MÉTHODES DE RELAXATION DIPOLAIREORIENTATIONAL DEFECT RELAXATION BY CLASSICAL AND TUNNELING PROCESSES

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MÉTHODES DE RELAXATION

DIPOLAIREORIENTATIONAL DEFECT

RELAXATION BY CLASSICAL AND TUNNELING PROCESSES

F. Lüty

To cite this version:

F. Lüty. MÉTHODES DE RELAXATION DIPOLAIREORIENTATIONAL DEFECT RELAX- ATION BY CLASSICAL AND TUNNELING PROCESSES. Journal de Physique Colloques, 1973, 34 (C9), pp.C9-49-C9-59. �10.1051/jphyscol:1973907�. �jpa-00215382�

METHODES DE RELAXATION DIPOLAIRE

ORIENTATIONAL DEFECT RELAXATION BY CLASSICAL AND TUNNELING PROCESSES (*)

Department of Physics, University of Utah, Salt Lake City, U t a h 84112, U. S. A.

RCsumB. - Nous donnons une revue des principaux aspects experimentaux et thtoriques et des recents developpements ayant trait a la relaxation de defauts de basse symetrie a basse tem- perature. Les systemes en question sont les defauts moleculaires substitutionnels (tels OH-, CN-, ...) et des ions ⁽⁽ off-center ⁾⁾ (tels Li-, AgL, C u t ) dans differents reseaux d'halogenures alcalins.

Quelques techniques experimentales importantes pour mesurer la relaxation a basse temperature (pertes dielectriques, techniques electro-caloriques, ITC et RPE) et des resultats marquants sur les defauts OH- dans differents reseaux seront discutes.

Abstract. - A survey is given on the main experimental and theoretical aspects and recent developments, related to the orientational relaxation of low symmetry defects at low temperatures.

The systems in question are substitutional niolecular defects (like OH , CN , ... ) and off-center point-ions (like L i . , A g . , C u t ) in various alkali-halide lattices. Some important experimental techniques for measuring low temperature relaxation (dielectric loss. electro-caloric, ITC, and PER techniques) and some representative results on OH defects i n various host lattices will be discussed. The results will be conlpared to the various available relaxation models (classical rate process, phonon-assisted tunneling, ⁽⁽ dressed ⁾⁾ tunneling). The latter model will be particularly used to explain the peculiar static and dynamic off-center properties of A g - ions in RbCI and RbBr, measured recently. I t will further be shown how a combination of stress and field application can be used to produce ⁽⁽ stress-tunable Orbacli relaxation ^)), which allows an accurate determi- nation of the relative magnitudes in the rates of relaxation through diferent possible angles between dipole states. Trends in the observed of-center effects of Ag' and C u . ions and in the transition from quantum-mechanical to classical reorientation behavior will be discussed.

Relaxation phenomena due to the reorientation of low symmetry defects have originally, and for a long time, been a domain of classical rate processes, typically with activation energies around a n d above 0.5 eV, which means experimentation at high tempe- ratures. The prototype classical dipole defect which has governed this field, and - as can be seen from several papers a t this conference ^- is still of great importance today, is the defect coniplex formed by a n aliovalent substitutional ion a n d a charge-compen- sating vacancy o r interstitial ion. The s t ~ l d y of the dielectric and elastic relaxation of these defect- complexes in rnany ionic c o l n p o ~ ~ n d s has produced a wealth of information which has contributed deci- sively to o u r understanding of the related ionic transport phenomena like diffusion and ionic conduc- tivity.

In the last decade a new development has taken place which has moved the field of orientational defect relaxation t o low a n d ultra-low temperatures,

( * ) Supported by NSF Grant G H 33703 X.

to new experimental techniques, a n d t o new physical concepts. We shall attempt t o give a short survey on the main aspects of this new development, illus- trating them by some selected results f r o m recent work. In n o way will completeness be attempted, nor can it be achieved in such a brief survey.

The systems we will be concerned with a r e of two types : substitutional i ~ l o l c c l r l a r defects (like OH -, CN-), and substitutional atotnic defects (like L i f , A g i ~ , Cu') occupying off-center lattice positions.

The orientational properties of both these types of defects are determined by the multi-well potential produced by the strong electric and elastic interaction of the defect with the s ~ ~ r r o u n d i n g lattice. F o r a potential of reasonable strength, the minima of the potential define the orientational states of the defect

;it low temperatures which for a cubic crystal may be the six equivalent

<

>,

<

>,

<

>

tlons.

