The Theory of ESR Hyperfine-Structure Narrowing as Applied to Wide-Gap Semimagnetic Semiconductors

JOURNAL OF MAGNETIC RESONANCE, ARTICLE NO.

Series A 123, 16–25 (1996)

0209

The Theory of ESR Hyperfine-Structure Narrowing as Applied to Wide-Gap Semimagnetic Semiconductors M. N. ALIEV, L. R. TAGIROV,*

AND

V. S. TAGIEV †

Hacettepe University, Beytepe 06532, Ankara, Turkey Received November 21, 1995; revised June 24, 1996

of the Mn 2/ ion (nuclear moment I Å 52 ) is observed (8). Upon increasing magnetic-ion concentration, the hyperfinestructure lines showed increasing broadening and then merged into a single line. In (9–11) much attention was given to the magnetic behavior of SS at relatively high Mn 2/ content (0.1 £ x £ 0.6), and a drastic increase in a linewidth at low temperatures side was interpreted in (10) as the indication of transition to the spin-glass state. The authors of the works cited above did not analyze in detail the evolution of the ESR hyperfine structure and did not use this structure to extract quantitative information about magnetic-exchange interactions, important for the physical properties of SS. This is due to the absence of a detailed quantitative theory of ESR hyperfine-structure exchange narrowing. As we shall show below, ESR spectra provide information about the exchange interaction dependence on the distance between magnetic moments, and give an independent tool for estimating the gap value in the band structure of these compounds. First, we derive, using the method of the nonequilibrium statistical operator, the system of kinetic equations which describes the dynamics of ESR hyperfine-structure lines. Then we develop a procedure for numerical solution of equations with application to Mn 2/ ESR spectra in Cd10xMnxTe SS. Finally we discuss in detail the values and concentration dependencies of physical parameters extracted from the fitting of the theory to experimental data and the significance of the spatial dependence of the exchange interaction and gap value in the SS under investigation.

The theory of ESR hyperfine-structure exchange narrowing is developed. The kinetic equations describing the nonequilibrium magnetization dynamics of hyperfine components of ESR spectra are derived by using the nonequilibrium statistical-operator method. Numerical solution of these general equations is applied to the simulation of the experimental spectra of (Cd, Mn)Te crystals with a Mn 2/ ion content of between 0.05 and 0.5%. The hyperfine-splitting constant, homogeneous ESR linewidth, and meansquare value of the exchange fluctuation frequency are obtained from the fitting of the spectra for different concentrations of manganese ions. The concentration dependence of the latter quantity is discussed in relation to the spatial dependence of magnetic-ion exchange interaction and the gap value in the band structure of semimagnetic semiconductors. q 1996 Academic Press, Inc.

INTRODUCTION

Semimagnetic semiconductors (SS), based on SnTe(Se), CdTe(Se), and PbTe(Se) semiconducting hosts with substitution of the part of metal ions by transition magnetic ions, for example, manganese ions, have attracted interest due to a variety of anomalous physical phenomena, such as negative magnetoresistance, a large Faraday effect, and spin-glass magnetic ordering at low temperatures (1–5). These magnetic properties and the possibility of varying the band-gap value by changing or mixing the halcogenide component of these compounds open promising applications of semimagnetic semiconductors in scientific instruments and industrial techniques (1–4). Since the work by Holm and Furdyna (6) on Hg10xMnxTe, the ESR method has been applied to investigations of Hg10xMnxSe (7) and then to Cd10xMnxTe(Se), Pb10xMnxTe(Se), and Sn10xMnxTe(Se) wide-gap SS in (8–15). It was pointed out that, at the lowest concentrations of manganese ions investigated (x £ 0.001), the hyperfine structure

THEORY

Derivation of kinetic equations describing the exchange narrowing of ESR hyperfine splittings. Assume that the system of localized moments (LM) of magnetic ions, dissolved in a SS, is subjected to the action of a DC magnetic field H0 and a weak oscillating magnetic field h(t). Suppose that magnetic ions are randomly distributed over the metalion sites of the SS lattice with concentration n Å NLM /V, where NLM is the number of LMs in a volume V. The nucleus

* Physics Department, Kazan University, Kazan 420008, Tatarstan, Russian Federation. † Institute of Physics, Azerbaijan Academy of Sciences, Baku 370143, Azerbaijan.

