Direct observation of the Jahn-Teller splitting in ZnSe: Cr2+

Solid State Communications,

Vol, 11, pp. 35-38, 1972.

Pergamon Press.

Printed in Great Britain

DIRECT OBSERVATION OF THE J A H N - T E L L E R SPLITTING IN ZnSe: Cr 2" Bertil Nygren and John T. Vallin Department of P h y s i c s , Chalmers University of Technology, S-402 20 Gothenburg 5, Sweden and Glen A. Slack General Electric Research and Development Center, Schenectady, N.Y. 12301, U.S.A.

(Received 8 March 1972 by S. Lundquist)

The optical absorption spectrum of Cr 2+(3d') in single crystals of ZnSe has been studied in the wave number region 5 0 0 - 4 0 0 0 c m -I . Our results are interpreted in terms of a strong J a h n - T e l l e r coupling between the impurity ion ground state and a vibrational mode of E symmetry. From the data obtained at 5 K we estimate the J a h n - T e l l e r energy to be 370 cm -~ and the vibrational energy hc~ to be 75 cm -~ .

Cr concentration. Low temperature measurements (80 and 5 K) have also been performed on a sample with a Cr concentration of I019cm -~. The samples were obtained from E a g l e - P i c h e r Industries, Inc., and measured in a _Beckman IR 10 spectrometer. The present study is a complement to earlier microwave s's and near infrared 7 studies of Cr 2÷ in I I - V I compounds. In Fig. I is shown the absorption coefficient (a) as a function of wave number (zT) for a Cr concentration of 10'gCr/cm 3. This broad absorption band cannot be accounted for by simple crystal field theory but requires an introduction of vibronic coupling ( J a h n - T e l l e r coupling). The crystal field of tetrahedral (Td) symmetry acting on substitutional Cr 2+ in ZnSe will split the free ion ground state (5D) into an orbital doublet (~E) and an orbital triplet ( 5T2 ), the latter being the ground state. Earlier experiments s-v on Cr 2÷ in ZnSe show that the J a h n - T e l l e r coupling is with an E-mode, and that this coupling in the orbital doublet ( ~ ) is probably small compared with the coupling to the ground state triplet ( ~2 1. The operator describing 3 the coupling between the orbital triplet ground state and an E vibrational mode of normal

THE IMPORTANT role of vibronic coupling for defect centers in crystals has recently attracted increased interestJ '2 This coupling may show up as a dynamic J a h n - T e l l e r effect, in which matrix elements of orbital operators among the lowest vibronic levels of the system are reduced by appropriate 'reduction factors' without any change in apparent symmetry. 3 Or, in the limit of strong coupling the more familiar situation of a static J a h n - T e l l e r distortion appears. These distortions have been directly observed by EPR experiments for Cr 2÷ in several I I - V I compound semiconductors. 4,s In the present work we have studied optical transitions between the J a h n - T e l l e r split levels of Cr 2÷ in ZnSe. To the best of our knowledge this is the first time a direct measurement of the J a h n - T e l l e r splitting of the electronic energy levels has been o b t a i n e d from which the J a h n - T e l l e r stabilization energy is easily derived. The room temperature optical absorption spectrum for pure and Cr doped ZnSe has been obtained for the infrared region 500-4000 cm -1 . The spectrum has an absorption band that peaks at about 14000cm -~ and is proportional to the 35

36

J A H N - T E L L E R SPLITTING IN ZnSe: Crz÷

IS

5K

~t

Vol. 11, No. 1

o

I _

o o

u. 1.0 ?

O5

0.0

o 0

ot 500 PHOTON

I 1000

I 151)0

x. 2000

WAVENUMBER, cm"~

FIG. 1. The measured optical absorption coefficient as a function of photon energy for three different temperatures. coordinates QO, Q• is:*

Hj. r

= V(Q~

(1)

+ Q•~•).