The .stcrtic. low symmetry properties of thcse defects are ch:~racterii.ed by t w o cluantitics : t l l c olcc.t/.ic.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1973907

dipole vector p (produced by the intrinsic electric moment of the molecule and/or the off-center dis- placement of the charged defect), and the elastic dipole tensor hi (produced by the non-cubic lattice distortions around the non-spherical defect). Appli- cation of an electric field E or stress S will split the equivalent orientational levels of the defect, with values for the energy splitting AU cc p.E, or AU cc u . S (where the stress-splitting coefficient u is a linear function of the elastic dipole components Li).

The magnitude of AU depends strongly on the sym- metries of dipole states and applied perturbation.

The orientational ordering of dipole defects under applied static fields or stress has been studied for a large variety of crystal/defect systems using various experimental techniques, like dielectric [I], [2], [3], electro-optic [4], [5], elasto-optic [6], [7], electro- caloric [5], [8] and ultra-sonic techniques [9], [lo].

From the anisotropy, magnitude, and/or saturation behavior of such measurements, the symmetry and magnitude of the electric and elastic dipole can be derived. As an example, for OH-, Li+, and Agf dipoles, one finds electric moments close to 1 eA, with most of the OH- dipoles oriented in

<

>

<

>

<

>

The knowledge of these static dipole quantities (which, in the folliwing, we assume to be available) is the precondition for the study and understanding of the dyizan~ic relaxation belrauior, which is the objec- tive of our discussion here.

How does one nieasure the relaxation behavior of these defects ? First, one can apply the same expe- rimental techniques used for high temperature rela- xation studies of classical defects, if they can be adapted to low temperatures. Thus, measurements of the dielectric loss (as done extensively in Kiinzig's laboratory) [I], [2], [I31 and of the eIri.stic loss (as done by Sack and co-workers) [14], [ I 51 are standard techniques. A new technique, closely related to these loss measurements, which became possible only a.t low temperatures, is the eleciro-ca101.i~ c:f]i.ci. During the electric polarization or depolarization, a dipole system will emit or absorb thermal energy in order to establish Boltzmann equilibrium, as indicated schematically in figure 1A. Two limiting cases can be differentiated : If the rise- and decay-time T, of the field is long compared to the dipole-lattice relaxation time ^7, thermal equilibrium exists between dipole and lattice systems at every moment (left side, Fig. I A ) . As a consequence, phonon emission and absorption (i. e., heating and cooling of the crystal) is a reversible process. For the alternative extreme t, < r, relaxation takes place after field-application in the fully split level system, after field-removal in the unsplit level system (right side Fig. 1A). As a consequence, the phonon energy emitted in this latter irreversible case should be doubled compared to the reversible case, while the phonon absorption should go to zero.

- -- -

A ) D C Electro-calortc E f f e c t , 5 Variation

AT

fseld -on

H e o l ~ n g

G ^Tr.1..

-

-

W

C) A C D i e l e c t r i c L o s s 1

FIG. I . - A) Electro-caloric effect for a two-level dipole system (relaxation time 7) in field-switching experiments (field rise- and decay-time 7s). The level splitting and subsequent phonon absorption and emission is shown for ^{T E} > r (left side) and ^{T E} < 7 (right side). In the middle the expected varia- tion of the heating and cooling effect between these two extremes is shown. B) Electro-caloric effect with a n AC field o r variable Frequency ( , I . The curve and expression show the variation of the net heating per AC cycle AT/co with the frequency. C) Debye ~"(co) dielectric loss curve, as obtained from a loss angle

measurement.

Thus, under continuous variation of the field switching time ^{T E ,} the observed heating and cooling effect should display a dependence as indicated in figure 1A.

Figure 2 suliimarizes experimental results on the

t, variation of the electrocaloric effect for a variety of crystal/defect systems [16], [17]. The measured electro-caloric effect f AT, normalized to its value AT,,, at s, -t a, displays exactly Ihe behavior pre- dicted in figurc IA, with the heating effect doubling and tlic cooling erect approaching zero for ^{T, -+} 0.

The full lines in figure 2, calculated for a single rela- xation time r, and fitted to the measurement by choice of the r paramcter, show perfect agreement with the experimental I-esults.

