Exciton dispersion in MnI2 and MnBr2

~

Solid State Communications, Vol.51,No.9, pp.657-66], Printed in Great Britain.

1984

0038-i098/84 $3.00 + .00 Pergamon Press Ltd.

EXCITON DISPERSION IN Mnl2 and MnBr2 H.J.W.M. Hoekstra, H.F. Folkersma and C. Haas Laboratory of Inorganic Chemistry, Materials Science Center of The University Nijenborgh 16, 9747 AG Groningen, The Netherlands Received by S. AMELINCKX - May 25, 1984

The line shape of some of the optical d-d transitions in Mnl2 and MnBr2 is asymmetric and depends strongly on temperature and magnetic field strength. It is shown that these effects are due to the hopping of excitons from one Mn 2+ ion to another. The hopping integral depends strongly on the correlation between the directions of the spins of the two ions involved. The line shape in the magnetically ordered crystals depends in a sensitive way on the type of magnetic order. We also present a simple method to calculate the line shape in crystals where the magnetic moments are not fully ordered.

Mnl 2 and MnBr 2 crystallize in the Cd(OH) 2 structure, with hexagonal layers of metal ions separated from one another by two layers of anions. The interlayer distance between the Mn 2+ ions is much larger than the intralayer distance. The crystals have magnetic moments on the M-n2+ ions that order antiferromagnetitally at low temperature. From neutron diffraction a helix I type of magnetic order has been deduced for Mnl2; the spins in the hexagonal layers are ordered in rows of parallel spins, which make angles of y = 67.7 ° between spins of adjacent rows z (see inset of Figure 3b). The magnetic structure of MnBr 2 consists of pairs of rows of parallel spins (see inset of Figure 3c). The magnetic exchange interactions between the Mn 2÷ ions in Mnl 2 and M_nBr2 are weak, as follows from the low values of the paramagnetic Curie temperatures e = -8K for Mnl23 and 8 ~ -4.7K for MnBr24. The N~el temperatures T N depend on the magnetic field; for Mnl2 TN = 3.6K at H = O, T N = 3.OK at H = 2T and T N = 2.5K at H = 4.75T 5, and for MnBr2 T N = 2.16K at H = 0, TN = 2.0K at H = 2T and Tr~ = 1.8K at H = 2.7T 6. The d-d transitions of the high-spin manganese compounds Mnl2 and MnBr2 are parity and spin forbidden for electric-dipole transitions. Some of these transitions are non-vibronic and exchange-induced s'?, and for these transitions the line-shape will show the effects of magnon and exeiton dispersion. The magnon dispersion is expected to be small because the magnetic interactions are weak. The Mn2+(3d s) ions in Mnl 2 and MnBr2 have a 6 A I ~ ( S6 ) ground state; all excited states of The 3d s configurations are quartets or doublets. Thus the transfer of excitations involves exchange interactions and the hopping integral depends strongly on the relative orientation of the spins of the two ions involved. As a consequence the density of exciton states and also the line shape of an optical transition will depend strongly on the magnetic structure of the solid.

Optical absorption spectra of Mnl2 and MnBr 2 have been measured at low temperature on crystal plates cut perpendicular to the crystallographic c-axis; the direction of the incident light and the applied magnetic field were parallel to the c-axis. In this paper we report measurements of the line shape of the non-vibronic exchangeinduced 6AI~(6S)= ~ ~Eg(~G) transitions of Mnl2 and MnBr2. (Figs. I and 2). The spectra

zero-phonon ,' 6Alg..~Eg /~ T= 2.9 K

'

H=4.75 T

"~

T= 2 . 5 K 5T

T = 2./+ K ~ ~ S T g

T=2.0 K

J~

T=I.5K H=OT I

22000

22250 o-(cm -1)

Figure I: Optical absorption of the zero-phonon exchange-induced 6Alg(6S) ~ gEg(gG) transition of IvLrll2 .

