The Jahn-Teller theory of Mn3+ : Al2O3

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The Jahn-Teller theory of Mn3+ : Al2O3

C.A. Bates, P. Brauns, J.R. Fletcher, P.C. Jaussaud

To cite this version:

C.A. Bates, P. Brauns, J.R. Fletcher, P.C. Jaussaud. The Jahn-Teller theory of Mn3+ : Al2O3.

Journal de Physique, 1976, 37 (6), pp.763-767. �10.1051/jphys:01976003706076300�. �jpa-00208472�

THE JAHN-TELLER THEORY OF Mn3+ : Al2O3

C. A. BATES

(*),

^{P. BRAUNS}

(t),

J. R. FLETCHER

(*)

and P. C. JAUSSAUD

(t)

(*) Department

^of

Physics, University

^of

Nottingham, University ^Park, Nottingham

^NG7

2RD, England (t)

^Service

B.T., C.E.N.G.,

^B.P.

85,

^{Centre de}

Tri,

³⁸⁰⁴¹

Grenoble,

^France

(Reçu

^{le 8 décembre}

1975,

^{révisé le}

16 février 1976, accepté

^le

17 février 1976)

Résumé. ²⁰¹⁴ Une étude expérimentale ^{des ions Mn3+} par

spectroscopie infrarouge

(Aurbach ^et

Richard)

^dans

Al2O3

vient d’être publiée. ^Nous présentons ^{ici une} analyse

théorique

détaillée de ces

résultats, ainsi que de ^ceux que ^nous avions obtenus en résonance

paramagnétique acoustique.

^Nous

montrons que ^tous les résultats

expérimentaux

^{connus à ce} jour ^{sur l’ion Mn3+ sont}

expliqués

^de

manière satisfaisante à l’aide d’un effet Jahn-Teller sur l’ion dont le fondamental est un 5E. On obtient ainsi les paramètres ^{de réduction}

Jahn-Teller q

⁼ 0,50, r ⁼ 0,70. p est ^{différent de} zéro, ^{mais ne} peut être déterminé. Ces résultats sont tels que 2 q - p 1, ^et donc que le modèle de Cluster n’est pas valable.

Abstract. ²⁰¹⁴

Following

^{the recent}

publication

of infra-red measurements on crystals ^of

Al2O3 containing

^Mn3+ ions, ^a detailed theoretical analysis ^{of the} system has been undertaken. From these results and our earlier APR measurements, it is shown by computer

analysis

that all the data for this 5E ion can be

satisfactorily explained

by ^a multimode lattice model of the Jahn-Teller effect. The

values q

⁼ 0.50, r ^{= 0.70} and p ^{finite are found} for the Jahn-Teller reduction factors, ^{so that}

(2 q 2014

p)

is less than unity.

Classification

Physics ^Abstracts

8.822 ^- 8.630 ^- 8.512

1. Introduction. ^-

(3d)4

ions in octahedral or near-

octahedral sites are well known to exhibit

strong

Jahn-Teller effects as

they

^have

^’E (S

^{= 2 T =}

2)

ground

^{states. The}

A’203

lattice is a

particularly important

^{host and}

many experimental

measurements and theoretical calculations have been made. The results obtained have been summarized in three papers

[1-3]

and in references contained therein.

Experiments using

far infra-red

spectroscopic

^methods

have

subsequently

been carried out on

crystals

^of

Mn3+ : A1203. Preliminary

measurements were

reported by Stoelinga et

^al.

[4]

^and

by

^{Aurbach et}

al.

[5]

^{and more detailed results in the}

frequency

range 3 to 30 cm -1 and with

applied magnetic

^fields up ^to 55 kG have

just

^been

given by

Aurbach and Richards

[6].

While a set of values for the parameters ^{for the}

Cr2 + : A’203

^{system were} deduced in

[3]

^{from the APR}

data,

the lack of accurate information on the

Mn3 + : A’203

^system

precluded

^{a detailed}

analysis

^{for that}

system. However,

^{the new results}

[6]

have indicated that the

problem

^of

^{Mn3 + :} A’203

should be re-

opened

^{and it is the} purpose of this note to

give

^the

results obtained.

2.

Theory.

^{- Consider a}

single ^3d4

^ion

coupled

^to

E-type displacements

^{of its}

surroundings.