It is interesting to note the connectio~i between dielectric loss and electro-caloric effects and the consequences of this interrelation. For ^T, > r, or AC frequencies o,,, < 2 nlr, an AC field will pl-oduce a reversible heating and cooling effect in each cycle, so that no net energy loss is dissipated. When to,,,

approaches 2 n/r, heating exceeds cooling in each

ORIENTATIONAL DEFECT RELAXATION BY CLASSICAL A N D TUNNELlNG PROCESSES C9-51

tlectroca

r r ^{- - -}

lorlc tttect

- -- -. - - - -- - -

1 I

heating cooling Field

on

.--

.--

0

o

I a K 1 O H - J

I

-.. - ^A R b I OH-

1

I 1 1 1 1 - /

1'

8 10'' 1 6 6 105 ^la4 lo3 , 10' ^10'

TE Rise and Decay Time of

- - -- - . -

-2 .

--.\.

cooling ,

-- . ^{.-0 RbCl} 4' _-. ~ j

the Applied Electric Field

FIG. 2. - Electro-caloric heating and cooling effect AT (normalized to the AT value at ^{T E +} m), as a function of the field rise- and decay-time ^{T E} for different dipole systems and host lattices. The solid lines are calculated on the basis

of a single relaxation time, the value of which was used as a fitting parameter.

cycle, s o that an energy loss results. The net energy dissipation (heating) per A C cycle, given in figure I B as a function of w, should directly follow the Debye- curve for c"(w). Due to the sn?all value of the specific heat at low temperatures, this caloric technique becomes superior to standard loss angle bridge techni- ques, as was shown by a study of O H - relaxation in KBr [IS]. Moreover, the frequency range can be easily extended beyond that of normal bridge techni- ques, becoming limited only by the rallge of the AC generator, the thermal stability, and the accuracy of the temperature measurement.

One of the most direct techniques to monitor tlie re1:lxation of the dipolc system is tlic clc~c~/r.o-o/~/icc~l I C Y . / I I I ~ Y I I C , . As all d i ~ o l e s wit11 measurable optical ahso~.ptions (1'1.om electronic, vibrational, o r librational excitation) are expected to have different absorption bclia\ior par:~llel ( / / ) and perpendicular (1) to the dipolc axis, stress o r field alignnlent of the dipole s),stems will product charactcristic absorption changes Ah'(/:'. S ) which arc difli-rent for- polarization // and I

to the i~ppliccl pcr[urhation [ 5 ] . M e a s ~ ~ r e m e n t s of the time-dependence of this ,fic.lr/ 01. .\tl.cJ.s.r. tlic~lr~.oi\/i~

after a ~ l ~ ~ i c k change i n E o r S tliercfbrc allo\\s a direct monitoring of the dipole reln\ation. 1-igu1.c 3 sho\\s. :IS n n example, the time-dependence of' the U V elsctro-dicllroism of O H - in KBI-, follo\\ing a quick ( - lO-'s) licld-application ci). field rc~iio\.al h ) . a n d change of field-polar-it!. c ) measured at diflkrcnt temperatures [I 91. [I 71.

I n many cases (like for C N - molecules) a suitable optical absorption is not accessible, because the elec- tronic absorption of the defect may be hidden under the fundamental crystal absorption. These inaccessible absorptions, however, will introduce Kramers- Kronig related ~.cf/jfi.rrt.tir.cj inclc,.~. cl7a17gc..s, which extend due to their chal-acteristic dispersion pattern far beyond the spectral region of the absorption, particu- larly into the low energy (visible) range. Field alignment of the dipolcs \vill therefore cause a bircfringe~lce

1 1 - / I ~ ( ( I ) ) in tlie crystal, which can be measured with high sensitivity (e. g., field-modulation technique) using the nor111a1 Kerr-effect geometry (see illustration in Fig. 4). This K ~ I . I . - c ~ / / C ( , ~ 111~tl70tl hi^^ been applied to O H dipoles, testing a n d confirming the Kramers- Kronig relation of tlic Kcrr-effect to the well-known clectro-dichroism of tlie O H - UV-band [20]. It is now s u c c e s s f ~ ~ l l j ~ being applied to C N - defects

\vhich don't exhibit an electronic absorption in the rransp;went crystal I-nlige [21]. Measurements of the time-constant of' the Kerr-elTect (either with field- motlulatiori or step-I'~~nction technique) allow a direct monitoring 01' the dipole relaxation. As this can be done outside the defect absorption in t11c

\isiblc or near U\/ ranye sing, for example, stable a n d high-intensit!. I;tscr light, this tecll~iique a171x;tr-s to he the most sensitive a n d generally applicable I'or the < t ~ ~ d ! 01' dipolc rel:~\ation.