1064-1858/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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WIDE-GAP SEMIMAGNETIC SEMICONDUCTORS

of the magnetic ion has an intrinsic moment of the value I Å 52. In a DC magnetic field, the electronic LMs experience the action of local magnetic fields. This defines local precession frequencies 0 gsmB H0 / AIl , where 0 gsmB H0 is the Larmor precession frequency of LMs, gs is the spectroscopic splitting factor of LMs, mB is the Bohr magneton, A is the hyperfine interaction constant, and the nuclear moment projection on the DC field direction takes the (2I / 1)-fold set of values Il Å 0I, 0 I / 1, . . . , I 0 1, I. Thus, all LMs can be divided into 2I / 1 groups randomly distributed in space. Each group has a precession frequency in a local internal field and interacts by dipole–dipole and exchange interaction with other groups. We shall call these groups spin packets and assign a unique precession frequency to each. The Hamiltonian can be written as

H (t) Å H 0 / H 1 (t) / H h f / H ex / H d / H L / H SL ,

[1]

where

H 0 Å vs ∑ ∑ S zkl , vs Å 0gsmB H0; l

[2]

kl

gsmB ∑ ∑ [S k/l h 0 (t) / S k0l h / (t)], 2 l kl

H 1 (t) Å 0

h { (t) Å h exp( {ivt);

[3]

H Å ∑ v ∑ S , v Å AIl ; hf

I l

z kl

l

I l

tively. Hereafter indices a Å l, m, n will refer to spin-packet labels, but doubled indices like ia run over LMs, pertaining to the ath spin packet. (d) H ex mm describes the exchange (dipole–dipole) interaction between LMs belonging to the same spin packet m, the (d) with l x m corresponds intrapacket interaction, but H ex lm to the interpacket interaction. We dropped conduction electrons and their interaction with LMs in [1], because in SS of Cd10xMnxTe type, to which we intend to apply our theory, the carrier concentration is rather low (n £ 10 19 cm03 ) and their contribution to the hyperfine-structure dynamics can be neglected. The influence of carriers can be incorporated easily into the formalism which is developed below. First, we refer to the monograph by Zubarev (16), in which the general method of the nonequilibrium statistical operator (NSO) is formulated. In specific derivations we shall follow the Zubarev–Kalashnikov (17) formulation of the perturbation theory for a NSO. According to the general prescription (16), it is necessary to build a NSO, choosing, first, the set of dynamic variables for the description of the nonequilibrium state. Actually, the latter state appears as a result of microwave field absorption by the spin packets. That is why it seems natural to choose as dynamic variables operators of the spin-packet magnetizations Ml Å gsmB (jl Sjl , assigning to each one a thermodynamically conjugate nonequilibrium parameter Al (t), as well as the operator of the lattice energy with conjugate equilibrium parameter b Å 1/kBT. Then the Hamiltonian Eq. [1] can be represented in the form

[4]

kl

H (t) Å H0 (t) / V,

H Å ∑ H , ex

ex ml

where H0 (t) is the Hamiltonian of the noninteracting subsystems, and V is the perturbing interaction between them:

m£l

H ex mm Å

1 ∑ Ji j Si Sj , 2 imxjm m m m m

H ex lm Å ∑ Jjm kl Sjm Skl ;

H0 (t) Å H 0 / H 1 (t) / H h f / H L , [5]

V Å H ex / H d / H SL .

jm ,kl

H d Å ∑ H dml , 1 ∑ Di j S zi S zj , 2 imxjm m m m m

H dlm Å ∑ Djmkl S zjm S zkl . [6] jm,kl

Specific expressions for H L and H SL are not important for our purposes. In Eq. [1], the term [2] represents the energy of LMs in the external DC magnetic field H0 , Eq. [3] describes the interaction of LMs with a weak alternating field of the strength h, Eq. [4] refers to the hyperfine interaction, which splits all LMs into 2I / 1 spin packets, and Eqs. [5] and [6] are the exchange and dipole–dipole interaction energies with interaction potentials Ji a i*a= and Di a i*a= , respec-

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[8]

We use the general integral equation for a NSO with e Å /0 obtained in (17):

m£l

H dmm Å

[7]

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r(t) Å rq (t) 0 1

*

0

dt1e e t1 U / (t / t1 , t)

0`

H

Ìrq (t / t1 ) 1 / [ rq (t / t1 ), H0 ] Ìt1 i\

/

1 [ r(t / t1 ), V ] U(t / t1 , t). i\

J

[9]

Here rq (t) is the quasiequilibrium statistical operator (QSO), defined by

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ALIEV, TAGIROV, AND TAGIEV

rq (t) Å Q 01 exp[ 0 S(t)],

Q Å Tr{exp[ 0 S(t)]},

S(t) Å

1 ∑ [A l/ (t)M l0 / A l0 (t)M l/ ] 0 b ∑ Hl M zl / bH L , 2 l l Hl Å H0 0 AIl /gsmB .

