Infrared absorption on impurity excitations near the upper edge of spin-wave band of the antiferromagnet

Solid State Communications, Vol. 34, pp. 629—633. Pergamon Press Ltd. 1980. Printed in Great Britain. INFRARED ABSORYFION ON IMPURITY EXCITATIONS NEAR THE UPPER EDGE OF SPIN-WAVE BAND OF THE ANTIFERROMAGNET M.A. Ivanov and Yu. G. Pogorelov Institute of Metal Physics, Academy of Sciences Ukrainian SSR, Kiev 180, USSR (Received 22 January 1980 by E.A. Kaner) The dependence of infrared (IR) light absorption on external magnetic field is considered when the impurity level approaches the upper edge of spin-wave band of antiferromagnet. It is explained a strongly nonlinear field dependence and rapid decrease of corresponding line intensity been observed experimentally in Fe1~Co0F2.Such behaviour is due to existence of some subthreshold range near the band edge and corresponds to the incoherent collective rearrangement (ICR) of the system spectrum. AN INVESTIGATION of electromagnetic radiation absorption in antiferromagnets with the impurity levels near the spin-wave band edge is of considerable interest now [1—4]. In such a system the band edge ~.)m(H)and impurity level wo(H) may be drawn closer when the external magnetic field H is applied. As a result different kinds frequencies of the spectrum rearrangement are observed in these crossing the vicinity [5]. So far the main attention was given to the case when the impurity level approaches lower band edge corresponding to the uniform precession frequency (AFMR frequency). Thereby the strong increase of intensity of the impurity line (the “burning up”) was observed. Then either this line confluence with the AFMR one [21 or cross splitting of these lines [1—4] occur depending on the ICR or the coherent rearrangement (CR) of spectrum taking place, respectively [61. An interesting experiment [7] concerning IR absorption in the crystal Fe 1_~Co~F2 (impurity concentration c 1%) was carried out recently. The impurity 2~level (~ 85.5 cm’) is located slightly above the Co upper band edge (~78 cm’ [81) in this system. In the fieldH II C 4 (C4 is the antiferromagnetic axis) an essentially nonlinear field dependence of frequency was observed for the line which approaches the band edge. Accordingly the intensity of this line decreased (linearly in the band edge vicinity); that is “burning down” took place instead of “burning up”. An essential difference of an impurity line’s behaviour as compared to [2] is due to the upper edge being inactive optically. Note that analogous “burning down” effect is known for the exciton line in molecular crystals [91. When the frequencies ~,0(H) and ~~m(H) approach themselves the impurity state radius i(H) = av’E(~2/Iwo(H) Wm(H)I)] (a lattice parameter and £2 parameter determined by the band width) increases —

—

—

and may exceed the mean distance between impurities ac’ “s. Then a collective spectrum rearrangement occurs in the close vicinity of wo(H) and Wm(H) crossing so that the single impurity approximation (SIA) fails here. The kind of rearrangement is determined by the relation between c and some 6concentration (m is the characteristic systeminteraction ce,. = ~(m/cz) matrix elementforofthe resonant of impurity and host spin-wave excitation and ~ are some numerical factor). The ICR occurs at c ce,.. The impurity line behaviour far from its crossing with the band edge (when the SIA is true) depends nevertheless essentially on the kind of collective rearrangement taking place in the crossing area. Hence, studying the frequency and the intensity of impurity line in a wide range of fields, a conclusion may be drawn about the kind of rearrangement which have to occur in given sample. Particularly the nonlinear dependence ~0(H) and linear intensity fall would have to to be [7].observed for ICR near the upper band edge similar To describe the impurity level behaviour near the upper band edge we shall use the simplest model considering only the resonant interaction of the upper (from the two split in the field) branch of host spin excitations with that impurity level which frequency decreases with the field increasing (since the impurity ions can substitute both magnetic sublattices there are two impurity levels at H II C4): ~C = ~ [w(k) + pgH] a~ah+ ~ (“.‘o k

