Squares, D 1 ; circles, D 2 , and triangle, D 3 . The lines are plotted for a better view.

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FIG. 4. B-exciton line positions for the sample with x5x 3 at H'c. Experimental points: circles stand for Ei c ~full, up branch; open, down branch! and triangles ~full and open! stand for Ei Y and Ei X, respectively. Lines 1 and 4, E 1 and E 2 @Eq. ~22a!#; lines 2, and 3, E 3 and E 4 @Eq. ~22b!# calculated with the parameters obtained.

Note that for all the crystals investigated the sign of D 1 2D 2 is negative. This result is in qualitative agreement with the results of our previous paper.9 The parameters D 1 ,D 2 ,D 3 determined in the present work seem to us to be more reliable in comparison to the results of Ref. 9 because we obtain them without A-exciton GSS data. In accordance with Table II and Fig. 3, the dependence of the valence-band parameters D 1 ,D 2 ,D 3 on x, observed in Cd12x Mnx S crystals, has no tendency to change the sign of D 1 2D 2 in the limit x→0. In this case the values of D 1 ,D 2 ,D 3 in a compound with minimal x5x 1 correspond to those for pure CdS. As it is known, the only measurements of A-, B- and C-exciton energy positions do not give enough reasons for the choice between solutions with positive and negative values of the D 1 2D 2 difference. Both these possibilities were discussed for CdS crystals. The models were proposed by Thomas and Hopfield13 with D 1 2D 2 ,0 and by Birman14 with D 1 2D 2 .0. After the measurements of the influence of pressure on the CdS excitonic spectra reported by Langer et al.,15 which were described in the framework of Birman’s model, the last choice was preferred in the literature. The question of the compatibility of experimental data15 with both D 1 2D 2 .0 and D 1 2D 2 ,0 sets of parameters was analyzed in Ref. 9. It was obtained that such a compatibility is possible under some changes of the CdS deformation potential within the experimental error. At the same time the GSS measurement data give the unique possibility for the choice of sign of D 1 2D 2 . Let us discuss briefly the limits of applicability of the proposed method to the description of the experimental results on exciton splitting in hexagonal DMS’s. A comparison of the experimental results for the crystal with x5x 3 and the calculations of B-exciton line positions on the basis of Eqs. ~22a! and ~22b! with J e 50.23 eV, R54.2, and J h 521.7 eV is shown in Fig. 4. We can see that the splitting between B-exciton components coincides very well with

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FIG. 5. Comparison of the experimental data with calculations on the basis of diagonalization of the 636 matrix for valenceelectron subband top energies with parameters obtained in the paper from B-exciton splitting. Experimental data: circles stand for the B exciton at Ei c ~full, up branch; open, down branch!; triangles up ~open! and down ~full! stand for Ei X and Ei Y , respectively; full and open squares stand for the A exciton in Ei X and Ei Y polarizations, respectively. Lines 1–4 are calculated positions of the B-exciton components; lines 5,5a and 6,6a are for the A-exciton positions.

those in the theoretical curves, but this model does not describe the joint shift of experimental points with increasing magnetic field. This shift is connected, in our view, with A, B, and C subband mixing by the exchange field ~first of all A and B for H'c!. To check this supposition we calculate the B-exciton line positions by exact diagonalization of the 636 matrix for valence subbands with the same parameters J e ,J h ,D 1 ,D 2 ,D 3 ~the exact MFA solution!. This calculation should not exactly describe the A and B subband mixing effects because the MFA is not valid for a description of the exchange interaction between the A subband valence electrons and the magnetic ions in Cd12x Mnx S crystals. But we suppose that such a description will be substantially better for B-exciton component positions than that of Eqs. ~22a! and ~22b!, which completely ignores the subband mixing. The comparison of the calculations mentioned with experimental results is shown in Fig. 5. We can see that for B excitons in this case we have not only excellent agreement for the values of the line splitting but the acceptable agreement for joint line shifts too. At the same time the A-exciton line shift is not described well enough. Thus we have put forth an approach for the description of carrier-ion exchange interaction effects in different valenceelectron subbands with respect to their effective masses, i.e., to the effectiveness of carrier multiple scattering on magnetic ions. This in turn permits using the MFA for a description of B-exciton GSS. The aforementioned approach permits in describing with good accuracy the B-exciton GSS value and obtaining a correct J h value. It also permits determining the crystal band-structure parameters D 1 , D 2 , and D 3 with the

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help of A-, B- and C-exciton energy position data. That is the main purpose of the proposed approach. But for a reliable description of split B-exciton line positions ~and for the A exciton as well! for H'c, the effects of intersubband mixing by an exchange field turn out to be important. These effects cannot be considered exactly in the framework of the proposed approach, although they may be taken into account approximately with sufficient accuracy for a B-exciton component description. For the crystals with larger x the need for the linear part of the DE (B) u H'c(DE (e) ) dependence extraction may arise. This is not a problem. However, for a complete description of all split exciton line positions ~including the A exciton! it is necessary to develop the theory beyond the MFA with respect to multiple scattering of all subband electrons with different effective masses on the magnetic ions. It is very interesting to apply the proposed approach to H'c GSS data for Cd12x Mnx S crystals with larger-x values.