I I' t he d ipole dct'ccts sho\v /)(r~.a/i~r~,yi~c,lic. ~.c,.coircrt~c~~.

the dill'crent dipolc orientations zind their rclati\.c

I I

1

f (11 T 50-K n m b b t d w f r

g=I

O l T : 3 1 * K '

.

T z 2 I . K o " "

@ Field off

1

1

0 -

I

e l l ~ p l l c o l l y

4

polarized polarized

~~~~~~ iB)

~ ¹

u o

-- A B C I l n e 3 r l y

o o l a r ~ z e d

FIG. 4. -Principle and geometry of Kerr-effect measure- ments for < ^I00 > dipoles. A) E = 0 : Degenerate energy levels for dipole orientations A, B and C, isotropic refractive index contribution 110, and isotropic optical behavior. B) Eool applied : Level splitting and dipole alignment produce optical birefringence "11 # ^"I. The incident light (polarized under 450 to E o o l ) becomes elliptically polarized after traversing the

crystal.

An interesting r(1wan1ic lirat capacity trclitiiq~ie was used by Seward r t 01. [26] when studying the relaxation of C N - ~nolecules in NaCI. After appli- cation of a short heat pulse to the sample, the time dependence of tlie crystal temperature was monitored.

Following the initial quick temperature rise due to the heating of the lattice system, the temperature was found to decay exponentially, due to heat-flow to the weakly coupled ⁽⁽ auxiliary bath )) of the dipole system. Thus, the dipole-lattice relaxation time is directly measured here as the coupling time-constant

.,I

,

5 10 15 20 25 XIO'W

Tlmo . ^,..,- systems.

Extended experimental material on the relaxation FIG. 3. - Time dependence of the optical transn~ission at

225 mM, in RbBr OH-, measured a t diKerent temperatures behi~vior of various dipole-lattice systems has been after a quick (10-3 s) field switch at t = 0 , ~h~ points are accumulated using the above described techniques.

computed for a simple one-phonon-assisted 900 tunneling model. We will discuss some selected examples from this material (which is not in all cases fully understood population can be directly monitored by measurements

of the different EPR spectra. While this method has been successfully used in the pioneering work of Kanzig and co-workers on tlie relaxation of the paraelastic 0; defect [22], none of the electric dipole defects which we are considering here is paramagnetic.

On the other hand, several paraelectric defects exhibit the electric analog t o EPR, puru-c.lcctric r.c.srjttutzc.cJ (PER), i. e., microwave-induced transitions between the orientational dipole levels split by an electric field 1231, [24], From the saturation of the inhonio- geneously broadened PER spectra at low temperatures, and from their life-time broadening at highel. tempe- ratures, the dipole-lattice relaxation time can be measured, as it was done for the off-center Li' ion in KC1 [25].

and interpreted), examples which are most suitable to illustrate the underlying physical phenomena.

This is particula~-ly valid for O H defects, for which tlie static para-clcctr-ic and para-elastic properties as well as tlie relaxation behavior 1i:)ve been studied

~iiost compreliensively in a large variety of Iiost- materials. Three basic results will be considered :

n ) The t c ~ ~ ~ ~ ~ ~ c ~ ~ ~ c ~ ~ i r ~ ~ c ~ - c I ~ ~ ~ ~ c ~ t ~ ~ I ~ ~ t ~ ~ ~ o of the relaxation time, measured by electro-optical and electro-caloric

techniques for different crystal systems, displays at low temperatures ( T < 5 K ) a T -

'

6 ) The rlectric~fie~/r/ ~i'c.l,~~rttk~rrct~ o f T, as nieasureci in figure 6 for O H - in RbBr [I71 and for Ag+ in RbBr [12], is found to be approximately constant in

ORIENTATIONAL DEFECT RELAXATION BY CLASSICAL A N D TUNNELING PROCESSES C9-5:

I

10.' ! . Relaxation of OH--Dipoles

,

^\ .

" o a o optical

I- ^{. -. .}

2 i - - -

!

I 1 , , . I , 1 , I , I , ; , ^{, , / I ,}

2 3 4 5 7 10 'K 15

Temperature

FIG. 5. - Temperature dependence of the relaxation time of OH- in different host lattices, obtained from optical and caloric

measurements.

I

RbBr: OH-

T =2.1°K EIOO-

one phonon process

..>.

"., \.

\ '

.

_{. -.}

--

E I ^I I ^I 2 3 ^I 4 ^I 5 ^I 6 ^I

I * ^{E l l l -}

\\ \ ^A T - 1 . 8 - K