[10]

H *

U(t / t1 , t) Å exp 0

i \

t/t1

J

H0 (t * )dt * .

t

, V] Å 0

* dte

[S(t), V ]e

( t01 ) S ( t )

|

[11]

[12]

|

∑ ∑ Dinjm S zjm M i{n ∑ Dinjn S zin M j{n / [M n{ , H SL ].

Because the kinetic equations for both circular components appear to be independent in the high-temperature approximation, we write down equations only for » M n0 … t components. Substituting [16], [17], and [14] into [15], expressing the thermodynamic parameters A n0 (t) in terms of dynamic variables » M n0 … t by use of formula [A5] of Appendix A, and keeping the terms only up to the second order in interaction V, we obtain d » M n{ … t i Å 0 Tr{[( vs / v nI )M n0 0 h 0 (t)M zn ] rq (t)} \ dt 1 / Tr [M n0 , V ] \

1 ∑ (A l/ (t)[M l0 , V ] / A l0 (t)[M l/ , V ]) 2 l

∑ Hl[M zl , V ] / b[ H L , V ],

*

0

0`

1 U / (t / t1 , t) ∑ k

[13]

1

H

*

dt1e e t1

0`

*

0

1

d tr tq (t)U / (t / t1 , t)

0

∑ Hl[M zl , V ] / b[ H L , V ] l

t 1 U(t / t1 , t) r 10 (t). q

J

[14]

Then we must average with the density matrix operator [14] quantum-mechanical equations of motion for dynamic variables considered not to be in equilibrium. As a result, the set of kinetic equations for the transverse magnetization of the spin packets » M n{ … t can be obtained,

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J

t 0 xkh (t / t1 ))U(t / t1 , t) r 10 (t) . q

[18]

1 ∑ (A l/ (t)[M l0 , V ] / A l0 (t)[M l/ , V ]) 2 l

0b

t q

0

b [M k/ , V ]( » M k0 … t/t1 2xk

we obtain, after substitution into [9], 0

* d tr (t) 1

dt1e e t1

l

r(t) Å rq (t) 0

[17]

inxjn

and evaluating the commutator in [12] as

0b

∑ Jinjm (S zin M j{m 0 S zjm M i{n )

H

0

[S(t), V ] Å

[16]

m ( xn ) in, jm

1

[e

[M n{ , H0 (t)] Å | [( vs / v nI )M n{ 0 h { (t)M zn ], m ( xn ) in, jm

Taking into account the weakness of the external microwave field, we assume longitudinal magnetizations of spin packets » M zl … to be in equilibrium with the lattice thermodynamic parameter b Å 1/kBT (i.e., the absence of ESR saturation). Then we replaced the NSO r(t) by the QSO rq (t) in the last term of Eq. [9] to restrict ourselves to the firstorder deviation from the equilibrium state. Straightforward inspection of the first two terms of Eq. [9] in view of the formulas [24] – [27] of Ref. (17) shows that these terms do not contribute to kinetic equations, so only the last term in NSO [9] is applicable. Using the Kubo identity 0 tS ( t )

[15]

with r(t) given by Eq. [14]. The commutator in [15] can be easily evaluated:

[M n{ , V ] Å { ∑

The evolution operator is

0S ( t )

d » M n{ … t i Å 0 Tr{[M n{ , H0 (t) / V ] r(t)} \ dt

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The first-order deviation of » M n0 … t from equilibrium is sufficient for our problem. Therefore we change the QSO rq (t) [10] in the second term of [18] by the equilibrium result r0 [A4]. To avoid divergences in calculations of kinetic coefficients in [18], the spin packets should have nonzero frequency width. Of course, any spin–lattice interaction, for example, as represented by our H SL term in [1], will broaden the hyperfine-structure lines, but our crude estimates show that the dipole–dipole interaction is more likely responsible for the width of the spin packets. In order to remain in the frame of approximations adopted when deriving Eq. [18], we eliminate H SL in our further derivations, and take into account the dipole–dipole interactions outside of the NSO scheme that will lead to the temporary damping of correlation functions in Eq. [18], as shown below. The dec-

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WIDE-GAP SEMIMAGNETIC SEMICONDUCTORS

rement is given by the Anderson statistical theory of ESR dipole line broadening (18), 2 2p q (gsmB ) 2\ Sc, 3 3

dÅ

[19]

where c is the concentration of magnetic ions. Equation [18] becomes, with \ Å 1,

represented » M zn …0 Å xn ( vs / v nI ), where xn is the static susceptibility of nth spin packet [A6], [A8], introduced in Appendix A. The kinetic equations, given by [20] and [24], together with expressions for the kinetic coefficients [21] and [25], are the principal results of this section of the paper. The analytical evaluation of the kinetic coefficients is outlined in Appendix B.

