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JAHN-TELLER AND ”FANO” SYSTEMS : ON THE
APPROXIMATION SITUATION OF RESONANCE
STRUCTURES
E. Sigmund
To cite this version:
JOURNAL D E PHYSIQUE Colloque Cl, supplément au n° 12, Tome 37, Décembre 1976, page C7-117
JAHN-TELLER AND «FANO » SYSTEMS :
ON THE APPROXIMATION SITUATION OF RESONANCE STRUCTURES
E. SIGMUND
Institute of Theoretical Physics, University of Stuttgart, 7000 Stuttgart 80, Pfaffenwaldring 57, F R G
Résumé. — Trois méthodes différentes sont présentées, qui décrivent de manière tout à fait satisfaisante le comportement résonant des systèmes non adiabatiques électron-phonon. Il s'agit des méthodes de Transformation Exponentielle (ETM), de l'Opérateur (OM), et des Moments (MM). A titre d'applications, on donne les spectres d'absorption optique des cas Jahn-Teller purs E-e et T-t, ainsi que d'un système Fano. De plus, on discute la dynamique interne d'un système Jahn-Teller E-e.
Abstract. — Three different methods are presented, which describe the resonant behaviour of nonadiabatic electron-phonon systems quite well. These are the Exponential Transformation Method (ETM), the Operator Method (OM) and the Method of Moments (MM). For an applica-tion the optical absorpapplica-tion spectra of the pure E-e and T-t Jahn-Teller cases and of a Fano system are given. Furtheron the internal dynamics of an E-e Jahn-Teller system is discussed.
1. Introduction. — One of the most fundamental problems in non-relativistic many-body physics is the separation between the motion of fast (electrons) and slow (nuclei) particles. Since the fundamental work of Born and Oppenheimer it has become conventional to adopt the adiabatic principle for this basic decoupling procedure. However, this theoretical description breaks down in such coupling regions, in which the energetic separation of the electronic states is of the same order of magnitude as the elementary excitations in the nuclear system. In this case there is the possibility of a resonance effect in the sense of an energy exchange between the fast and slow system. Hence, the separation
between electronic and nuclear motion is impossible. In this context 3 theoretical methods are presented, which lead to very good results also in the region of these dynamical resonances.
We will test these methods by calculating the optical absorption spectra of E-e and T-t Jahn-Teller systems. The structural form of the vibronic spectrum is contained in the optical absorption function. It depends strongly on the coupling of the system to the external light field. Under the influence of this coupling, which is characterized by the electronic dipol-operator P( l ),
the system undergoes an optical transition from the total groundstate W0 to the final state fif) [1, 2] :
(1)
Here we have confined ourselves to T — 0 K. m counts the number of phonon quanta, which are involved in the transition. coZ is the energy of the final state, co the frequency of the lightfield (h = 1).
The optical response function is the Fourier transform of (1) :
(2)
where 3t is the total Hamiltonian.
The first method considered is the exponential transformation method [2, 3, 4]. The calculation of the absorp-tion funcabsorp-tion is done in a transformed space :
(3)
i.e. each single operator of the system is transcribed into the new basis by means of the exponential transformation:
(4)
For an illustration the optical absorption spectra of the pure E-e and T-t Jahn-Teller systems are considered.
9
C7-118 E. SIGMUND
The second method is an operator method [5]. Here we directly start our calculations at the optical response function. Applying Feynman's disentangling theorem we get an expression, which exhibits time-ordered products :
G(t) =
(
P oI
P")(
exp[-
i1:
Hl(tl) dr']]
P")/
P o ).
Ho is the diagonal part of the Hamiltonian and H,(t)
the nondiagonal part in interaction representation
For this ordered exponential a closed solution can be given. We again calculate the optical absorption spec- trum of the E-e J. T. system [5]. As a further illustration we treat the internal dynamics of this system.