I q 2 . 6 . K

- _3.8.K

5 . 3 - K

I 8.O.K

-

\

\ '. *\ -

FIG. 6. ^- Electric field dependence of the relaxation time of OH- and Ag' in RbBr, obtained from electro-optical nieasure-

ments, plotted agains ( p E / k T ) Y ~ - ,. ,,,,, ,.

the low field range (pE/lcT < I ) , and steadily decreas- ing in the high field range pE > k T .

c) The variation of the O H relaxation witli the /1o.~t-n7arerinl (which was already evident in Fig. 3 and 5 ) is plotted in figure 7, d~splaying a variation over 12 orders of magnitude [ 1 7 ] . A clear trend of increasing T values with increasing lattice parameter is noticeable.

A

29 3.1 3.3 3.5 3.7 Po 3

Nearest Neighbor Separation

FIG. 7. - T h e relaxation time of different OH dipole syh- tems plotted as a f~lnction of the nearest-neighbor separation

of the host crystals.

In figure 8 we consider different relaxation rneclia- nisms for a dipole with two possible orientations . I and 3, which have different energy due to an applied field E. The most common relaxation mechanisnl.

thermally activation motion over the potential barrier (Fig. 8 A ) , leads to the clas.sical 1.aro pt.ocr.s.s with an exponential dependence of r 011 tlie inverse tem- perature and a particular field dependence of' r which is given by the dotted line in figure 6. Clearl!

both the measured T ( T ) and s ( E ) dependence (Fig. i and 6) disagree with this behavior, excluding ~ v i t l i certainty relaxation by classical rate processe\.

Instead, q~rni1~ri11i-1i1c~c11~1t1ic.NI r ~ i t ~ t ~ ~ , l i t l g (case B ill

Fig. 8) provides the basis for the obsened l-corientatioli behavior : A slight overlap between the localizcil dipole states in the potential wells leads to a periodic motion of the particle between tlie two \veils \\ it!?

a frequency Alll, o r in a steady state descript~oti.

to delocalized eigenstates of the dipole \ \ ~ i t I i a t ~ ~ n n c l i n g splitting A . This well-known picture for the two-\\.ell potential can easily be extended to the 1.ealistic 6, S.

or 12 well potentials in the crystal [ 3 7 ] . Under ~1pp11- cation of field 01- stress (witli pE o r y.5' 9 A ) the ~ ~ L I ; I - lity of the m~~lti-well potential is remo\,cd, thc tunrlelirlg states become mixed. and tlic eigeristates ol' r h ~ , particle become ⁽⁽ localized clectr;,,- or cl:~stic-dipol~, states )). The transition betwcen tllc.;c Iocali;/ecl state\

(case C in Fig. 8 ) can be clescrihccl :I\ a ( ( /~hotloti-

4

- t -

a t - .

- t -

. - --.

B ) Pure Tunneling

T u n n e r m ~ r a t e 'r ' = A C ) ~ h o n o n - ~ s s l z e d Tunneling

J' T ' - A 2 0' k T (for pE k T )

-- - -- - - -

Elastlc Dlpole "~ress~ng"

\ r\ ,

FIG. 8. - Illustration of the different reorientation mecha-

nisms, discussed i n the text.

assisted twvieling process ⁾⁾ [ 2 8 ] , involving tunneling through the barrier and emission o r absorption of phonons to accomn~odate the energy difference pE.

Theoretical treatments [ 2 9 ] , [30] show that the rela- xation rate of this process should be proportional t o the squared tunneling matrix element A 2 and ^- at low temperatures, where one-phonon processes are dominant - should be proportional to IiT. This agrees with the observed low tempel-ature behavior in figure 5. Moreover, tlie theoretically 131-edicted T ( E ) dependence for pE > IiT is well I-epl-oduced by the low temperature experiments in figure 6. The observed T - " dependence above 5 K (Fig. 5) can be well accounted for tlieoretically by multi-plionon relaxation processes, which become dominant at higher temperatures [31], [33]. TIILIS, the simple phonon-assisted tunneling model is in good agreement with the observed 7(T, E ) dependence.

The defect-lattice coupling is provided by the coupling of the elastic dipole moment to the strain field of the pl~onons. (The latter can be conside~.cd t o be h o m o g c n e o ~ ~ s over the range 01' the defect for the long-wavelength phonons involved in a one- phonon process.) Therefore, within tliis approximation, the same stress splitting parameter ^Y, appearing in para-elastic alignment experiments. provides the