—

p,gH)a;a~

P

i +

-~

~ (m e~ a~a~ + h.c.),

V

k,p

—

where 4 is the magnon creation operator and the 629

(1)

630

INFRARED ABSORPTION ON IMPURITY EXCITATIONS

Vol. 34, No.8

e~) (cni~)

88 86

—

° n

84 82

— —

8O

— —

78

— •

I

I

I

‘ II

I I

I

I

8)

~ 1.0 ~,

-~-~

0.8 0.6

~0.4 ‘— 0.2

TITT~1 1

2

3

4

5

0 7H~~ H(T)

6

Fig. 1. Field dependence of frequency (a) and intensity (b) of impurity line when it approaches the upper band edge in Fe1_~Co~F2. 1. The dots are experimental data of [7], full lines theoretical curves at ~S= 15 cm”, 2j.~gH,. 15.6 cm dispersion law near the upper band edge is w(k) + ~ e~_~”((a~Ia;’)Y~], (2) Wm (47T)2~’3cz(z2(k km)2, p denotes the impurity N~ sites with the spin directed opposite to the field and N is the number of magnetic cells in crystal. The resonant where A is a parameter weakly depending on frequency. interaction constant m is non-zero for oppositely Using the Hamiltonian (1) an expression for a(co) near polarized excitations only if the single ion anisotropy is the upper band edge Wm(H), when 1w j4gHI ~ taken into account [10]. According to the experimental w w(0) pgH £2 and c ~ 1, may be obtained: data for Fei..~Co~F 2 g-factors of impurity and host m were considered to be equal [3, 7] (g 2.2). o(w) = cA 1 + w—w(0)—~gH) The coefficient of absorption of light polarized in the plane I C4 may be written as usual by means of [‘~/2 + Im P(co, 2 + H) [r 2 retarded Green’s functions [11]: [w w0 + pgH Re P(w, H)] 0/2 + Im P(w, H)] a(w) = A Im [<(aaIa~))’~ Here F 0 is an intrinsic width of impurity line, and (3) k 0 P(w, H) is the polarization operator due to impurity— host interaction; the form of P(w, H) depends on the + ~ (e~’(” + e ~ ((ak?, I a~)’°) choice of relevant representation (renormalized or non-renormalized [6]). As c ~ 1 we neglect elsewhere the clusters of neighbouring impurities which do not —

—

—

—

—

—

(

,

—

—~

—

—

—

INFRARED ABSORPTION ON IMPURITY EXCITATIONS

Vol. 34, No.8

631

—S

•1~

CL) (cn~ ‘.5— ‘

.~. ~ ~.

88

‘5—

1.0

S..-

86/’

0.8

-

_~‘

,/

—, ---

//

——

,‘

,/ /

,/

-

‘

,___

,

‘S

,f

/

~

4~

~

/

.‘ ,

5;,

,.

--

/ /

/

/~ /

/

-,

--~S ,

/

-

“ /

82

,

__,-,

, /

,

/

0.6

‘

//

“,

,,

84

—

,,‘‘

,

——

/

/5

~ __5

/

, ,‘

,~

, /

0.4

/

/

3

4

/

‘

-S.-

80 78 1

2

5

6

7H,18

/1(7’)

9

Fig. 2. Dependence of light absorption a(w) on frequency at various field values. appreciably affect the light absorption near the single impurity level, (a) First we will consider the impurity line behaviour for the case when c
The impurity level frequency c4.,0(H) is determined by the solution of equation:

Let the condition I’

2l~Hr = w

~-

~

w—w0+~.~gH—ReP(o.,,H) = 0.