V is the coupling coefficient and ~8 and ~ are electronic orbital operators belonging to E having the following matrix representation with respect to the orbital Tz functions:

E~=

~

x/(3)/2

0-

.(2)

0

The full Hamiltonian for our combined VIBRational and electrONIC (vibronic) problem is:

H = H, + H r T + H r .

oscillator frequency, and P8 , P• are the momentum operators conjugate to QO, Q•). The vibronic eigenfunctions of Equation (3) are: i

~

Ig~,~(r__,Q_)= Vi (r)F~o(Q~ + VE e~) F%(Q• + VEei•

(5) where eio and ei• are the appropriate diagonal components of the matrices e~ and e• (Equation2). VE = V / p ~ : and F~ (y) is a standard harmonic oscillator function defined by the generating relation: exp(-S 2 + 2 S a y - a2y*/2) = XY('rr/aa)~=° Fn (y) [Sx/(2)],~ =

4(n!) (6)

(3)

Here He is the electronic operator, HjT the vibronic operator, and H v the vibrational operator. In the harmonic approximation H v is a two-dimensional harmonic oscillator operator:

Hv = ~-~'[p~ + pa• + /.zoJ~(Q~ + Q~)] (4) is an effective mass, a~E the c h a r a c t e r i s t i c * 0 and £ is used to d e s i g n a t e partner functions (or operators) belonging to E and transforming r e s p e c t i v e l y a s 2z 2 - (x z + y2) and X/3(x 2 - yZ), while ~r 77' ~ designate t h o s e belonging to T2 and transforming as yz, zx, xy.

)

witha 2 = g/~/h. The e i g e n e n e r g i e s corresponding yo g2~ are: Ei~

= Eo-

EjT ÷ (n e ÷ n• + i) I ~ E

(7)

where Ej. r = V2 /2~zo~ and no, ng = O, 1, 2 . . . . are harmonic oscillator quantum numbers. We attribute the absorption band in Fig. 1 to transitions between the vibronic s t a t e s of Equation (5). Notice: In the static limit (P0 = P• = 0) we obtain for the distortion Qo (Q• = 0). Eg = E o + 2EjT E~ = E o - EjT

= E.~

(8)

Vol. 11, No. 1

JAHN-TELLER

S P L I T T I N G IN ZnSe: Cr 2÷

3E3T r e p r e s e n t s the s t a t i c J a h n - T e l l e r s p l i t t i n g of the ground s t a t e orbital triplet. T h e maximum in the a b s o r p t i o n band c o r r e s p o n d s a p p r o x i m a t e l y to t h i s energy. It i s now c o n v e n i e n t to c a r r y out t h e i n t e r p r e t a t i o n of the e x p e r i m e n t a l r e s u l t s in terms of the l i n e - s h a p e function f ( E ) and i t s moment analysis, a f ( E ) i s d e f i n e d by the r e l a t i o n : c~(E) = E f ( E )

(9)

where c~(E) i s the a b s o r p t i o n c o e f f i c i e n t . Although the c o u p l i n g is with a double d e g e n e r a t e mode we need only c o n s i d e r the overlap b e t w e e n two harmonic o s c i l l a t o r functions. F o r i n s t a n c e we obtain:

Similar e x p r e s s i o n s c a n b e o b t a i n e d for other o v e r l a p i n t e g r a l s by s u i t a b l e c o o r d i n a t e transformations. We can therefore u s e the s t a n d a r d t r e a t m e n t c o n c e r n e d with n o n - d e g e n e r a t e e l e c t r o n i c s t a t e s and b r o a d e n i n g through c o u p l i n g to a s i n g l e mode. 9 Under t h e a s s u m p t i o n that s p i n - e f f e c t s c a n be n e g l e c t e d one o b t a i n s ( r e f e r e n c e 9 p.425)

f(E)

= I < i l R l i > l 2 pn 8(E

-

nhooE). (11)

Here R i s the e l e c t r o n i c operator c a u s i n g the t r a n s i t i o n and for n > 0 p= = e x p ( n x - y c o t h x ) l ~ ( y / s i n h x) x

= ha~E/2kT

y = 3EjT / h ~ E

= fjf(E)

d E = cll a



= A-' f j E

f(ED d S



= A -I f o ' ( S - < S > ) " f ( e ) a S .

be i n d e p e n d e n t of T with < E > = 3EjT • We a l s o obtain:

= 3 E j T ho~E coth (hc~E/2kT)

From our e x p e r i m e n t a l r e s u l t s at T = 5 K we can t h u s obtain both EjT and haJE. F o r Cr 2+ in ZnSe we o b t a i n EjT = 370cm-' and ha~E = 75cm -~. T h e s e numbers have been used in our c a l c u l a t i o n s [from e q u a t i o n (11)] plotted in F i g . 2. We have n o r m a l i z e d the c a l c u l a t i o n s so that the peak height at OK c o i n c i d e s with the e x p e r i m e n t a l p e a k height at 5 K. In T a b l e 1 we summarize a c o m p a r i s o n b e t w e e n the e x p e r i m e n t a l r e s u l t s and the c a l c u l a t i o n s . Tablel. Temp. (K) Theor. Exp.