657

EXCITON DISPERSION

658 ,

-

•

'

•

i

,

•

I 6A1g._,~Eg

-4 1.0

E(ke) = E 0 + t {2 cos(kxa)

T= 1.8 K

<

Vol. 5], No. 9

We first discuss the hopping of excitons on a hexagonal lattice, with all magnetic spins on this lattice fully ordered in some ferromagnetic or antiferromagnetic pattern. If only hopping between nearest neighhour sites (distance a) is taken into account and if it is assumed that all spins are parallel (ferromagnetic order, one excited ion with M S = 3/2 and all other spins with M~ = 5/2 ) the wave functions of the excited s~ates are linear combinations of wave functions in which the excitation is localized on one site. The energy of an exciton with wave vector k e for a simple hexagonal lattice of parallel spins is7:

,

Lxzero-phonon

~

IN Mnl 2 AND MnBr 2

+ 4 cos(½kxa)

0.5

cos(½kya/3)}

(2)

Using the formalism of the Green's function, the density of states per formula unit is given by

23000

g(E)

23500

o-(crnA)

4~/a/3 × ~

Figure 2: Optical absorption of the zero-phonon exchange-induced 6AIg(6S) ~ ~Eg(~G) transition of MnBr2. show clearly a dependence on the magnetic field and the temperature. In order to explain the observed effects we discuss in some detail the consequences of exciton transfer on the line shape of exchange induced optical transitions. In an exchange-induced transition an exciton (wave vector ke) and a magnetic excitation (magnon, wave vector k m) are produced simultaneously. The selection rule for optical transitions ~s k e + k m ~ 0, the energy relation is E = E 0 + Ee(ke) + ~L~m). Exchange interactions in Mnl2 and MnBr 2 are weak and therefore the energies of magnetic excitations Em are small (a few cm-1). The dispersion of ~Eg(4G) excitons in Mn 2+ compounds can be appreczable . Therefore we attribute the width of the zero-phonon ~AIR(6S) ~E~(~G) line to exciton dispersion ~nd neglect the contribution of magnetic excitations. The shape of the absorption band will refTect the density of exclton states. The hopping of an excitation from a site a to another site b is given by a hopping integral h = <~a]~l.[~>, w h e r e ~ h represents the exchange~in~er~ctions between the electrons on ions a and b 9. Because of the large interlayer distance, we consider only hopping of excitons between sites on the same hexagonal layer perpendicular to the c-axis. The hopping integral depends on the resultant spin S' of the two lens a and b as h = (112o) t

S'

(S'+l)

a2/3 - ~

=

(1)

For a pair of ions, with one of the ions excited to a S = ~2 quartet state and one in the S = ~2 ground state, S' can take the values 1,2,3,4. The transfer integral for hopping between parallel spins (S' = 4) is t.

2w/a Im ~ dk x

[E + is - E(ke)]-i

dky

(3)

Here s is a measure for broadening effects, e.g. life time broadening. The integral can be evaluated by integrating first over k. and then numerically over k x. The result Is given in Fig. 3 for E 0 = 0 and s = 0.0St. The shape of g(E) shows a great resemblance with the line shape of the non-vibronic exchange-induced transitions of MnI 2 and MnBr 2 (Figs. I and 2), especially the asynm~etry is reproduced quite well. The parameter t for the hopping of excitons may be estimated from the total line width, which is 9t if all spins are parallel. This is approximately the situation in a high field (H = 5T) at a temperature T N 3K (Figs. | and 2). We find t ~ 9 cm -I and t ~ 10 cm -I for the transfer of t h e 4 E ~ 4 G ) excited state of Mn 2+ in Mnl 2 and MnBr2, respectively. Next we discuss exciton hopping in the antiferromagnetically ordered states. For the magnetic structure of MnI2, with a helical ordering of the spins (Fig. 3b), the "hopping" unit cell is identical with the crystallographic unit cell and each Mn 2+ ion feels the same environment. This is in agreement with the fact that the spectra of MnI 2 below T N consist of a single sharp peak. The hopping integral in one direction is equal to t; for the hopping integral in the other direction we take the classical value t cos2(½y) = 0.72t (T is the angle between neighbouring spins) I° . For the exeiton dispersion in the magnetically ordered state of MnI2 (at T = OK) we obtain E(k e)

=

E0 + t [2cos(kxa )

+ 4 × 0.72 The with the is shown observed moderate. that the

cos(½kxa)

cos(½kya/3)]

(4)

density of states is calculated again Green's function method; the result in Fig. 3b. The resemblance with the line shape given in Fig. 2 is only Differences will be due to the fact magnetic moments are not yet fully

Vol. 51, No. 9

EXCITON DISPERSION

•

•

'L

659

IN Mnl 2 AND MnBr 2

22-2-¢

•

!