The third method is the method of moments [6, 71. We again start with the optical response function and evaluate it in terms of the moments of the absorption band : G(t) = e-iwot
"
(- it)" P m m = O where + 00 ,urn=1
G ( o ) ( w - w 0 ) " d o . - m (7)oo is the energy of the electronic subsystem. This procedure is applied to the calculation of the spectra of the E-e J. T. system 161 and a Fano system [7], both in the strong coupling limit.
2. The exponential transformation method. -The aim
of this method is to transform the considered system into a new basis of coordinates, in which the trans- formed operator is diagonal or at least nearly diagonal. 2.1 In the case of the pure E-e J. T. system [ 2 , 4 ] the Hamilton operator consists of a double degenerate electronic oscillator, which is coupled to a double degenerate vibration mode :
2 2
= wo a t ai
+
01z
b; bj+
K(
(a: al-
a; a,) (b:+
b,)+
(a: a,+
a; a , ) ( b i+
b2))
(8)i = 1 j = 1
o0 is the electron energy, o , the phonon energy and K the coupling parameter. The transformation (4) of this system is defined by the exponent
K
S = - ( ( a: a ,
-
a: a,) (b:-
b,)+
(a: a ,+
a; a , ) (b;-
b,))
.
0 1 (9)
It turns out to be exact in both extremal coupling cases and leads to rather good results for the intermediate region.
After rather lengthy calculations the optical absorption function can be given in a closed form. For brevity here only the even-numbered phonon excitation lines are written down : (2 =
do,)
G2(o) =
z
($)
'
Pm
[
($1
Q(li)] 6 ( o-
EL.
-
2 m o , )m
E ; ~ is the energy eigenvalue of
%
for the state a+I
0>.
We see that the characteristics of the optical response are completely derivable from the analytic properties of the function Q(2) ( ~ ( 2 ) ) is related to the Error- function). This function and its square are drawn in figure 1. The crosses denote the zero points of the derivatives of Q(2). Q2 is the zero phonon line.
The intensities of the single excitation lines as a function of
L
are drawn in figure 2. All lines (except the zero-phonon line) are zero forL =
0. They increase to a maximum and decrease to zero again. Finally theyFrc;. 1.
-
DynamicaI resonance effect of the zero-phonon Iine intensity in the dynamical E-e Jahn-Teller system. Q 2 is the intensity of the zero-phonon line and 1 a measure for the coupl-ON THE APPROXIMATION SITUATION OF RESONANCE STRUCTURES C7-119
reach a second maximum, beyond which they decrease
asymptotically.
i
FIG. 2.
-
E-e Jahn-Teller system : Appearance of the dynamical resonances in the intensities of single excitation lines depending on the coupling parameter. 12m is the intensity of the (2 w;)-pho-non line and 1 the coupling strength.
This resonance effect manifests itself also in the whole spectrum. In figure 3 the coupling parameters are choosen in such a way that the intensity of one excita- tion line is exactly zero.
FIG. 4.
-
T-t Jahn-Teller system : Manifestation of dynamical resonances in the structure of the optical spectra. G(w) is the intensity of the absorption and o is a measure for the energy.3. The operator method.
-
In the operator method we expand the time-ordered exponential operators and rearrange the terms by partial integrations. Then, most of the terms can be summed up again in a closed form If one looks at the series expansion of the ordered exponentials it turns out that there appears a specificFIG. 3. - E-e Jahn-Teller system : Manifestation of d~namical commutator [5] : resonances in the structure of the optical spectra. G(w) is the
intensity of the absorption and o is a measure for the energy.
[St
o H,(t? dtr
,
[It
O W , ( t f ) dt',
~ , ( t ) ] ].
(1 1)For increasing coupling strength ;1 the resonance point is shifted to higher excitation lines.
2.2 Similar investigations can be done for the more complicated T-t J. T. problem [8, 91. There a triply degenerate electronic oscillator interacts with a triply degenerate vibration mode.
In figure 4 some spectra for different coupling para- meters are given [lo]. For interaction values between 1.6 and 2.2 the spectra show one resonant dip. For greater parameters two dips arise.