(( coupling constant ⁾⁾ between dipole and phonons.

Thus one would expect ^- for otherwise constant conditions ^- an increasing relaxation I-ate with increasing elastic dipole values ( T - I z Y', Fig. YC).

Experimentally it is found that the elastic dipole o f O H - increases with increasing lattice parameter of the host (about a factor of 3 from NaCl -. RbBr).

The observed variation of ^T with the host lattice parameter (Fig. 7 ) is thus opposite to the one expected from the z(a) dependence (and very much stronger !).

A qualitative explanation can be found if we proceed one step further with our model (Fig. 8D), following the theoretical ideas of Pirc a n d Gosar [33]

a n d of Shore 1341 : Instead of a ⁽⁽ naked ⁾⁾ dipole in a cubic crystalline field we have to consider a dipole which is ⁽⁽ dressed ^D by strong non-cubic lattice distortions (elastic dipole) around the defect. This dressing increases the effective moment of inertia of the dipole and hinders its reorientation ^(t( polaron

1?70&/ ofreorioltation ))) [35]. The calculations show that the effective squared tunneling matrix element A should consist of the squared tunneling matrix ele- ment in a cubic surrounding A , , multiplied by a

(( dressing o r renorn~alization factor ⁾⁾ exp(- D o ) which takes account of tlie dressing effect from the non-cubic lattice distortions. The dressing parameter D , contains ^- besides constants of the plionon- spectrum and host nearest-neighbor masses - the square of the elastic dipole moment ² (Fig. 8 0 ) . Calculations of tliis parameter D,, using the measured elastic dipole moment, yields values between 7 and 27 for O H - in various host lattices [17]. Thus, the renol-malization factor exp(- D,) from the contri- bution of tlie lattice distortion is very strong

(lo-"

to 10-13) and significant for tlie defect reorientation, displaying a clear trend to very small values with increasing host lattice parameter. T ~ I L I S the major part of the observed trend in tlie variation of the O H - relaxation time with liost material (Fig. 7 ) originates in a strong increase of the elastic dressing (polaron) ell'cct with the size of the host ions 1\71, A quantitative analysis is difficult, because the value of the undressed tunneling element d o , and its \,aria- tion with liost materials, is not known.

An alternative system, in which (he dominating influence of tliis polaron-effect on the reorientation is evident in a dramatic and peculiar wiiy, is the A&' clefect in RbCl ant1 RbBr. The lirst indication for.

olr-center behavior of thesc systems \\,as obtaincrl from s t ~ ~ d i e s of the A&" parity-forbidden U\' tran- sitions by the gro~113 i n F r i ~ n k f ~ ~ r t [MI. ASier s~'ver;11.

partly conflicting, measurements with various tcchni-

C ~ L I C S , electro-optical measurements clariliud a

<

>

ofl-center behavior with clipole moment5 01' 0.7s.

a n d 0 . 9 8 c ~ for A g ' in RbCland R ~ B I . . r~spccti\~cl!. [I?]

A very puzzling result was ohtnincd \vhcn at~~d!,ing the orientational relaxation ol' the s!.:;tcrn. Figurc 0 shows the measured time-dcpcndcnce 01' thc A g ' UV-electro-dic111-oism p r o d ~ ~ c c d by a cl~~icli ⁽

-

application, polarity change, and re~no\~:ii of an clectr-ic licltl [IZ]. While espe~.imcnts of this tylx for other dipole systems (e. g., OH , see Fig. 3 ) show essentially isotropic behavior, tlie opticall).

ORIENTATIONAL DEFECT RELAXATION BY CLASSICAL A N D TUNNELING PROCESSES C.9-5i

-

FIG. 9. - Time dependence of the A-band electro-dichroism of A g in RbBr, n i e a s ~ ~ r e d for three switching operations (0 -> E, -:- E ⁱ - E, E + 0, see lower part) of an electric field ( I : 105 V/cm) applied in < ^{1 1 1} >, < ¹¹⁰ >, ^and

< ¹⁰⁰ > directions. The light is polarized 1 to E ; for El 1 0

the two dill'erent I directions are measured separately. Note thc dilferent time scalcs in the dill'erent parts and in the insert.

detected I-elaxation of the A g f dipole is found to be extremely rrni.roti~o/)ic ^: For

<

>

<

>

<

>