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H50. For all investigated crystals the sign of the D 1 2D 2 difference turns out to be negative. It is shown that it should be the same for CdS crystals too. Since in the majority of papers the CdS optical properties were described with the help of the parameters D 1 ,D 2 ,D 3 , corresponding to a positive D 1 2D 2 difference, the interpretation of some of the earlier results may require reconsideration. One more result seems to be important: the valence-band exchange constant J h 5 h J e 521.7 eV, which we have found, turned out to be substantially smaller than that obtained by the MFA application to the A-exciton GSS description. This value is closer to the J h 521.4 eV value obtained in Refs. 6–8 on the basis of the model that takes into account multiple scattering of valence electrons on magnetic impurities. Such agreement is probably evidence of the reasonability of approximations made both in the present paper and in those works.

CONCLUSION ACKNOWLEDGMENTS

The band-structure parameters D 1 ,D 2 ,D 3 of Cd12x Mnx S crystals, presented in Table II and Fig. 3, were obtained only from the spin-splitting part of the B excitons that are linear on G h and G e and values of the splitting of the top of the valence band on the A, B, and C components at

We are grateful to S. I. Gubarev for crystals with x5x 2 and x 3 and to G. S. Pekar for growing the crystal with x 5x 1 . This work was supported by INTAS Grant No. 933657.

S. I. Gubarev, Zh. Eksp. Teor. Fiz. 80, 1174 ~1981! @Sov. Phys. JETP 53, 601 ~1981!. 2 V. G. Abramishvili, S. I. Gubarev, A. V. Komarov, and S. M. Ryabchenko, Fiz. Tverd. Tela ~Leningrad! 26, 1095 ~1984! @Sov. Phys. Solid State 26, 666 ~1984!#. 3 S. I. Gubarev and M. G. Tyazhlov, Pis’ma Zh. Eksp. Teor. Fiz. 44, 385 ~1986! @JETP Lett. 44, 494 ~1986!#. 4 M. Nawrocki, J. P. Lascaray, D. Coquillat, and M. Demianiuk, in Diluted Magnetic (Semimagnetic) Semiconductors, edited by R. L. Aggarwal, J. K. Furdyna, and S. von Molhar, MRS Symposia Proceedings No. 89 ~Materials Research Society, Pittsburgh, 1987!, p. 65. 5 S. I. Gubarev and M. G. Tyazhlov, Pis’ma Zh. Eksp. Teor. Fiz. 48, 437 ~1988! @JETP Lett. 48, 481 ~1988!#. 6 C. Benoit a la Guillaume, D. Scalbert, and T. Dietl, Phys. Rev. B 46, 9853 ~1992!. 7 D. Scalbert, A. Ghazali, and C. Benoit a la Guillaume, Phys. Rev.

B 48, 17 752 ~1993!. J. Tworzydlo, Phys. Rev. B 50, 14 591 ~1994!; Mater. Sci. Forum 182-184, 593 ~1995!; Solid State Commun. 94, 821 ~1995!. 9 V. G. Abramishvili, A. V. Komarov, Yu. G. Semenov, and S. M. Ryabchenko, in Defect and Diffusion Forum, Vol. 103-105, Defect in Semiconductors, Proceedings of the First Russian National Conference on Defect in Semiconductors, St. Petersburg, 1992, edited by N. T. Bagraev ~Scitecpublications Ltd., Aedermannsdorf, Switzerland, 1993!, p. 181. 10 G. L. Bir and G. E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors ~Wiley, New York, 1975!. 11 S. M. Ryabchenko, Yu. G. Semenov, and O. V. Terletskii, Phys. Status Solidi B 144, 661 ~1987!. 12 S. I. Gubarev, Phys. Status Solidi B 134, 220 ~1986!. 13 D. G. Thomas and J. J. Hopfield, Phys. Rev. 116, 573 ~1959!; 119, 570 ~1960!. 14 J. L. Birman, Phys. Rev. 114, 1490 ~1959!. 15 D. W. Langer et al., Phys. Rev. B 2, 4005 ~1970!.

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