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DYNAMICAL RESONANCE PHENOMENA IN NON-ADIABATIC VIBRONIC SYSTEMS
M. Wagner
To cite this version:
M. Wagner. DYNAMICAL RESONANCE PHENOMENA IN NON-ADIABATIC VIBRONIC SYS- TEMS. Journal de Physique Colloques, 1973, 34 (C9), pp.C9-133-C9-136. �10.1051/jphyscol:1973925�.
�jpa-00215400�
DYNAMICAL RESONANCE PHENOMENA IN NON-ADIABATIC VIBRONIC SYSTEMS
M. WAGNER
Institut fiir Theoretische Physik, Universitat Stuttgart, Herdweg 77, W e s t - G e r m a n y
Rhumb. - Le processus de decouplage adiabatique est interrompu dans les systemes vibroniques o h I'energie de separation dans le systeme electronique est du m&me ordre que les excitations d e phonon.
L a methode de transformation canonique est un outil de choix pour etudier ces phenornenes de resonance (et anti-resonance). Des exemples seront donnes pour des degenerescences de syrne- trie electronique dans des environnements trigonal et cubique et pour des excitations electroniques degenerees a deux sites reticulaires differents.
Sur la base de nos resultats, quelques remarques peuvent Ctre faites sur I'impact tventuel sur des excitations electroniques dans des symetries du groupe d'espace non perturbees.
Abstract. - The adiabatic decoupling procedure breaks down in those vibronic systems, where the energy separation in the electronic systeni is of the same order as the phonon excitations. In those systems (( dynamical resonance )) effects are to be expected. Exaniples of such effects can be found in the optical spectra of single impurity centers in crystals (colour centers) o r in coupled vibronic centers.
The canonical transformation method will be shown to be a rather good tool to investigate these resonance (and anti-resonance) phenomena. Examples will be given for electronic symmetry degeneracies both in trigonal and cubic surrounding and also for degenerate electronic excitations a t two different lattice sites.
On the basis of our results some remarks can be made on the possible impact onto electronic excitations in undisturbed space group-symmetries.
In non-relativistic solid state a s well a s molecular physics o n e is continually confronted with t h e simul- taneous presence o f t h e electrons a n d tlie m u c h heavier nuclei. Hence, there is the f u n d a m e n t a l coupling problem between fast a n d slow moving particles.
1. Adiabatic decoupling. - F o r the sake o f mathe- matical convenience it is almost universally a d o p t e d that tlie cc,fust )) a n d rr . s l o ~ i y )) particles c a n be decou- pled by m e a n s o f the tr urlinhulic p1.it7cil~le )). T o illus- trate this principle a s well a s s o m e o f t h e PI-oblems t o be discussed later. we switch very briefly froni mole- cular t o celestial mechanics. S u p p o s e tliere a r e 2 pla- nets coupled in ;i non linear way by tlie law o f gravita- tion. T h e n there will be a resonance, if n , (11, = n, w,.
where l i i a r e integers a n d (11~ tlie circular frequencies.
This h a p p e n s t o be the case for J u p i t e r a n d S a t u r n , since t h e time o f revolution f o r J u p i t e r is 17 years a n d for S a t u r n 30 years. Hence 3 to, = 5 (I),. F r o m this it follows t h a t o n e o f tlie planets gains energy ( a n d angular m o n ~ e n t u m ) from the otlier. T h i s had been known t o the astl-onomel-s 1-01. a long time a n d it had been t h e reason tliat they did not believe Newton's ( ( l e x secunu'u)) until a b o u t 150 years ago. A t that time a physicist niadc a n exhaustive calculation. where
lie proved tlie energy transfer t o be a consequence o f Newton's law.
O n t h e otlier h a n d , if o n e wi is m u c h smaller t h a n the otlier, tlie faster circulating p l a n e t will (c see )) t h e slower essentially a t rest, whereas t h e slower circulating o n e will (c see )) a n average potential of t h e faster.
T h i s is the philosophy o f the (( adiabatic approxh~a- ti011 )) ( A A ) .
2. Dynamical resonances. - S u p p o s e however tliat tliere rtre 3 planets, t w o o f which a r e circulating very fast a n d the third very slow. i . e. o , , o, % w,. In this case we will have a n o t h e r possibility o f a resonance, if
I
?CII - toII
z O J ~ . a n d tliere will be a n energy transfer froni the cc,fust )) systems t o t h e rr S / O M ~ )) o n e o r vice versa. T h i s is against the very spirit o f t h e adiabatic a p p r o x i m a t i o n , whence tlie latter is n o longer appli- cable. These situations arise very frequently in micro- scopic solid state physics, but until very recently o n e has not been very careful in t h e application o f the a d i a b a t i c approxiniation.Even if t h e relation
I
t u z - t o ,I
= o, is n o t satisfied very well. the coupling between systems may provide;I variation o f to,, to2 in such :I way t h a t t h e resonance condition is fulfilled cKcctively. T h i s will also o c c u r
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973925
C9- 1 34 M. WAGNER
if the c( fast )) frequencies are degenerate, o, = w , ,
and the coupling constant k is of the order of o,.