This is just the first commutator in the expansion, we neglect in further calculations. Thus this form of commutator plays the role of a jundamental characte- ristic commutator for approximations. This immedia- tely suggests the approximation possibility to set the commutator equal to zero.
If for a given problem the commutator turns out to be zero, the solution can be given in a closed form. If the commutator is small, the closed solution still will be a good approximation.
C7-120 E. SIGMUND
shown that in the weak as well as in the strong coupl- ing limits the fundamental commutator vanishes [ 5 ] .
Furtheron it turns out that the character of the operator factor of the internal commutator
will not play any role in the very extremal coupling cases. But naturally the region of validity can be extended, if the properties of this operator factor are taken into account.
3.1 We apply the method to the optical absorption
spectrum of the E - ~ J. T. system in two different FIG. 6. -The envelope of the optical absorption spectruz of
approximations. lzirst, we set the inner commutator the E-e Jahn-Teller system in the second approximation. G(w) is equal t o zero : the intensity of the absorption and is a measure of the energy.
As a result we got a spectrum which consists of a sequence of &function like peaks. The envelope shows the mirror symmetric form of the semiclassical result (Fig. 5).
FIG. 5. - The envelope of the optical absorption spectrum of the E-e Jahn-Teller system in the crudest approximation. G(;) is the intensity of the absorption and
;
is a measure of theenergy.
Since the first approximation is very rough, we improve it by taking the main part of the inner commutator into account :
4
[Jb
Hl(tl) dt',
H,(r) (- 1+
cos w10
.
(14) We now arrive at an asymmetric absorption envelope (Fig. 6). The asymmetry is of the same form as in the result of the transformation method and as measured in experiments.In figure 7 two absorption line shapes as measured by Ulrici [11] are drawn (AgCl and AgBr crystals doped with Fez+).
We see that our second approximation and the transformation method are in good qualitative agree- ment with the experiment. In particular the calculated curves show the correct asymmetry, the higher peaks being on the high-energy side.
L5
3
' Oq4f
<
8
+
6 8 W A V ~ N U N ~ F R(qe'ci~7
a2
2
4
6 8WAVE NUMBER
[1e3cmcm3
IFIG. 7.
-
Experimental measurements of Ulrici (1). Lineshape1 : AgCI doped with Fez+ (700 ppm ; T = 20 K), lineshape 2 :
AgBr doped with Fe2+ (700 ppm ; T = 20 K) ; OD for optical density.
3.2 To get a deeper insight into the nature of these resonant systems, we calculate the internal dynamics of the E-e J. T. system [12]. We again use the operator method.
We assume that a t an initial time only a single state of the system is excited. At a later time t the probability that a distinct state of the system is excited is given by
This is the square of the projection of the time deve- loped, initially excited state
9r'
onto the arbitraryON THE APPROXIMATION SITUATION OF RESONANCE STRUCTURES C7-121
state
qy'
of interest. As an example the initially excited state is taken to be the state a:I
0>
and we are interested in the time-evolution of the occupation of this state. The result is given in the figure 8 by the full line. The dotted line characterizes the time depen- dent occupation of the one phonon state.FIG. 8. - Amplitude of the occupation probability of the
state a: 10
>
(10, full line) and the one phonon state (11,dotted line) in dependence of the time t for various coupling parameters L (Explanation see text).
We see that independent from the coupling para- meter the energy oscillates with a periodicity of T = 2
nlu),.
The amplitude of the occupation pro- bability depends greatly on the interaction parame- ter1.
For higher couplings the curves reflect the resonant nature of the system.4. The method of moments.
-
In the Method of Moments we confine us to the strong coupling limit. Then, referring to ref. [6], the moments are defined by (T = 0 K)4 . 1 In the E-e J. T. case the moments of the optical absorption spectrum in the strong coupling limit have been calculated by M. Wagner [6]. They are given by
With this expressions the response function can be summed up and Fourier transformed. Then the absorp- tion function reaches the final form
This result is identical to the semiclassical result of Toyozawa and Inoue [17]. The spectrum shows the same form as the one calculated by the Operator Method in the crudest approximation (Fig. 5).