I I

(cl

(bl

i

-2 0 2 l+

-2 0 2 4 6 E (units tl

-2 0 2 ~

Figure 3: Calculated density of exciton states for a hexagonal layer of spins with (a) ferromagnetic order (b) helical order, Mnl2-type (c) antiferromagnetic order, MnBr2-type.

aligned at 1.5K, so that E0 in eq. (4) is not a constant but differs from site to site due to diagonal exchange interactions. For the magnetic structure of MnBr2 the magnetic unit cell in the hexagonal plane is four times the crystallographic unit cell, but the hopping unit cell is only two times the crystallographic unit cell. The Mn 2+ ions are labelled as indicated in Fig. 4; odd (2p-I) and even (2p) numbered rows of ions should be

2p-1

,'

(a)

,'

t~/

\o

t

0.1t

,-'"B: 1c1

:

2p

2p+1

' \/

f'-.,

discriminated. The hopping integrals between sites with parallel and antiparallel spins are t and t' = 0.1t (eq. (I)). The eigenvalue problem for the exciton states in MnBr 2 may be solved by choosing exciton wave functions of the form ~(k)

Figure 4: Magnetic structure of MnBr2 in a metal layer perpendicular to the c-axis. (a) magnetic unit cell with 4 Mn 2+ ions, two with spin up and two with spin down. (b) unit cell for exciton hopping, with two Mn 2+ ions. The hopping integrals between parallel and antiparallel spins are t and 0.1t, respectively. (c) twodimensional reciprocal lattice, with first Brillouin zone for exciton hopping. The lattice points of the crystallographic unit cell are indicated by black dots.

~ [l~n,2 p exp {2~i(nkl + 2pk2)} n,p

+ ~i0n,2p_1 exp {2winkl + 2~i(2p-1)k2}]

(5)

k is the exclton wave vector, with k = kla* + k2be , a* and b* are lattice vectors of the twodimensional reciprocal lattice, defined by a.a* = b.b~ = 2~ and a.b* = b.a* = 0. The vectors k are confined to the two-dimensional Brillouin zone which corresponds to the exciton hopping unit cell with translational periodicities a and 2b (see Fig. 4). ~n an represents an exclton localzzed on szte na + 2pb. The eigenvalues for the exciton states of MnBr2 in the magnetically ordered state at low temperature are e(k)

:

=

=

2t cos (2~kl) ± [2{1+cos(2~kl)}

x {t 2 + t '2 + 2tt' cos(4zk2)}] ½

(6)

and correspond to two branches in the first Brillouin zone (Fig. 5). The elgenvalues were calculated for a number of points in the first Brillouin zone, and from these calculations we obtained the density of states given in Fig. 3c. The calculated density of states should be compared with the observed line shape of zerophonon exciton bands, i.e. the frequency region 23000 - 23150 cm -I in Fig. 2, for H = 0, T = 1.8K. We see that the calculations indeed explain the strong asymmetry and the spllttin~; of the band in two (or more) subbands in the magnetically ordered state (as compared to the paramagnetic state T = 3.2K, H = 0, which shows a single line). The calculated density of states shows three peaks, the observed band (Fig. 2 for T = 1.8K, H = 0) shows only two

EXCITON DISPERSION

660

UJ

/

0

-3

A

e

c 9 wavevecfor k

B

Figure 5: Dispersion curves for excitons in the antiferromagnetic MnBr2-structure, along directions in the first Brillouin zone, as indicated in Figure 4. peaks. The difference between calculated and observed line shape is caused by broadening effects, so that the first two peaks in Fig. 3c are not resolved, and by the fact that the crystal is still far from the fully ordered state. The observed energy difference between the two peaks at 1.8K (Fig. 2) is 49 cm -I. This is in good agreement with the value of 5.5t = 55 cm -I, which one obtains from the calculated density of states curve (Fig. 3c) combined with the value of t ~ 10 cm-1 deduced from the width of the band at strong magnetic fields (T = 3.2K, H = 5T in Fig. 2). If the spins are more or less randomly oriented, i1~ormation about the density ofz~tates can be obtained by the use of moments We consider an ensemble of hexagonal layers, each consisting of N atoms with a certain distribution of the orientation of the spins. The moments of the average density of states per atom, g(E), are defined by: +oo

~p =

~

dE E p g(E), p = 0, I . . . . . .