^{The Hamil-}

tonian for this

system

^can be written as

where YE

^is the orbit-lattice

coupling

^{constant for}

E-type displacements (Q±

⁼

Qo

±

iQe)

^{of the nearest}

neighbours, 3Çmagnetic is

^the

^magnetic

Hamiltonian as

given

ⁱⁿ eq.

(3)

^of

[3]

^and

Jelattice

is the lattice Hamilto- nian. An

important

contribution to

JClattice

^{is the}

anharmonic

term -1 B(Q 3

⁺

Q 3 ).

We

simplify

^our discussion

by taking

^the

magnetic

field H

parallel

^{to the}

crystal

^{c-axis so that} the effects of randon internai strains and

electric fields

^{can be}

neglected [1].

^That

is,

^{we take}

Q + = Q - = 0 in Ilmagnetic

^{We also}

^neglect

^the

^trigonal

distortions in

JClattice

^{and assume} therefore that the Jahn-Teller

coupling

^is

diagonalized by

^the

unitary

transformation

exp(iOT3) given

ⁱⁿ

[3]

^{where 0 = tan -1}

(QeIQo).

However, ^we assumed there that the

coupling

^was very strong ^so that the cluster model limit was used for which the reduction factors had the values p ⁼

0,

q ⁼ 0.485 and r ⁼

2-q.

^{However, our recent ana-}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01976003706076300

764

lysis [7]

of the Jahn-Teller

theory

^{of Ni 3 + ions in}

A1203,

which have a 2E

ground

state, ^{has shown that}

a multimode lattice Jahn-Teller model and not a

cluster model is needed to

explain

the results. It

therefore seems

appropriate

^{to use a} similar model for the

Mn3+ : A’203

system.

As a result of the

transformation,

^the

angular part

of the effective vibronic Hamiltonian can be written as

where

O:t, 0 , 0+ and 0 +

^are operators

transforming

as

Et

^{of the}

C3

group. The 0 operators ^are linear in S and H and the 0 operators ^are

quadratic

^{in S as}

follows

where the

signs

^{are taken in}

pairs

^as indicated. The parameters have been derived

by perturbation theory [3]

^and may be

approximated

^{to the values}

where

and

The relative

energies

of the excited levels

5T2(A), ST2(E)

^and

3T 1 (3H)

^are

Al, A 2

^{and E}

respectively,

A

- 13 (A1

⁺

2 d2)

^{and the}

trigonal splitting

v ⁼

d 2 - A 1.

^{ba is the}

off-diagonal trigonal

^field

parameter. ^In eq.

(2),

p

(-0.1 cm -1)

^{is the}

spin- spin

constant,

Ro

is the value

of Q + Q _ at

^{the mini-}

mum in the

potential

energy ^versus

.JQ+ Q-

^curve.

J e

^{is the momentum}

conjugate

^{to 0 and M is the}

ligand

^mass..

As the first two

spin-independent

^{terms in}

Jerot

^are

much

larger

^{than the}

remaining magnetic

terms,

they

give

^a lower vibronic doublet

15E >

^{and an} upper vibronic

singlet 1 5 A1 > (for

^the

anharmonicity

^cons-

tant B >

0) or 1 5A2 > (for B 0)

^{at a relative} energy of b ^as before. We

neglect

^{all other}

higher

^{states in the}

analysis.

^The

spin-dependent

^{terms in}

^Jè

^{remove the}

spin-degeneracy

^and

couple together

^the

1 5E >

^and

1 5A1 > (or SA2 >)

^{states. The fifteen states then}

^split

into three sets of five states, ^{the sets}

transforming

like

A, E+

and E- of the

C3

group

respectively.

^The

additions and modifications to the model mentioned above

modify

^the

original

^matrix

appearing

ⁱⁿ

table II of

[3]

^{to the three}

(5

^x

5)

^matrices

given

^here

in table I. The

problem

^{is to} obtain values for the

parameters (A, f, Do, D’, K2, ^b,

g, p, q and

r)

^which

account

satisfactorily

for all the

expérimental

^data.

As the

NI3 + : A’203 system

^{was solved}

by analytic methods, attempts

^{were made to use a} similar method here.

Unfortunately,

^{it did not succeed as there were}

many ^{more terms}

contributing

^{to each}

expression

than in the case

of Ni3+ (as

^the

spin degeneracy

^{is five}

instead of

two)

^{and each term was of more or less the}

same

magnitude making perturbation

calculations invalid. This means that numerical methods had ^to be used once

again

^{for the}

^3d4

system.

3.

Summary

^of

expérimental

^{data and} parameters obtained. ^- The data obtained from APR is

given

in table 1 of

[1].