( c m - ' ) Theor. Exp.

0

5

III0

III0

80

80

III0

1160

(cm -2) Theor. Exp. 83.10 a

83.10 a

140.10 ~ 98.103

From F i ~ 2 and T a b l e 1 we c o n c l u d e that we c a n r e a s o n a b l y w e l l account for the o b s e r v e d a b s o r p t i o n band and i t s temperature d e p e n d e n c e ( w i t h E j T = 3 7 0 c m - ' , h c o E = 75cm-1). Our r e s u l t s a r e in r e a s o n a b l e agreement with e a r l i e r e s t i m a t e s of EjT and hoJE from E P R s ' l ° (EjT > 3 6 0 c m -I , hCOE > 7 5 c m -I ) and near infrared 7 (EjT = 5 5 0 c m -I , ha~E = 69cm -I ) a b s o r p t i o n e x p e r i m e n t s . R e f i n e m e n t s of t h e t h e o r e t i c a l t r e a t m e n t i n c l u d i n g the e f f e c t of s p i n d e p e n d e n t t e r m s and t h e c o u p l i n g to a c o n t i n u o u s spectrum of p h o n o n s a r e d e s i r a b l e . We f e e l , however, that t h e p r e s e n t t r e a t m e n t is a d e q u a t e for the experi m e n t a l r e s u l t s p r e s e n t e d in t h i s paper. A c k n o w l e d g e m e n t s - We are i n d e b t e d to F.S. Ham for many h e l p f u l comments and d i s c u s s i o n s .

/~ i s a B e s s e l function of i m a g i n a r y argument. Defining the a r e a (A), f i r s t moment ( < E > ) , and higher moments ( < E = > , n /> 2) a s in r e f e r e n c e 9 we have: A

37

(12)

We then obtain from e q u a t i o n (11) A and < E > to

38

"~

J A H N - T E L L E R SPLITTING IN ZnSe: Cr2*

z° I

(a)

I

I

I

1

I

,s i

Vol. 11, No. 1

1

I

]

Z (b) 80K

z 0 I-

5K

o o 1.0

~a

o o

--

u,. o 2::

°

o

~o.s

"4

o o

-

tn

!'

0.0

,

1

500

0.0 S00

1000

ISO0

Z000

2S00

I000

I

I S~

Zll00 PHOTON

pHOTON WAVENUMSER, c m ~1

2000

.! WAVI~NUM~tER,

cm

FIG. 2. The spectral line-shape as a function of photon energy at (a) ,5 K and (b) 80K. The circles denote theoretical values obtained from relation (11).

REFERENCES 1.

STURGE M.D., Solid State Physics Vol 20, p.169 (edited by SEITZ F., TURNBULL D. and EHRENREICH H.) (1967).

2.

HAM F.S., Electron Paramagnetic Resonance (edited by GESCHWIND S.), Plenum Press, N.Y. (1972).

3.

HAM F.S., Phys. Rev. 138, A1727 (1965).

4.

ESTLE T.L., WALTERS G.K. and DeWIT M., Paramagnetic Resonance, Vol- I, p.144 (edited by LOW W.) Academic P r e s s , N.Y. (1963).

S.

VALLIN J . T . and WATKINS G.D., Solid State Commun. 9, 9S3 (1971).

6.

VALLIN J . T . , SLACK G.A., ROBERTS S. and HUGHES A.E., Solid State Commun. 7, 1211 (1969).

7.

VALLIN J . T . , SLACK G.A., ROBERTS S. and HUGHES A.E., Phys. Rev. B2, 4313 (1970).

8.

HENRY C.H., SCHNATTERLY S.E. and SLICHTER C.P., Phys. Rev. 137, A583 (1965).

9.

PRYCE M.H.L., Phonons in Perfect Lattices and in Lattices with Point Imperfections, p.403 (edited by STEVENSON R.W.H.) Plenum Press, N.Y. (1966).

10.

VALLIN J.T. and WATKINS G.D., to be published.

Das optische Absorptionsspektrum van Cr 2. (3d 4) in ZnSe Einkristallen wurde im Wellennummerbereich zwischen 500 und 4000 cm -~ untersucht. Unsere Resultate werden durch feste Jahn-Teller-Kopplung zwischen dem Grundniveau des Fremdeniones und einer Schwingungsmode mit E-Symmetrie erkl~rt. Van den an einer Temperatur van 5°K durchgefi~hrten Messungen bestimmen wir die Jahn-Teller-Energie als 370cm -~ und die Schwingungsenergie ha~ als 75cm -~ .