process, while for El,,,, tlie slow 60" reorientation is necessary too and is tlie only optically detectable effect).

What can be the physical reason for this strong preference of tunneling reorientation b e t w e n next- nearest neighbor dipole states through a larger angle ? Such a behavior appears highly unlikely on first sight : in fact, i t has been postulated from general arguments [37] that in a cl-ystal potential of octahedral symmetry it should always hold for the tunneling rate that A (90") ,< A (60"). An attractive pliysical

explanation was proposed [12], based on n dilTere11~

elastic dressing effect for the two types of motion,.

which has been tested very recently experimentally [38].

Figure 10 illustrates the 12 possible

<

>

IW OPTICAL ABSORPTION E L A S T I C DIPOLE TENSOR

DIPOLE MODEL I A,D, and 0, t r o n r ~ l ~ o n r l [Components A , . A 2 .A,)

Srrcss axis 8 Energy L e v e l s a,-f,(A,j

reor. _{o ,} . ^uo (+ +*>) E g

90' I

FIG. 10. - Illustration of the 12 equivalent sites of the A s

< ¹¹⁰ > dipoles, and of the optical absorption and clas~ic dipole components of one partic~~lar dipole (.. 3). Bclow ale

sketched the energy splitting of the orientational states unde~.

< 100 > and < 1 1 1 > stress, expressed in terms of thc elastic dipole coniponents 2.1, j.2, and i 3 .

tations of the A g f dipole and, for one particul;~~.

dipole orientation ( + 31, tlie three component\

of the elastic dipole tensol- I * , , h z , and IL,. U n d e r applied

<

>

<

>

,

s,,,

s , , ,

Figure I 1 shows the result, a normalized plot of the logarithniic population ratio of the two-level systenis

FIG. I l . - Plot of the logarithmic population ratio o f the t\vo-level system under S I ~ ~ O and SI 1 I. obtained I'~.oni ela\to- optical mcasurelncnts in the A and D hands of A g in Kh131- atid RbCl a[ verioi~s tenipcraturcs, plottctl :IS :I function of .SiT.

under S,,, and S , , , , measured at various tempe- ratures and in various optical transitions, as a function of SIT. F o r each stress symmetry, a perfect linear dependence is obtained, as expected for para-elastic alignment, with the slopes yielding the a , and a, values, i. e., the E, a n d T2, parts of the elastic dipole.

F o r both RbCl and RbBr the E, part is found to be about 2.7 times larger than the G, part.

Figure 12 illustrates the two types of orientational motions of the Ag' ions a n d their elastic dipole

9 0 ~ _ ~ & ~ n t g 1 o n ~n (100) Plane 60' R e o r ~ e n t o t ~ o n ~nll)mz

1 8) I

FIG. 12. - illustration of the two types of orientational motions and related elastic dressing effects for the < 110 > off-center Ag+ defect. A) 900 reorientation in a (100) plane, hindered by a pure T2g elastic dressing. B) 600 reorientation in a (1 1 1 ) plane, hindered by both E, and Tzn elastic dressing.

dressing, by looking into a (100) and ( 1 1 1 ) crystal plane. When the Agf dipole reorients by 90" jumps in the (100) plane (left side), the E, part of the elastic dipole moment 4-(2,

+

motion contains both E, and T,, parts, as indicated in figure 12. When adding the experimental result from figure 1 1 ( E , > Tz,, part), it becomes evident that the 60° motion (which should be predominant in the ⁽⁽ undressed ⁾⁾ case) is much stronger dressed by elastic dist01,tions than the 90° motion, t h ~ ~ s producing the peculiar ⁽⁽ inverted ⁾⁾ relaxation beha- vior : The very strong tetragonal E,, distortion ⁽⁽ traps ⁾⁾ the dipole in a particular (100) plane, allowing a relatively easy (90") rotation within this plane around the fixed distortion axis. 60° reorientation, on the other hand, requires a change of the plane, i. e., a reorientation of the strong E, distortion.

With the off-center Agt system we have a case in which the orientational motion of the ^.vrrnlc defect is dominated by t ~ t . o dilrerent types of polaron o r dressing effects, so that two very different reorientation

rates occur. The possibility of measuring both these static elastic dressing parameters and the two resulting dynamic motions, makes this system a particularly attractive model-case to test the dressed tunneling model. A quantitative analysis will be possible when the constants C,, and C,,, contained in Do (see Fig. 12), have been calculated from crystal and phonon parameters. Such a calculation is in progress now [39].