d » M n0 … t dt

RESULTS

Å 0i[( vs / v nI ) » M n0 … t 0 h 0 (t) » M zn …0 ] /∑ k

*

0 0 dt1[ » M k0 … t/t1 0 xkh 0 (t / t1 )]K nk (t1 ),

0`

[20] 0 where the kinetic coefficient K nk (t1 ) is

0 K nk (t1 ) Å

be d t1 2xk

* d t Tr{[M 1

0 n

, H ex ] r t0 e it1( H

0/ H hf)

0

1 [M k/ , H ex ]e 0 it1( H

0/ H hf)

t r 10 }. 0

[21]

The temporary Fourier transformation of time-dependent quantities in [20] is performed by the rule

*

» M n0 … t Å

`

dv 0 M n ( v )e 0ivt . 2p

0`

[22]

Introducing the dynamic susceptibilities by the relationship M n0 ( v ) Å x n0 ( v )h 0 ( v ),

[23]

we get a system of (2I / 1) equations for these dynamic susceptibilities, 0 [ 0 v / vs / v nI 0 id / iK nn ( v* )] x n0 ( v )

/i

∑ K nk0 ( v* ) x k0 ( v ) k ( xn )

Å xn ( vs / v nI ) / i

∑ K nk0 ( v* ) xk ,

[24]

k 0 ( v* ) is the Fourier transform of the kinetic coeffiwhere K nk cient given by Eq. [21]:

0 K nk ( v* ) Å

*

0 0 dtK nk (t)e 0ivt .

[25]

0`

The use of v* Å v / id indicates that we must keep the 0 dipole damping in the temporary behavior of K nk (t). We

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Numerical solution of the kinetic equations and fitting of the experimental data. For the solution of the system of (2I / 1) kinetic equations given by Eqs. [24], [B11], and [B15], we designed a computer program (available from T.L.R. upon request) in the FORTRAN-77 programming language, solving the system of linear equations with complex coefficients by the Gaussian elimination method. Because in the actual experiments the field derivative d x9 /dvs is measured, we calculated this value by numerical differentiation, and compared the calculated spectra with measured digitized spectra using the graphic capabilities of a PC-386/ 387 computer. The hyperfine interaction constant A, the homogeneous linewidth d, and the mean-squared exchange fluctuation frequency v 2ex , introduced in Appendix B, were considered to be fitting parameters. Four high-quality rods of Cd10xMnxTe were grown in the Institute of Physics of Polish Academy of Sciences by floating zone melting. The actual content x of Mn 2/ was 0.05, 0.1, 0.3, and 0.5 at.%. The ESR spectra were recorded at room temperature on a Bruker X-band (9 GHz) ESR spectrometer. Figures 1–4 show the results of the best fits, and the actual values of parameters are indicated in the figure legends. Figures 1, 2, and 4 show good agreement between calculated and measured spectra, including fine details like slightly more intense side lines as compared with the internal lines in Figs. 1 and 2. The fitting of the spectra for x Å 0.3% shows poorer agreement. The reason is unclear to us; perhaps the quality of this sample was not as good as that of the others. The set of Figs. 1 to 4 shows the progressive narrowing of the ESR hyperfine structure with increasing Mn 2/ ion concentration. The hyperfine-interaction constant appeared to be constant, A Å 59 { 0.2 G for all four samples, as was expected. The values of d which we attribute to dipole–dipole broadening can be approximated by a linear dependence with Mn content x in accordance with our conjecture [19] (see Fig. 5). Figure 6 shows the concentration dependence of the mean-squared exchange-fluctuation frequency v 2ex , which has pronounced nonlinear behavior, contradicting the simple approximation of linearity in c and x [B14]. This clearly shows that the magnetic-exchange

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ALIEV, TAGIROV, AND TAGIEV

FIG. 1. ESR spectrum of Mn 2/ ions in a Cd10xMnxTe sample with x Å 0.05%. The solid line is the experimental result and the dashed line is computed with the parameters H0 Å 3397 G, A Å 59.2 G, d Å 19 G, and v 2ex Å 0.

FIG. 3. ESR spectrum of Mn 2/ ions in a Cd10xMnxTe sample with x Å 0.3%. The solid line is the experimental result and the dashed line is computed with the parameters H0 Å 3395 G, A Å 59.0 G, d Å 33 G, and v 2ex Å 1250 G 2 .

interaction in SS does not scale as an inverse volume per magnetic ion upon changing the concentration c. Since the work by Bloembergen and Rowland (19), and a few more recent ones (20–24), it is known that in semiconductors with a gap between valence and conduction bands, the main spatial dependence of the exchange via the virtual excitations of carriers through the gap comes from the exponential damping factor depending on the gap value of the actual SS. It is clear that, if the average distance

between Mn 2/ ions scales as rav Ç ac 01 / 3 , but the spatial dependence of the exchange is r 0n exp( 0 gr), then the strength of exchange at the average distance will scale more strongly than c n / 3 . For a more definite conclusion, we shall investigate the distribution of exchange-fluctuation frequencies and analyze the mean-squared exchange frequency according to a more elaborated treatment. Calculation of mean-squared exchange frequency. Suppose that the magnetic moments of manganese ions interact

