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DYNAMICAL JAHN-TELLER EFFECTS ON THE
ZERO-PHONON LINE
E. Mulazzi, A. Maradudin
To cite this version:
C7-114 JOURNAL DE PHYSIQUE Colloque Cl, supplément au n° 12, Tome 37, Décembre 1976,
DYNAMICAL JAHN-TELLER EFFECTS
ON THE ZERO-PHONON LINE
E. MULAZZI(*+) and A. A. MARADUDIN 0
Department of Physics, University of California, Irvine, California 92717, U. S. A.
Résumé. — Nous présentons la théorie pour la forme de la ligne à zéro phonon dans le cas où
l'effet Jahn-Teller dynamique est important dans l'état électronique excité. Les résultats obtenus sont examinés en particulier dans le cas de la ligne à zéro phonon due à la transition électronique
*A2e -* 4T2g du V2+ en MgO.
Abstract. — We present a theory for the shape of the zero phonon line in the case that dynamical
Jahn-Teller effects on the degenerate excited state are important. The results obtained are used to examine the shape of the zero phonon line for the electronic transition 4A2g -*- 4T2gofV2+in MgO.
It is well known that the structure of the zero phonon line for the electronic transition 4A2 g ->• 4T2 g of V2 +
in MgO is determined by the dynamical Jahn-Teller (JT) effect on the excited electronic state [1]. The same effect has also been observed in zero phonon lines associated with electronic transitions from a singlet ground state to degenerate excited states of other impurity centres in polar crystals [1]. However, until now, no theory has been able to explain completely the form of the line as a doublet of the zero phonon line, the broadening of the two peaks, and the temperature dependence of these features.
In this paper we present a theory of, and theoretical evaluations for, the shape of the zero phonon line in the case that the electronic transition is from a ground singlet state to a triply degenerate excited state. For simplicity, the point group of the impurity site is assumed to be Oh, the perturbation is assumed to
be localized on the six nearest neighbors of the impu-rity in the ground as well as in the excited electronic states, and the electronic transition is considered in the electric dipole moment approximation. We show that the splitting of the zero phonon line is a dynamical effect, because the point symmetry of the impurity site is not lowered statically, as has been shown experi-mentally [1]. Specifically, we find that the effect is due to the fact that the linear electron-phonon (EP) interactions transforming according to the irreducible representations (irr. rep.) rf, rt of the point group Oh
make a contribution to the definition of the energy Ei of the zero phonon line which is dynamically different from the contribution to E\ made by the EP interactions transforming according to the irr. rep. i"1^ of Oh. The
energy E° is obtained by subtracting from the energy of the electron in the static lattice, Eu the energy
(*) Permanent address : Universita di Milano, Istituto di Fisica, Via Celoria 16, Milano (Italy).
(t) Work supported in part by ONR contract number N00014-69-A-0200-9003.
dressing arising from the phonon processes induced by the EP interactions. Moreover, while the splitting of the line is determined principally by the EP interac-tions transforming according to r^, the broadening and the shape of the doublet are determined by the EP interactions transforming according to both r$ and r5 +.
The Hamiltonian of a bound electron interacting with the phonons of the imperfect lattice of a polar crystal and making a transition from a ground singlet state to a degenerate excited state is given in [2]. Since in the present case the degeneracy of the electronic excited state is threefold, the index i used in [2] runs from 1 to 3, and the EP interaction matrices h{Ty) are those constructed in the three-dimensional space spanned by basis functions transforming as x, y, z [3]. (We use the notation that r labels the irr. rep. f^, rt, r5, while y labels the partner functions for multi-dimensional irr. reps.). In the 3 x 3 space the EP interaction matrices h(Fy) have the properties that h(ri+) and h(r$,y), h ( r ^ , / ) commute with each
other, while :
i) [h(r5+ y), h(r5+ / ) ] * 0
for y # y', and
ii) [h(r
3+y) , h(/t / ) ] * 0
for all y and / . Moreover, higher order commutators in both cases i) and ii) are also nonvanishing. The EP interaction coupling coefficients are called C(r) in the present paper.
We study the absorption response function for Jahn-Teller systems [2] in the framework of the Independent Ordering Approximation (IOA) [4]. The general theory and the theoretical evaluations, which also include the case of the broad structured band shapes due to the dynamical JT effect, will be given elsewhere. The response function Rt(t) for the electronic transition is given by [2] (here and in the following h = 1)
DYNAMICAL JAHN-TELLER EFFECTS ON THE ZERO-PHONON LINE C7-115
Xi(') = eirlt (T
(
expzry
1:
dsl(
ds' h ( r ~ ) ~ D0(r ; s-
st) h(Q),.0 (1)
where i = 1,2,3, Do(r, s
-
s') is the propagator for phonons ofr
symmetry [2], and T is the time ordering operator. The main point of the IOA method lies in time ordering the EP interaction matrices h(Ty), and h(Ty),,independently of the time ordering of the phonon propagators Do(T ; s
-
s f ) . Within this approximation eq. (1) becomeswhere
PV?
0') dw{
eimf(n(o)+
1)+
e-'"' n(o) - iot- (2 n(w)+
1)}
.