^For

example,

^{for an APR}

frequency

v - 9.3

GHz,

^the

edges

^{in the resonance lines occur}

at

magnetic

^{fields H}

of 4.8, 5.15, 10.8,

11.9 and 18.5

(all

±

0.2)

^kG for the lines labelled

B, D, F,

^{E and G}

respectively.

^{The new infra-red}

data [6]

^is

reproduced

in

figure 1 ;

the transformation

properties

^{of the}

states, from which the levels ^are

derived,

^{are also}

given. (The experimental points given

^{in the}

figure

were those obtained in the first detailed

experiments

^of

Aurbach and Richards and these results were used as a basis for

obtaining

^{the fit} described here.

They

therefore do not

exactly

coincide with the

points finally

included in the

corresponding figure

^{3 of}

Aurbach and Richards

[6].)

A

Telemechanique

^T1600

computer

has been used to

diagonalize separately

the three 5 x 5 matrices

by

successive

approximations by simultaneously varying

the

parameters ap,f, Do, D’, K2, b, g, q

^{and r. The}

program

stopped

when further corrections _gave the

TABLE 1

The matrix

of

^the

ground

rotational states

of jé

^when

H //

^{c. The}

(15

^x

15)

^matrix

splits

^{into three}

(5

^x

5)

^{matrices shown as blocks}

A,

B and C which

transform

^as

E+,

^{E- and A}

respectively

^{under the}

C3

group. The lower

half of

^{each matrix}

only

^is

given for simplicity.

^{The states}

labelling

the columns and

rows are in the

form vibronic

state;

spin

^{state in}

C3

symmetry

>.

The remaining parameters ^are defined in [3].

same results within the

experimental

^error.

Many

different

starting points

^{were used for the}

input

data to ensure that the fit obtained was

unique.

^The

energies

^{of the 14} levels relative to the

ground

^level

were

computed ;

the nine lowest levels are al*sô

6hpwn

in the

figure

^{and the}

good

agreement ^with

thé

infra- red data is

readily displayed

^{with the}

parameters :

From

[3],

^{this set of} parameters

gives

the additional results

so that k ⁼

0.06,

^{1 = 0.001 4.}

Also, d

^{= 5.52}

^cm-1

and e = 0.13

cm-’.

If we label the levels from the lowest

upwards,

^our

conclusions are in

agreement

^{with the new data}

[6]

for the transitions

1 -+ 3, 1 -+ 4, 1 -+ 5

and 1 -+ 6.

However,

^we

suggest

^{that the set of}

experimental points

^{labelled 1 - 9 in}

[6]

may be attributed to the transitions 2 -+ 7 and 2 -+ 8.

The other

pieces

of information

explained by

^our

analysis

^{are as follows :}

(i)

Five main APR lines

only

^{are seen}

[1]

^corres-

ponding

^{to the} transitions between the levels labelled

2,

3 and 4

only.

(ii)

^{As v decreases}

[ 1 ],

^the

magnetic

^{field at the}

edge

of the APR lines decreases for the

B,

^{G and F}

lines,

but it increases for the E and D lines.

766

FIG.1. ^- The _energy level diagram for ^{Mn3+ :} Al203 for zero-strain sites as a function of H parallel ^{to c. The} infra-red data of Aurbach and Richards [6] is also included. The levels are labelled 1 to 9 and

have the C3 symmetry properties ^shown.

(iii)

^{In order to} illustrate the closeness of the fit of the APR data

[1]

^we have followed

[6]

and calculated the _energy

separations

^{as a} function of H between all

possible pairs

^{of levels labelled}

2,

^{3 and 4 as deduced} from our parameters. The results are shown in

figure

^2.

On the

figure,

^{we have also}

superimposed

^{the APR}

data of

[1].

The closeness of this data to the calculated energy

separations clearly

demonstrates the

goodness

of the fit

given by

^our parameters.

(iv)

The maxima in the

absorption

in the APR

experiments

^{occur at}

(5.2

±

0.5)

^K for all lines with the G-line

arising

^{from a}

slightly higher

energy level than the rest

[1].

We need also

explain

the relative intensities of the infra-red lines. When H ⁼

0,

^{the levels 3 and}

4, transforming

^as

E+

^{and E_}

respectively,

^are

dege-

nerate. If the

trigonal

^{field were also} zero,

they

^would

also be

degenerate

with level 2 and

together

^.form

the

Tl triplet

^{in cubic}

symmetry.