FIG. 2. ESR spectrum of Mn 2/ ions in a Cd10xMnxTe sample with x Å 0.1%. The solid line is the experimental result and the dashed line is computed with the parameters H0 Å 3397 G, A Å 59.1 G, d Å 25 G, and

FIG. 4. ESR spectrum of Mn 2/ ions in a Cd10xMnxTe sample with x Å 0.5%. The solid line is the experimental result and the dashed line is computed with the parameters H0 Å 3393 G, A Å 58.8 G, d Å 71 G, and v 2ex Å 4800 G 2 .

v 2ex Å 40 G 2 .

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by means of exchange interaction with dependence on distance as J(r) } r 0m / 2 exp( 0r/l),

[26]

where m is a positive integer, and l is the spatial cutoff length. Then the probability of mutual spin flips per second (provided that they are not forbidden by some physical reason) for the given spin i interacting with surrounding spins is

vi Å v0

∑ k ( xi )

SD a rik

m

exp( 02rik /l),

[27]

where v0 , containing the strength of exchange, will be considered an adjustable parameter, and rik Å Éri 0 rkÉ is the distance between these spins. If we wish to find the number of spins having spin-flip rates in the range ( v, v / d v ), we should count them by applying

FIG. 6. The dependence of mean-squared exchange-fluctuation frequency v 2ex on Mn 2/ ion content. The solid line is a guide for the eyes.

F( v ) Å

*

1 ∑ 2p i

`

0`

H F GJ

1 exp( 02rik /l)

F( v )d v Å

S

∑ d v 0 v0 ∑ i

k ( xi )

SD a rik

m

D

exp( 02rik /l) d v,

[28]

where the d function selects the values of the rate [27] which are equal to v, and sums all the sites i for which the above conditions can be fulfilled. Then, using the integral representation for the d function we get for the distribution function F( v ) of the spin-flip rates

SD a rik

dy exp 0iy v 0 v0 ∑ k ( xi )

.

m

[29]

As the summation over i and integration over y are independent, we may change the order, thus obtaining

F( v ) Å

1 2p

*

dye 0iyvA(y),

[30]

0`

where

H

A(y) Å ∑ exp iyv0 ∑ i

k ( xi )

SD a rik

J

m

exp( 02rik /l) .

[31]

Summations in Eq. [31] can be performed independently over all sites occupied by magnetic Mn 2/ ions, because the selection is provided by the d function. Considering the exponential in the sum in Eq. [31] as the product of exponents, interchanging independent summation over i and multiplication over k, and passing from summation to integration, we get A(y) Å ∏ k

FIG. 5. The dependence of homogeneous ESR linewidth d on Mn 2/ ion content. The data for triangles are taken from the fittings in Figs. 1–4; the solid line is the fitting to linear concentration dependence according to d (x) Å (12.1 / 105x) G.

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H F

10

1 V 0 Va

*

dri

V0Va

S SD

1 1 0 exp iyv0

a rik

m

exp( 02rik /l)

DGJ

.

[32]

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After the integration over the entire volume over variable ri , excluding the volume per single site Va , the product becomes the NLM power of the curly bracket in Eq. [32]. After taking the limit NLM r ` , holding NLM /V Å n constant, we have

H * F S SD

A(y) Å exp 0n

according to Eqs. [30], [34], and [35] involve many computations. First, we check the case of m Å 6 and L Å ` in Eq. [34], corresponding to the r 03 dependence of the interaction, which scales as the inverse volume upon dilution. The integral in Eq. [30] can be evaluated analytically:

dr

F( v ) Å

V0Va

1

1 0 exp iyv0

a r

m

exp( 02r/l

DGJ

For isotropic exchange, with the dimensionless integration variable R Å r/a we obtain finally the distribution function, A(y) Å exp{ 04pc

*

sÅ

.