F o r all these resonance situations we henceforth will use the term (( clj~t~an~ical resotlance )) phenomena.
3. Nonlinear canonical transformation (NCT) - Several methods have been developed to deal with dynamical resonance phenomena. The one with the greatest elegance and at the same time with the highest accuracy has been the canonical transformation method [ I ] . In this method a unitary operator
is applied to the Hamiltonian as well a s to all other operators and also to the state vector of the system,
where L is any hermitian operator representing a physical quantity. The expectation value of this quan- tity is then given by
Specifically
E. g. if we consider L = H, we may try to choose S in such a way that [H, S ] just compensates the coupling between the subsystems. Thus, in
fi
the remaining coupling is one order beyond the original one. The drawbacks of this method are threefold : a ) there is no systematic way of finding S, 6) it is hard to show tliat the infinite series beyond [H, S ] in the expansion ( 3 ) forfi
yields a negligible contribution, and finally c) even if one has reached at a good solution of the transformed Schrijdinger equation, one lias to calculate the transformed operator of any physical quantity of interest. This, in general, leads to rather involved combinatorial problems.In spite of these shortcomings the author lias been stimulated to apply the method to the dynamical resonance problem, when he realized that the Hamil- tonian for the most simple vibronic system,
can be diagonalized exactly by the transformation exponential
yielding
It is assumed here that Q ;;!ow, whence double excita- tion processes a+ a+ and double annihilation ones aa in the c( fast )) system can be neglected. By means of
this the total Hamiltonian
Je
commutes with the HamiltonianJe,
of the fast system,[x, ~ e , ]
=o
(7)and it does not make much difference if a, a+ are either Fermi o r Bose o r excitonic operators. It should be noted tliat the diagonalization in classical corres- pondence means the solution of a non linear equation of motion.
4. Degenerate systems. - The application of a N C T is least complicated if relation (7) is satisfied, which is true for degenerate ((,fast )) systems. The most simple of these would be given by the Hamil- tonian
JC = ~ ( a : a ,
+
a: a,)+
which is a rather unrealistic one. It roughly would correspond to a double mass oscillator in free space, where to each of the (( slolzl )) masses a cc,fast )) exci- tation a: o r a: respectively is attached. This system again can be diagonalized exactly. Here tlie fallacy of the AA can be examplified drastically. Suppose Y(0) = a: 10
>
is the state vector at t = 0, which is an excitation of the heavy nucleus M , . In the AA this excitation would cc,follo~~ )) the motion of M , adiaba- tically, but there would be n o energy transfer to M,.The exact solution, in agreement with physical intui- tion, tells another story. The excitation a: decays and via a n intermediate excitation of the slow system (vibration of M , and M,) the excitation a: is built up, wliicli afterwords itself decays, etc. The expectation value of the slow system is found to be
More realistic are the systems E-e and T-t, appearing in trigonal and cubic symmetry respectively. In a n earlier paper [I] it has been shown tliat by a N C T found in analogy to ( 5 ) the E-e case can be diagona- lized in a quasi-exact manner, i. e. the remaining coupling can be shown to disappear both in the weak and strong coupling cases and, moreover, is of an oscillating nature. Recently, a numerical analysis has proven [2] that also in the intermediate coupling region the inaccuracy of tlie diagonalization never exceeds I 'j/,. Thus, indeed, tlie NCT is highly success- full in this case. In the T-t case the N C T also seems to lead to a quasi-exact diagonalization, but the neces- sary combinatorial problems to prove it have not been solved yet.
5. Optical absorption. - The structural form of a vibronic spectrum strongly depends on tlie coupling of the lightfield to the system. T o be specific, we
A
assume the active dipole to be given by p=p(a: + a , ) , where a: is one of the 2 o r 3 c( fust )) excitations in tlie
systems
E-e
andT-t
respectively. We consider a transition from the total groundstate (i. e. T = 0) to the excited state a: with v I+
\12 o r I - ,+
v z+
v 3(( slo~z~ )) quanta in the
E-e
andT-t
cases respectively.C(
The intensity of this transition is given by oj
where Tand r a r e the afore-mentioned initial and final
states respectively. These intensities have been dis- 0.01 cussed in great detail in a paper of Sigmund and the
author [3] for the
E-e
case and one of Grevsmiihl and the author [4] for T-t. Again, it should be emphasized0.001 that in the evaluation of exp[- S ] a: combinatorial
problems have to be solved. In figure 1 it is shown.
how in the
E-e
case the zero-, two- and six-phononlines depend on the coupling parameter 1.' = (li/02). 0.0001
As expected, a resonance effect is exhibited in the 0.1 1 10 100
unity region of 1 for each single phonon line. Quite
l 2 -
the same is true in the
T-1
case, as shown in figures 2 FIG. 3. - T-, case. Intensity of the one-phonon line [4]. For and 3. The whole absorption spectrum of theE-e
case comparison the adiabatic result is given also (dotted line).for several coupling parameters is shown in figure 4. I. is a measure for the coupling strength.