Taking into account that the second term in equation (5) is real out of the band where w> wm + pgH, we obtain: wo(H) = wm + pgH+ 2pg(H,. —H) + 6/2

—

k

3m2/~l be fulfilled. Then the

(6)

(7)

+ 2,.~g(H,.—H)6], H
_\/[62/4

0 > c’~’

0 + P0 indirect interaction via the magnons exchange between impurities (i.e. concentrational broadening of impurity The frequency wo(H) at first decreases with H, level) may be neglected even in the collective rearrange- achieving the minimal value Wmifl = C~)m + ugH,. 6/8 ment area and the expressions (3,4) ate
—

—

—

6

=

—

—

m4/~23.

(5)

Here the terms of next orders on (w pgH)/fZ ~ I are omitted. The quantity 6 determines a width of subthreshold range [61 and just this quantity scales such phenomena as nonlinear field dependence of frequency and the fall of intensity of the impurity line near the band edge. —

—

removed by using more precise approximation for P(w, H) (4) compared to expression (5), far from the band edge. An effective g-factor: g~~(H)=

I dwo(H) 11

~

632

=

INFRARED ABSORPTION ON IMPURITY EXCITATIONS g (1 —

—

.,/[2pg(H,.

6 11)6 +

—

62/4]

)

(8)

varies strongly near the band edge in accordance with the experiment [7]. At H

---

1 —

i 1 +2

(H))

=

=

6

—

wo(H)

s,/[2~g(H,.

—

—

-~

H) +

—

1

—1411

—

6/41

—

—

113m2/~ it covers ICR in this case. However the linesowidth at Hthe Hr is range I’~/6when F0 > c When F 3m2/1Zboth peaks would merge in the fields ~g(H~ 0
~(H) w(0) —~~H) m

—

=

below the noise level may not be registered in real experiment conditions. So the additional maximum is not displayed and only the intensity 11(H) of “main” line is measured. As H tends to H,. both peaks draw closer and merge at ~~g(H,. H) F0. In the fields greater than H,. the impurity level becomes virtual and only the broad maximum within the band remains. The frequency dependences of light absorption o(w) at various fields (H

---

H,.) are displayed in Fig. 2. 1~’3m2/~ The ICR would occur at pgIH,. —Hi C in the band edge vicinity 1w wm ILgfil c2~’3~2 [6]. —

—

Vol. 34, No. 8

v’~l2< 1

(9)

remains invariable, including the nonlinear field dependence impurityand line’s its narrowing and intensityofdecrease, alsofrequency, the existence of supplementary of 0(w)the within the band. (b) Now letpeak us analyse behaviour of lines in the

—

K(H) 4j.~g(H,.—H)/6 when 2zg(H,. H) ~ 6/4 [12]. Thus integral intensity 1 1(H) of the impurity line does fall withH increasing and the dependence I,(H) is linear in the H,. vicinity. The comparison with the experimental data is shown in Fig. 1(b) (the parameter m is chosen here as m = 22 cm’, but 11(H) is of small sensitivity to this value). The theoretical curve 1~(H)is normalized so to provide the best agreement with experimental data near H,. when wo(H) i~H~ £2. Similar to wo(H) the agreement in the small fields may be achieved using the exact values ofP(w, H) instead of equation (5). It(9) should also be remarked that as from equation the narrowing of impurity lineseen occurs when H Hr but the height of its maximum —

—

—

-+

remains invariable. An account of inhomogeneous broadening does not change these results. Besides the narrow peak at wo(H) there is also a broad asymmetric in a(w)spectrum given byrange equation (3) withinmaximum the continuous (w < Wm + pgH) [13]. The spectral distribution here

may be derived easily taking into account the second term in equation (5) becomes purely imaginary. This distribution maximum is located at the frequency ~H) = + MgH— [2i~g(H,. —H)]2/6 when 2,ug(H,. —H) ~ 6 and its2 height equalsH)]. cA[1 + ml 6/ [2/2g(H,. A long tail (c.~(H) w(0) pgH)] of this distribution is extended far into the band and its intensity decreases as (wm + j.~gH—w)1~’2up to Wm + MgH— w 6 and then as(wm + 14H w)3”2. In result the total absorption intensity 1(H) f a(w) dw conserves and equals ~ cA{l + m/ w(0)1}2. However the values of a(w) which are —