[33]

`

RdR

1 [1 0 exp(iyv0 R 0mexp( 02R/L)]},

*

vmax

v 2 F( v )d v,

[35]

0

where vmax is the maximum exchange spin-flip rate corresponding to two nearest Mn 2/ neighbors in the Cd lattice, i.e., vmax Å v0 exp( 02R/L). Equations [30], [34], and [35] solve the problem of finding v 2ex . DISCUSSION

Concentration dependence of mean-squared exchange frequency. According to our derivations, the exchange interaction between manganese ions enters only into the meansquared exchange frequency v 2ex . The dependence of exchange on the SS gap structure is seen in the concentration dependence of v 2ex . The simple approximation [B14], neglecting the strong spatial fluctuations of v 2ex , predicts a linear dependence of this quantity on Mn 2/ concentration, which contradicts the nonlinear concentration dependence observed in the experiment. Thus the analysis in the subsection Calculation of mean-squared exchange frequency, which takes into account the spatial fluctuations of the squared exchange frequency, makes it possible to establish the correct concentration dependence of v 2ex and to estimate a gap value in the band structure of Cd10xMnxTe. Actual calculations of the v 2ex concentration dependence

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v 2ex á

[34]

where c ! 1 now is the number of magnetic ions in a unit cell of the lattice (i.e., the dimensionless concentration c in [B14] of Appendix B), L Å l/a. With distribution F( v ) the mean-squared exchange-fluctuation frequency can be calculated as

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D

,

1 (2p ) 3 / 2 c. 3

[36] [37]

Distributions similar to that in Eq. [36] had been derived by Anderson (25) in his analysis of diffusion in random lattices, and by Rozman (26) for the quenching of fluorescence in solutes. The straightforward analytical integration in Eq. [35] then gives for s ! 1,

1

v 2ex Å

S

s / v0 1 v0s 2 exp 0 3/2 2v 2p ( v / v0 )

q

S

D

3 v 20 3 q s 1 0 s2 . 2 2p 2

[38]

As s depends linearly on concentration according to Eq. [37], we conclude that for this marginal spatial dependence of exchange as r 03 the mean-squared value of the exchangefluctuation frequency is linear in concentration, as in the simple approximation [B14]. With an interaction potential of the general form of Eq. [26], v 2ex cannot be calculated analytically. So we undertook a numerical analysis, taking first the integral in Eq. [35] analytically and then integrals in Eqs. [30] and [34] numerically. We established the nonlinear concentration dependence of v 2ex when the range of exponential cutoff becomes of the order of the lattice spacing a. We consider this behavior a clear indication of a gap in the band structure of CdTe. We observed a weak sensitivity of the calculation results in Eq. [26] and some dependence of them on the number of integration points in numerical routines. A double-precision calculation regime and the increase in the number of integration steps did lead to an unacceptable increase in calculation time; this is why we do not discuss the value of the gap and its concentration dependence in detail. Further analytical development and advanced numerical procedures will be the subject of a following paper. APPENDIX A

Relation between Dynamic Variables and Thermodynamic Parameters in the NSO Method

Observation of the general form of kinetic equation [15] containing NSO, given by formula [14], shows that both dynamic variables of the problem » M n{ … t and conjugated thermodynamic parameters A n{ (t) enter into Eqs. [14] and

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WIDE-GAP SEMIMAGNETIC SEMICONDUCTORS

[15], so that the total number of variables appears to be redundant. That is what lets us find the relationship between dynamic variables and thermodynamic parameters. According to the ideology of the NSO method (16, 17), a nonequilibrium state goes in time to equilibrium through the infinite sequence of quasiequilibrium at actual time moment states. This means that at any given time moment, » M n{ … t Å » M n{ … tq Å Tr{M n{ rq (t)},

[A2]

and expanding the rq (t) exponentials in Eq. [A1] up to the first order in deviation from equilibrium, we collect terms on the right-hand side of Eq. [A1] up to the first order on that deviation, thus obtaining » M n{ … t á Å

b » { ∑ [ dH m/ (t)M m0 / dH m0 (t)M m/ ]}M n{ …0 2 m

b dH n{ » M n| M n{ …0 , 2

[A3]

»rrr…0 Å Tr{rrrr0 },

r0 Å Q

01 0

[A4]

where r0 is the equilibrium density matrix. First, Eq. [A3] together with Eq. [A2] establishes the desired relationship between dynamic variables and thermodynamic parameters, A n{ (t) Å 02

» M n{ … t . » M n| M n{ …0

xn Å

» M n{ … t , dH n{ (t)

[A6]

we then obtain from Eq. [A3] b b » M n| M n{ …0 Å (gsmB ) 2 » S n| S n{ …0 . 2 2

[A7]

In high-temperature approximation ( vs Å É gsmB H0É ! kBT )

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dt

0`

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e 0ivt/dt xk

* d t Tr{[M 1

0 n

, H ex ]

0

0

0

t 1 r t0 e it HH [M k/ , H ex ]e 0it HH r 10 }, 0

[B1]

˜ 0 Å H 0 / H h f . To shorten the writing of the equawhere H tions we introduce the following notation, 0

0

t }, (An ; Bk ) Å Tr{Anr t0 e it HH Bke 0it HH r 10 0

*

0

dt

e 0ivt/dt 2xk

[B2]