(7)

--co

The moments pp are connected with the Hamiltonian by ~p

N -I Tr [ < ~ > ]

-2J Sa.S b + 2g~B~0 H(Sz, a + Sz,b)

Vol. 5|, No. 9

it is straightforward to calculate the distribution p(S), where p(S) is the probability of finding a resultant spin equal to S (= 0, I, .... or 5). A precise calculation of the distribution p(S'), with S' (= I, 2, 3, or 4) being the resultant spin of one excited ion and one ion in the ground state, is lengthy and of an accuracy beyond the scope of this simple model. We assume that the resultant spin does not change for S = S' = I, 2, 3, 4 if one of the ions becomes excited by absorption of light or by hopping of an excitation from a neighbouring site. For S = 0 or 5, S' becomes | or 4, respectively. It is assumed that the created magnon does not affect the distribution p(S'). We further neglect energy differences due to interactions with other magnetic ions; i.e. we consider only pairs of magnetic ions. We obtain for the first five moments7: ~0 = I ; ~i = 0 ; P2 = ~ 6p(S') h2(S ') ; S'=I 4 p~ = E , 12p(S~)h(S~)p(S~)h(S~)p(S~)h(S~)

s;,s~,s~=l

4 4 p~ = ~,=16p(S')h4(S' ) + I0~2 E p(S')h2(S ') S'=1 4 + 2~3E p(S')h(S') S'=I

(10)

The observed moments of spectrum I of Fig. I, recorded at 2.9K, i.e. above the N4el temperature, are I, 0, 465.5(cm-Z) ~, 7.862 × 103(em-Z) 3, 6.22 x |05(cm-Z) 4 for ~0, .-.~, respectively. With the pair approximation mentioned above we calculate for the moments at T = 2.9K, H = 4.75T and J = -0.15 em -I (calculated from @ = -8K 3 and k@ = 2z JS (S+1)/3; z = 6, S = s/2) values of I, 0, 5.82t z, 11.3t 3, 84.4t~ respectively. With t = 9.0 em -I we obtain form the observed moments the values of I, 0, 5.74t 2, I0.8t 3, 94.8t ~. The agreement with the calculation is quite good. We remark that the values of the moments reflect the topology of the crystallographic structure of the metal layers.

(8)

If we consider again the case of complete ferromagnetic order the moments are easily calculated with eqs.(2),(3) and (7) (with s + 0; E0 = 0 for convenience) leading to ~0, ~], P2, ]~3, P~, .... , Pz2, .... = I, O, 6t 2, 12t 3, 90t ~, .... , 6 × I07t z2, .... Eq. (8) gives of course the same results; the numbers I, 0, 6, 12, 90, .... are the numbers of ways in which one can hop from one site to neighbouring sites in a hexagonal two-dimensional lattice, in 0, I, 2, 3, 4, .... steps, ending at the starting point. In order to calculate the first few moments of the density of states for a more or less disordered system of spins we approximate the correlation between the spins in the ground state by taking into account interaction between pairs of spins only. With the aid of the Hamiltonian ;~=

IN Mnl 2 AND MnBr 2

(9)

Table I. Comparison of observed moments pp ~p=2,3,4) of the line shape for the 6Azg(bS) E~(G) transition of MnI 2 in units t (t=9am-) an~ calculated values for the fully ordered magnetic state; Po and pz are chosen to be 1 and O, respectively.

Pz

observed H = 4.75T T = 2.9K

observed H = 0 T = 1.5K

calculated H = 0 T = OK

5.74

4.35

3.90

P3

10.8

12.8

~

94.8

86.7

5.71

The discrepancy between the calculated density of states of the ~Eg(4G) transition of Mnl2 (Fig. 3) and the observed line shape for temperatures below T N (Fig. I) is caused by the

;

EXCITON DISPERSION IN Mnl 2 AND MnBr 2

Vol. 51, No. 9

66]

Mnl2 and MnBr2 is attributed to the effect of exciton diffusion. The line shape reflects the correlation between the spins of neighbouring ions and may be calculated fairly accurately if the solid is magnetically ordered. A moment analysis shows that discrepancies between calculated and observed line shapes are mainly due to differences of the higher moments, which are caused by the fact that in our model we have neglected energy differences due to diagonal exchange interactions. The lower moments of the observed line shapes are reproduced quite well by using a simple Hamiltonian.

fact that the solid is not fully ordered at T = 1.5K, so that the value E0 of eq. (4) is not a constant for all sites. This will affect especially the higher moments of the absorption curve. This is in agreement with the result given in Table I: e.g. the third moment of the observed line shape is much larger than the calculated value with E0 = 0 for all sites. However, the observed second moment 4.35t 2 is in a good agreement with the calculated value ~2 = 2t2 + 4 (0.69t) 2 = 3.90t 2. CONCLUSIONS The asymmetric line shape of some of the zero-phonon exchange-induced transitions of

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