FIG. 1. -
of single
- Appearance of dynamical resonances in the intensities excitation lines depending on the coupling strength [3]
E-e case.
FIG. 2. - Intensity of the zero-phonon absorption for the T-1 case [4]. For comparison the adiabatic result is given also (dotted line). 1. is a measure for the coupling strength.
FIG. 4. - Manifestation of dynarnical resonances in the struc- ture of the optical spectra [3]. System E-e. I is a measure for the
coupling strength.
6. Many-electron systems. - Looking at an ideal crystal, the variety of symmetry-degeneracies increases enormously. Therefore. the application of the AA almost never can be justified. Moreover, the energy separation between neighbouring electronic states is very small. Tlius, one very well may expect huge energy exchange and dynurnical resonance effects between the electronic and phonon systems. Also in
C9-136 M. WAGNER these cases the NCT method may prove to be quite
useful. Studies of this kind, although very cumber- some, are of great urgency, since the lack of them throws heavy shadows onto the basis of both the electronic band calculations and the a b iriitio lattice dynamics. It might very well have a severe impact onto the electron- and phonon-transport properties of solids.
7. Outlook. - T o come back to localized vibronic systems in solids, one would wish to exploit the known
N C T for the pure point-symmetry cases more comple- tely. In particular, the dynamics of the energy transfer between the electronic and vibrational systems should be investigated in great detail, since this would provide valuable experience for the space-group problem.
One also would like to have experimental information about the influence of local centers with dynamical resonance behaviour onto electron and phonon transport. All the indicated problems have t o be left to future investigation ; their solution will require huge amount of scientific work.
References
(11 WAGNER, M., Z. PI?ys. 256 (1972) 291.
[2] RUEFF, M., Diploniarbeit, 1973, University of Stuttgart, unpublished.
[3] SIGMUND, E. and WAGNER, M., P ~ J J S . s f a t . sol. 57 (1973).
[4] GREVSM~~HL, U. and WAGNER, M., Phys. stat. sol. 58 (1973).
DISCUSSION
A. M. STONEHAM. - YOU suggest that the usual electronic dipole operator is not correct for the F- centre. In the past the likely corrections have been described by (a) deviations from the Condon appro- ximation and (b) local field corrections. Since the Condon approximation has been verified both expe- rimentally (e. g. Schnatterly) and theoretically (e. g.
Gilbert and Wood) and the local field corrections seem small (Smith and Dexter, in Prog. Opt. X ) 1 should like to ask (i) what type of modification d o you have in mind -could you write out an exam- ple ? (ii) D o you know of any real evidence where detailed measurements o r calculations demand a more sophisticated treatment ?
M. WAGNER. - T O (i) : Indeed I had in mind mainly the two corrections ( a ) and (b) you have mentioned. There are others, e. g. from the direct coupling of the light to the optical phonons o r to the other electrons, which both yield contributions to the effective F-centre dipole operator via the phonon- electron o r electron-electron couplings respectively.
But these latter probably are of minor relevance.
As to the Condon approximation I d o not deny that it works perfectly, which does not proof that it is correct. I am not familiar with the work of Gilbert a n d Wood, but a real proof certainly would have to be extremely involved. Therefore I doubt that it would increase the real physical insight.
As to the local field corrections, one knows; very well that even in a locally disturbed continuum theory the difficulties are enormous. Therefore, I would be very astonished, if it could be done for a
locally disturbed discrete lattice in the case of any arbitrary instantaneous displacement configuration.
Since you are aware yourself of the two main corrections to the dipole operator which one would expect in principle, I think it unnecessary to write down an example.
T o (ii) : There is no need for any sophisticated treatment of the F-centre. It has been experin~entally proven that with highest accuracy the optical res- ponse of the F-centre can be described by the Hamil- tonian H = Qa'a
+
ob'b+
K a f a ( b+
b'), ifthe dipole opzrator is assumed to be D = p ( a
+
a') (corresponds to Condon approximation). This in effect is a one-parameter theory. So the theory is perfect. However, one does not understand the meall- ing of the operators u and b. Certainly, they are local quasi-particle operators. But how can they be des- cribed in terms of the original electronic and vibra- tional coordinates ? This, 1 think, 1s the real challenge of the F-centre.A. M. STONEI.IAM. - Are there any simple rules for predicting the position of the resonances in the N- phonon lines ?
M. WAGNER. - T h e zero-phonon line in the E-e case is given by [SZ(i./\