* d t(A ; B ), 1

n

k

[B3]

0

where An and Bk are the first and second commutators in Eq. [B1], which can be evaluated using Eqs. [17]. With the 0 above definitions, K nk ( v* ) can be written as 0 ( v* ) Å 0 (gsmB ) 2 ∑ K nk

∑ {(Iv( S zin S j0m , S zik* S j/l* )

m ( xn ) in, jm l ( xk ) * * i k, j l

/ Iv( S zjm S i0n , S zj l* S i/k* ) 0 Iv( S zjm S i0n , S zik* S j/l* ) 0 Iv( S zin S j0m , S zj l* S i/k* )},

[A5]

Second, if we define the static susceptibilities of spin packets by

AID

0 K nk ( v* ) Å

0`

l

xn Å

To complete the derivation of the kinetic equations, describing the ESR spectra hyperfine structure, we evaluate the kinetic coefficients, taking into account all remarks under Theory. For this purpose let us rewrite Eqs. [21] and [25] in detail,

exp( 0 bH / b ∑ M Hl ),

Q0 Å Tr{exp( 0 bH L / b ∑ M zl Hl )},

NLM , [A8] 2I / 1

Evaluation of Kinetic Coefficients

z l

l

Nn Å

APPENDIX B

Iv( An , Bk ) Å b L

S(S / 1) , 3kBT

and all xn are equal to each other.

[A1]

with rq (t) defined by Eq. [10]. Denoting A n{ (t) Å 0 bdH n{ (t)

xn Å Nn (gsmB ) 2

[B4]

which is valid for both n Å k and n x k cases. v* Å v / id takes into account the discussion before formula [19] of the main text. As our theory is applied for the high-temperature case ( bvs ! 1, which means the temperatures T @ 1 K for 9 GHz X-band ESR), we shall consider the average single-site spin polarization in an external field to be equal to zero: » S zi … Å 0. With this approximation we can write, for n Å k, (S zin S j0m ; S zik* S j/l* ) Å (S zin S j0m ; S zin S j/m ) dii *dml dj j *, (S zjm S i0n ; S zj l* S i/k* ) Å (S zjm S i0n ; S zjm S i/n ) dii *dml dj j *, (S zjm S i0n ; S zik* S j/l* ) Å (S zin S j0m ; S zj l* S i/k* ) Å 0;

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AP: Mag Res, Series A

[B5]

24

ALIEV, TAGIROV, AND TAGIEV

and for n x k,

ions, because of the linear dependence on c of the sum [B13],

(S zin S j0m ; S zik* S j/l* ) Å (S zjm S i0n ; S zj l* S i/k* ) Å 0, v 2ex á c

(S zjm S i0n ; S zik* S j/l* ) Å (S zjk S i0n ; S zjk S i/n ) dij *dnl di * j dkm , (S zin S j0m ; S zj l* S i/k* ) Å (S zin S j0k ; S zin S j/k ) dij *dnl di * j dkm .

[B6]

(S zin S j0m ; S zin S j/m ) S(S / 1) 2S(S / 1) 0 btvm ivmt e . e 3 3

[B7]

Then, substituting [B7] into [B3] and performing integrations we get Iv( S zin S j0m , S zin S j/m ) Å fm ( v* ) Å

b xn

F

S(S / 1) 3

G

2

fm ( v* ), [B8]

d 0 i( vm 0 v ) . ( vm 0 v ) 2 / d 2

[B9]

Using expression [A8] for the static susceptibility xn we obtain finally Iv( S zin S j0m , S zin S j/m ) Å

S(S / 1) fm ( v* ). 3Nn (gsmB ) 2

[B10]

Applying similar calculations to the second contribution from [B5], and substituting the calculated Iv(rrr) expres0 sions into [B4], we find for K nn ( v* ) 0 K nn ( v* ) Å 0 v 2ex [ ∑ fm ( v* ) / 2Ifn ( v* )],

where i runs over all lattice sites of the crystal. This means that we neglect fluctuations of v 2ex , which originate from the randomness in spatial distribution of other manganese ions, surrounding the specific ion, considered the origin in summation over the site indices in [B13]. In the fitting of the experimental data, v 2ex will be considered the adjustable parameter. Applying the same argument, as above, we get, for the nondiagonal on the spin-packets indices kinetic coefficient 0 K nk ( v* ), the expression 0 ( v* ) Å v 2ex [ fn ( v* ) / fk ( v* )]. K nk

[B15]

Finally, the equation set [ 24 ] and the expressions for kinetic coefficients [ B11] and [ B15 ] together with definitions [ B9 ] for the function fn ( v* ) and [ B12 ] for v 2ex complete our derivation of kinetic equations, describing the dynamics of the ESR hyperfine structure in a magnetodilute SS. ACKNOWLEDGMENTS The authors are grateful to Professor R. R. Galazka for kindly having grown the samples and to Drs. I. A. Garifullin and N. N. Garifyanov from Kazan Physicotechnical Institute of RAS for recording the ESR spectra. T.L.R. gratefully acknowledges support of the Turkish Scientific and Re¨ BITAK) and the kind hospitality of the Department of search Council (TU Physical Engineering of Hacettepe University, Ankara.