^{Levels 5 and 6 are} also

degenerate

^{when H =} 0 and transform as E in

FiG. 2. ^- The calculated _energy separations between levels 2 - 3, 2 -+ 4 and 3 -+ 4 as a function of H. The APR data [1] ^{is also}

included.

both cubic and

trigonal

symmetry. ^{Thus IR} transitions from the A

ground

^state appear ^to be allowed for both the A -+ E cubic

(and

^{thus for 1} ---> 5 and

1 -+ 6)

^and

the A -+

T1

^cubic

(i.e.

^{1 -+}

2, 1

^{- 3 and}

1 -+ 4)

transitions in contradiction to

experiment

^{if the}

transition is induced

by

^an operator of the form

E;r r. ^However,

it is well known from

thermally-

detected electric-field-induced EPR

[9]

^{that the}

r f

electric field ^acts via an ionic effect in which the

neighbours

^are

displaced

ⁱⁿ

A’203 crystals.

^{We then}

suppose that the infra-red transitions are

similarly

induced. The main contribution then arises from

E-type

^ionic

displacements

^{and so the A - E tran-}

sition is allowed whereas the A -+

Tl

^is

forbidden,

in agreement with the observations

([6], ^Fig. 4).

^The

transitions induced

by Eu,

via the ionic effects have the

same

properties

^as acoustic transitions

[9].

^Thus

in thermal

conductivity experiments, only

^{the A -+ E}

transition is allowed and this is observed

[10].

^{As H}

is increased from _zero, the Zeeman term

(dominating

the

trigonal

^field

mixing)

^{mixes the}

E+

^{and E_ states}

thus

enabling

^{the 1 - 3 and 1 - 4} transitions to be observed.

However,

^{the 1 --+ 2} transition remains almost forbidden and

gives only

^{a small amount of}

absorption,

ⁱⁿ agreement with observation.

In our

analysis

^{we have not} used the infra-red data obtained for H

perpendicular

^{to c. For ions}

strongly coupled

^{to their}

surroundings,

^the

signal peak

^comes

from those

Mn3+

ions

having

^{a non-zero value of}

strain

[1].

^That

is, Q+

^and

Q_

^{are non-zero. The}

energy level

diagram

for such sites is different from that for sites with zero strain and

unfortunately

^we

have insufficient information to relate them.

4. Discussion of results. ^-

Taking

^{Â - 80 cm-1 1}

gives

^{4 - 20 000}

(see [3]

^for

details). Assuming

^the

off-diagonal trigonal

^field parameter ^{v’ to be 500}

cm-1,

gives

^{ba N - 0.025 so}

that, using

^{the value of}

K2

calculated we get

vld - -

^0.07. Therefore the dia-

gonal crystal

^field parameter v ^{is - 1 400}

cm -1

and

Ki - -

^{0.012 so that b - 0.10 cm -1}

and j -

^0.001.

As far as the Jahn-Teller

theory

^is

concerned,

^the

important

parameter ^{to obtain is} p. If we use the value of

fp

^deduced

by computer,

and the above value of ba

we

obtain p -

^{1. If ba is}

increased, p

is reduced

by

^the

same factor.

Alternatively,

^{if we use the}

computed

value of _ap and the formula

given

ⁱⁿ

[3]

^{we find} p is much smaller. This lack of agreement ^{it not}

surprising

as

jè

omits fourth order terms in the

spin

operators and second order in H. As in the case of

Ni3+ [2],

^the

magnitude

^{of a is}

critically dependent

^on

higher

^order

terms but such corrections are very difficult to incor-

porate

^for

(3d)4

^ions.

Unfortunately therefore,

^it

is

impossible

^{to deduce an accurate value}

for p

^{but it is}

undoubtedly

^{non-zero. This means that}

(2 q - p)

^is

less than

unity

but the difference cannot be’

reliably

found.

However,

^{we know from}

experiment [3, 10]

that thete is a

large

^{amount of}

phonon scattering

^from

Mn3+

ions in

A’203

^so that the Jahn-Teller

coupling

is strong. This indicates that while

(2 q - p)

^{is defi-}

nitely

^{less than}

unity, implying

^{that a multimode}

full-lattice model of the Jahn-Teller effect should be

employed [8],

the difference from

unity

^is

expected

^to

be

quite

^small.

Anharmonicity

^{effects are also}

important

^as

ô - 33

cm -1.