4 . 2 We also apply this method to a peculiar elec- tron-phonon system of two electronically coupled
i
excitonic states (a:, a, ; a z
,
a,), one of which (stateno. 1) interacts with the vibrations (bf, b,) of the surrounding crystal
X = E~ a: a,
+
E, a; a,+
v(a: a,+
a+, a,)+
+
a: a, Sj(bf+
bj)+
oj bf bj.
(18)j j
Examples of such a situation are found in the optical spectrum of V2+ in octahedral fluoride coordina- tion [14]. Because of the equivalence to the atomic systems discussed by Fano, we call these systems
Fano systems [15].
For the special system (18) the moments for strong couplings are given by [7] (T = 0 K)
K,,+ ,,,(Z) is a function of the family of Bessel functions. The response function can be Fourier transformed and one gets (see Fig. 9)
FIG. 9.
-
Optical-absorption line shape of a two-center antiresonant electron-phonon system. G(w) is the intensity of theabsorption, and w is a measure of the energy.
Sturge et al. [14] have measured this system exten- sively. Since our calculations are only valid in the strong coupling region, it is only possible to make a qualitative comparison. We see that in the experimen- tal spectrum a resonant dip arises at the same energy value as in the theoretical one.
5. Discussion. - The three methods we have con- sidered allow the calculation of resonant and nona- diabatic systems.
It can be shown that the Transformation Method yields the best fit [6]. The greatest benefit of the Ope- rator Method lies in the possibility of a simple gene- ralisation to arbitrary temperatures 1161. Such a gene- ralization is rather cumbersome in the Exponential Transformation Method.
C7-122 E. SIGMUND
valid in the strong coupling limit. An extension to Acknowledgment. - The author would like to
arbitrary interactions is possible, but requires further thank Prof. Dr. M. Wagner for critical reading of the approximations. manuscript.
References
[I] WAGNER, M., Z. Phys. 244 (1971) 275. [lo] SIGMUND, E.,BRuHL,S., Submitted to. Phys. StatusSolidi(b).
[2] SIGMUND, E., WAGNER, M., Phys. Status Solidi (b) 57 [Ill U~Rrcr, W., Phys. Status Solidi 27 (1968) 489 ; P h ~ s .
(1973) 635. Status Solidi (b) 62 (1974) 431.
[3] SIGMUND, E., WAGNER, M., 2. Phys. 268 (1974) 245. [12] SIGMUND, E., Submitted to. Z. Phys. B.
[4] WAGNER, M., Z. Phys. 256 (1972) 291. [13] KRISTOFFEL, N. N., SIGMUND, E., WAGNER, M., Z. Natur- [5] SIGMUND, E., WAGNER, M., Phys. Status Solidi (b) 76 forsch. 28a (1973) 1782.
(1976) 325. [14] STURGE, M. D., GUGGENHEIM, H. J., PRYCE, M. H. L.,
Phys. Rev. B 2 (1970) 2459.
[6] WAGNER, M., Z. Phys. 230 (1970) 460. [15] FANO, U., Phys. Rev. 124 (1961) ; 1866 Rev. Mod. Phys. [7] SIGMUND, E., Phys. Rev. B, July 1976. 40 (1968) 441.
[81 GREVSMUHL, U., WAGNER, M., Phys. Status Solidi (b) [16] RUEFF, M., SIGMUND, E., Phys. Status Solidi (b) in print.
58 (1973) 139. [17] TOYOZAWA, Y., INOUE, M., J. Phys. SOC. Japan 21 (1966) [9] SIGMUND, E., Z. Naturforsch. 31a ( 1 976). 1663.
DISCUSSION
J. DURAN.
-
1) Have you tried to apply your method to the well known case of equal couplingTO (E,
+
T,,) of a triplet with E, and T2, modes ?2) What is the upper limit for the electron-lattice coupling strength which you can calculate ?
E. SIGMUND.