REFERENCES 1. J. K. Furdyna, J. Appl. Phys. 53, 7637 (1982).

[B11]

m ( xn )

2. N. B. Brandt and V. V. Moshchalkov, Adv. Phys. 33, 193 (1984). 3. I. I. Lyapilin and V. V. Tsidilkovskii, Sov. Phys.-Uspekhi 28, 35 (1985).

where

4. P. I. Nikitin and A. I. Savchuk, Sov. Phys.-Uspekhi 33, 974 (1990).

v

2 ex

Åv

v 2ex Å

2 ex

S(S / 1) , 3(2I / 1)

∑* J 2ij .

[B12]

In v 2ex the prime indicates that the summation on i runs over the all sites occupied by manganese ions. If we assume every lattice site to be populated by manganese ions with equal probability, then formulas [B11] – [B13] will predict the 0 linear increase of the kinetic coefficient K nn ( v* ) with an increase in the dimensionless concentration c of manganese

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5. N. I. Zheludev, M. A. Brummel, R. J. Harley, A. Malinovskii, S. V. Popov, D. E. Ashenford, and B. Lunn, Solid State Commun. 89, 823 (1994). 6. R. T. Holm and J. K. Furdyna, Solid State Commun. 15, 844 (1974).

[B13]

i ( xj )

AID

[B14]

i ( xj )

Let us calculate, for example, the contribution of [B5], the 0 ( v* ). Adopting first line trace, to the kinetic coefficient K nn z the set of S a operator eigenfunctions {É Ma… } as the basis functions, we calculate first the trace according to Eq. [B2]:

Å

∑ J 2ij ,

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7. D. P. Mullin, R. R. Galazka, and J. K. Furdyna, Phys. Rev. B 21, 355 (1981). 8. M. Inoue, H. Yagi, T. Muratani, and T. Tatsukawa, J. Phys. Soc. Jpn. 40, 458 (1976). 9. T. Tatsukawa, J. Phys. Soc. Jpn. 50, 515 (1981). 10. S. B. Oseroff, Phys. Rev. B 25, 6584 (1982). 11. M. N. Aliev and L. R. Tagirov, Phys. Status Solidi B 127, K61 (1985). 12. D. J. Webb, S. M. Bhagat, and J. K. Furdyna, J. Appl. Phys. 55, 2310 (1984).

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WIDE-GAP SEMIMAGNETIC SEMICONDUCTORS 13. R. E. Kremer and J. K. Furdyna, Phys. Rev. B 31, 1 (1985). 14. J. C. Wolley, S. F. Chehab, T. Donofrio, S. Manhas, A. Manoogian, and G. Lamarche, J. Magn. Magn. Mater. 66, 23 (1987). 15. N. Samarth and J. K. Furdyna, Solid State Commun. 65, 801 (1988). 16. D. N. Zubarev, ‘‘Nonequilibrium Statistical Thermodynamics,’’ Nauka, Moscow, 1971. [In Russian] 17. D. N. Zubarev and V. P. Kalashnikov, Sov. Phys.-Math. Phys. 5, 406 (1971). 18. A. Abragam, ‘‘The Principles of Nuclear Magnetism,’’ Clarendon, Oxford, 1961. 19. N. Bloembergen and T. J. Rowland, Phys. Rev. 97, 1679 (1955).

AID

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25

20. A. A. Abrikosov, J. Low Temp. Phys. 39, 217 (1980). 21. A. Manoogian, B. W. Chan, R. Brun del Re, T. Donofrio, and J. C. Wolley, J. Appl. Phys. 53, 8934 (1982). 22. S. F. Chehab, G. Lamarche, A. Manoogian, and J. C. Wolley, J. Magn. Magn. Mater. 59, 105 (1986). 23. B. E. Larson, K. C. Hass, and H. Ehrenreich, Solid State Commun. 56, 347 (1985). 24. J. C. Wolley, S. F. Chehab, and T. Donofrio, J. Magn. Magn. Mater. 61, 13 (1986). 25. P. W. Anderson, Phys. Rev. 109, 1492 (1958). 26. M. P. Rozman, Opt. Spectrosc. 4, 536 (1958). [In Russian]

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AP: Mag Res, Series A