—

—

“-‘

=

—

absorption spectrum in the case c > ce,. when the CR occurs in the crossing range. Thereby the renormalized representation [6] is to be used in the definition (4) of P(w, H), particularly the concentrational shift of the band edge must be taken into account. As known [6] at CR a novel (impurity) band arises having the wave vectors and polarization similar to those of the nearby edge of the main band. The width of this additional band is ‘../~mat H ~ H,. and it is separated from the main band by the quasigap of the same width. Thus a distance between limiting frequencies of these bands equals 2-%/~mat the cross splitting range. Letas us 1 “3m2/~2>6. As far the assume ~/~rn> F0 > c upper edges of both the main and the impurity bands are optically inactive in the crystal Fe 1_~Co~F2, the light absorption occurs in the range between them on the localized states corresponding to “unperturbed” impurity frequency wo(H) w0 ugH. The(and concen3m2/~2 trational width of impurity line c’ ~‘ then F 0) exceeds at c > c,~,.the subthreshold width 6, so the effects of nonlinear wo(H) dependence become irrelevant. The impurity line being of about constant intensity penetrates atitH> Hr farresonance into the band. At 2~’3~2 becomes with the 2~g(H H,.) > c width F 0 + sJ[2~g(H—H,.)6] growing slowly with (HHr). The analysis carried out provides a conclusion that a picture been observed in [7] corresponds to the case of ICR, i.e. to the impurity concentration in the sample C C 1%. Note that the upper impurity level frequency depends on H linearly (in accordance with [7]). This is ~

—

—

-~

-~

‘~

Vol. 34, No.8

INFRARED ABSORPTION ON IMPURITY EXCITATIONS

due to this level interaction with the upper spin-wave branch which have the same polarization is field independent (for impurity and host g-factors are equal) while its interaction with the lower branch is weak.

5. 6.

633

M.A. Ivanov, Yu.G. Pogorelov & V.M, Loktev, Mat. 20th Nat. Conf Low Temp. Phys. NT.20, II. Moscow M.A. Ivanov(1979). & Yu.G. Pogorelov,JETP76, 1010

(1979)

7.

U. Diirr & B. Uwira,Abstr. Joint. Intermag-MMM Conf New York (1979). [J.Phyn C12, L793 Acknowledgements We are thankful to A.S. Prokhorov (1979)]. drawing our attention to the paper [7] and V.M. 8. M.T. Hutchings, B.D. Rainford & H.J. Guggenheim, Loktev for useful discussion. /. Phys. C3, 307 (1970). 9. E.I. Rashba, Fir. Tverd. Tela4, 3304(1961). REFERENCES 10. V.M. Loktev & Yu.G. Pogorelov, Fir. Nizkikh Temp. 5,483 (1979). 1. A.S. Borovic-Romanov & V.F. Mescheryakov, 11. R.J. Elliott, J.A. Krumhansl & P.L. Leath, Rev. JETPLett. 8,425 (1968). Mod. Phys~46,465 (1974). 2. AS. Prokhorov & E.G. Rudashevskij,JETPLett. 12. Note that K(H) = hl’012, where ji~ is that part of 22, 214(1974). wave function of impurityexcitation with eigen3. R. Sanders, V. Jaccarino & S. Rezende, Solid frequency wo(H) which is localized on the State Commun. 28,907 (1978). impurityspin [61. 4. M.A. Ivanov, Yu.G. Pogorelov, V.M. Loktev, 13. The existence of additional maximum for the local K.N. Kocharyan, A.S. Prokhorov & E.G. level near the band edge was mentioned in Rudashevskij, Solid State Commun. 33,623 P.H. Dederichs & R. Zeller, Phy& Rev. B14, 2314 —

(1980).

(1976).