Transitions between the

ground

^states

and excited states are thus

expected

in thermal

conductivity experiments

^at

energies

between 30 and 50 cm -1.

Experimentally,

^{a mean}

separation

of - 38

cm -1

is found

[10].

^The

corresponding

^inver-

sion

splitting

^for

^{NI3 + :} A’203

^{is found to be}

57 ± ⁵ cm -1

[7] again indicating

that both the Jahn- Teller

coupling

^{constant and}

anharmonicity

^effects

are

larger

^for

^Mn3+

^{than for}

^NI3+.

^In

addition, q

is found to be

positive

^after

diagonalization implying

that B is

negative [3].

5. Conclusions. ^- The _energy

level diagram

^obtain-

ed for

Mn3+ adequately

^accounts for both the IR data and the observation of

only

^{five APR lines.}

As

only

^five strong ^{APR lines were} also observed in all

Cr2 + : A’203 samples,

it is also our belief that the

Cr2 +

transitions must have an

origin

very similar to those for Mn3+ as the line

shapes

^and

positions

^had

an identical pattern. ^{We thus} suppose that the

Cr2 +

energy level

diagram

^{must be} very similar to that of

figure 1, suitably scaled,

^{and not that shown as}

figure

^{1 in}

[3].

^{It has not} yet ^been

possible

^{to do a}

revised calculation for

Cr2+ using

a non-zero value for _p, as at the moment

corresponding

^{IR data}

(or

any other suitable

alternative)

^is

lacking.

Acknowledgments.

^{- We wish to thank Pr. P. L. Ri-}

chards and Dr. R. L. Aurbach for

bringing

^{their work}

to our

attention,

for discussions and

correspondence,

and for

sending

^us their results

prior

^to

publication.

^We

would also like to thank our

colleagues

^for many useful discussions on the

problems

^of

(3d)4

^{ions in}

A1203.

^{We are}

grateful

^{to the} Commissariat à

l’Energie Atomique

^{and to} the Science Research Council for financial

support

which enabled this

investigation

^to

proceed

^{as a}

joint

programme.

References [1] ANDERSON, ^R. S., BATES, C. A. and JAUSSAUD, ^P. C., ^J. Phys. ^C.

Solid State Phys. ⁵ (1972) ^3397.

[2] ANDERSON, ^R. S., BATES, C. A., JAUSSAUD, ^{P. C. and} RAMPTON, V. W., ^J. Phys. ^C. Solid State Phys. ⁵ (1972) ^3414.

[3] BATES, C. A., JAUSSAUD, ^{P. C. and} SMITH, W., ^J. Phys. ^{C. Solid}

State Phys. ⁶ (1973) ^898.

[4] STOELINGA, J. H. M., WYDER, P., CHALLIS, L. J. and DE GÖER, A.-M., ^J. Phys. ^{C. Solid State} Phys. ⁶ (1973) ^L486.

[5] ^{AURBACH, R.} L., RICHARDS, P. L. and FORMAN, R. A., ^{Bull. Am.}

Phys. ^{Soc. 18} (1973) ^1572.

[6] AURBACH, ^{R. L. and} RICHARDS, ^P. L., Phys. ^{Rev. B 12} (1975)

2588.

[7] ABOU-GHANTOUS, M., BATES, C. A., CLARK, ^I. A., FLETCHER, J. R., JAUSSAUD, ^{P. C. and} MOORE, W. S., ^J. Phys. ^C.

Solid State Phys., ⁷ (1974) ²⁷⁰⁷

and

ABOU-GHANTOUS, M., JAUSSAUD, ^P. C., BATES, ^C. A., ^FLET-

CHER, J. R. and MOORE, W. S., Phys. Rev. Lett. 33 (1974)

530 and

ABOU-GHANTOUS, M., BATES, C. A., FLETCHER, ^{J. R. and} JAUSSAUD, P. C., ^J. Phys. ^{C. Solid State} Phys. ⁸ (1975) 3641.

[8] HALPERIN, ^B. and ENGLMAN, R., Phys. Rev. Lett. 31 (1973)

1052.

[9] MOORE, ^W. S., BATES, ^{C. A. and} AL-SHARBATI, T. M., ^J. Phys.

C. Solid State Phys. ⁶ (1973) L209 ^{and L277.}

[10] RIVALLIN, ^{J. and} SALCE, B., Proc. 2nd International Conference

on Phonon Scattering ⁱⁿ Solids, Nottingham (Plenum, London) 1975, p. 184.