Go2
In eq. (3) p(T, 0') is the perturbed phonon density of states of
r
symmetry projected on the space of the pertur- bation, and M is the mass of the nearest neighbor ions. Note that only linear terms in h(ry) appear in the argument of the exponential function in eq. (2). The time ordering is then completely taken into account by considering the exponential of the sum of these matrices. The x(Ty) are independent variables or time independent integration paths, one for every irr. rep.r
and partner y. The band shape function Ri(SZ), which is the Fourier transform of Ri(t) with respect to time, is given byWe first evaluate the zero phonon line shape (R,(Q)),.,. from eq. (4) by keeping only that part of q ( r , t ) that determines the frequency of the zero phonon line and its intensity, but taking into account all the EP interac- tions of r:, r i , I': symmetry. In evaluating the commutators of these interaction matrices of all orders we initially assume only the property i), but ignore property ii) :
where the energy E; of the zero phonon line is given by ,
with
The Huang-Rhys factor S in eq. (5) is given by
s
= s(r:)+
s(r:)+
s(r:) withs ( r ) = c 2 ( r )
I
~ (w2) ~ (2 n(w) 9+
l ) dm ?M
o2where n(w) is the phonon population. On integrating eq. (5), one finds
The principal characteristic of eq. (10) is that the zero phonon line is actually two lines whose energies are E: and E:
+
AQ(~;), the energy of the second line depending on the EP interaction coupling coefficient c(T:).C7-116 E. MULAZZI A N D A. A. MARADUDIN
is different from zero there is a dynamic effect which gives rise to two different redefinitions of the energy of the
zerophonon line. Note that the intensities of the doublet depend on the temperature through the factor e-S. Note
further that the intensity of the peak at 52 = E: is twice that of the peak at 52 = E:
+
AQ(r:). No additional structure at the energy 52 = E?-
AQ(T:) arises when the temperature is increased, in agreement with the experi- mental results. Thus the structures observed on both sides of the zerophonon doublet with increasing temperatureare due only to one- and two-phonon processes.
By applying the Baker-Hansdorff formula [5] to eq. (4) in order to take into account the commutation rela- tions given in (ii) in the evaluation of the line shape function of the zero phonon line, one finds
where
Fm(t) = exp[-
pm
t2 ASZ(~:) A52(r:)] e x ~ [ i t ~ ( ~ ~ ( r : ) ) ~ ASZ(T:)]m
= 1 , 2 , (12)with
B1
= 1 andfiz
=3.
Note that the effect of including the first commu- tator between the matrices h(~:, y) and h(~:, y) is to change the shapes of the zero phonon lines from the simple 8-function peaks given in eq. (10). In figure 1 we display the results of the evaluation of eq. (11) for different values of AQ(r:) and 852(r:). We see from eq. (7) that the values of AQ(T:) and AQ(F :) depend on the coupling coefficients C(T) and on the effective phonon frequencies describing the densities of phonon states transforming according to T: and
r:,
when the latter are approximated by 8-functions. From figure 1FIG. 1. - Normalized line shapes (RS(Q))~.~. from eq. (11) with
AG?(I'z) = 0.275, where 5 is the average phonon frequency ;
(a) A Q ( r 9 = 0.5 AQ(~:), (6) AQ(r:) = 0.7 AQ(rz) and
(c) AQ(T:) = ASZ(T:). The frequency of the absorbed light is given with respect to
EP
= 0 in units of the average phononfrequency G .
we see that when AQ(T:)
2
ASZ(T:) the doublet disappears and only one zero phonon line is observed. Moreover, since the A!2(T) depend on the temperature through the coupling coefficients via the lattice thermal expansion and through the phonon frequencies via anharmonicity, the doublet can be observed only in the region of low temperatures for which AQ(T:) does not exceed AQ(T:) and the Huang-Rhys factor S does notbecome too large. Note that as before the structure of the zero-phonon line still consists of two peaks and no additional structure at 52 = E: - ASZ(~:) arises with increasing temperature. Furthermore, the ratio of the intensities of the two peaks is again 2 : 1. Thus the main effect introduced by the commutators given in (ii), considered in all orders, is a broadening of the shape of the doublet, a broadening which depends on both 652(r:) and ~52(r:).
The curve (b) of figure 1 with the appropriate ratio between AQ(T:) and AQ(T:) can be regarded as giving the line shape of the zero phonon line for the transition 4Azg _t 4T2g of VZ+ in MgO. In fact, by considering
the values given in [I] for the coupling coefficients (c(T:)
-
+
c ( T ~ ) ) and for the average phonon fre- quenciesw,;
=3
=+
Z, where the average phonon frequency Z is 200 cm-l, we have that in this case AQ(T:)-
0.7 AQ(~:) and ASZ(T:) = 40 cm-l.References
[I] STURGE, M. D., Phys. Rev. 140A (1965) 880 ; [3] HAM, F., in Electron Paramagnetic Resonance, edited by See also STURGE, M. D . in Solid State Physics, edited by Geschwind S. (Plenum, New York) 1971.
Seitz I?. and Turnbull D . (Academic Press, New York) [4] VEKHTER, D . B., PERLIN, Yu. E., POLINGER, V. Z., ROSEN- 20 (1967) 91. FELD, Yu. B. and TSUKERBLAT, B. S., Cryst. Lattice [2] MULAZZ~, E. and TERZI, N., Phys. Rev. 10 (1974) Defects 3 (1972) 69.