The A g f system demonstrates a n important point : For setting up realistic tunneling o r reorientation models, one has to make an assumption about the relative rates of tunneling through the various possible angles between dipole states. For nearly all tunneling models [27], [40], calculated so far, one has considered nearest-neighbor tunneling as b e ~ n g highly predorni- nant, and neglected other contributions. The Agi system shows that t h ~ s can be co~npletely wrong.

Though for the

<

>

<

>

Lif system it is well established that nearest-neighbor tunneling is predominant [19], [I I], it is desirable for realistic tunneling models to determine the exact contributions from other reorientations. A scheme to d o this was proposed (Fig. 13), which uses both

II.I2

kFT;;.

4

042.3.4 3.1

-

-

FIG. 13. - Splitting of the orientational levels of < 100 >,

< ^{1 10} >, and < ^I I I > dipoles under a large ~ ~ n i a x i a l pres- sure S a n d an additional small clectric field (1 to S) as indicated.

For the resulting lowest rnuliiplet of levels, the possible reorien- tation processcs (direct process within mc~ltiplet and Orbach

process via higher lcvcls) arc indicated.

theelectric and elasticdipole properly ofthe defects [41].

F o r a dipole system of known symmetry, a stress and field direction (crossed) can be selected in such a way that dielectr~c relaxation within the low multi-

ORIENTATIONAL DEFECT RELAXATION BY CLASSICAL A N D T U N N E L I N G PROCESSES C9-57 plet of states is no longer possible via nearest-neighbor In figure 15 the results of the dielectric loss shift reorientation. I n figure 13, three cases are illustrated under S ,

,,

<

>, <

<

>

<

>

directed perpendicular to the stress, i. e., the dipoles of the opposite orientations [OOI] and [OOI]. Direct relaxation within this lower multiplet would be possible only by 180" reorientation, as illustrated in figure 13. F o r the normally predominant 900 reorien- tation process, thermal activation to the higher multi- plet of states is necessary. Thus, the stress is expected to shift the 90') I-elaxation rate exponentially to smaller values ((( .vtrr.s.r-tu~~a~l(.bI Oi.bucll i ~ r l a s u t i o ~ i pr0ce.r.s D), until eventually a direct 18O0 reorientation rate may become measurable. 111 the same way (see Fig. 13) for

<

>

<

>

Besides this expected .sll!'fi of the dielectric loss, its itlten.vit!~ should increase under the applied stress due to the higher dipole polarizability in the lower state multiplet.

Figure 14 shows first results obtained with this technique for O H - in KBr and KC1 [41]. F o r both cases the dielectric loss curve is found t o be shifted to lower frequencies under a simultaneous initial increase of its integrated value, as expected [42].

KCI: OH-

I

FIG. 15. -Shift of the relaxation time of OH- in KC1 at 1.22 K as a function of the applied stress Sl1 0 (1 to the field Eon!). The measured behavior (points) is compared to the calculated behavior of a stress-tuned Orbach ell'ect for relasa- tion by classical rate processes (A) and by one-phonon assisted 900 tunneling ( B ) . [These curves have been calcillated using thc

known valc~es of T (S = 0) and the stress-splitting

1.1

tunneling process. Good agreement with the latter process is found in the measured range. The absence of any saturation effect after 4 orders of' magnitude stress-tuning indicates that the next-nearest neighbor (1800) reorientation rate is at least 4 orders of magni- tude smaller than the 90" rate. M e a s ~ ~ r e m e n t s of this

t

0 7 type will be ~lsed to measure with high accuracy

0 6 K C I : O H -

~ ~ 1K 2 2 different angles for various dipole systems. thus

2.0 allowing one to set up tunneling models with realistic

0 4 - relative tunneling parameters.

Z o o Several expel-imental and tlieol-etical aspects related

0 3 to low temperature dipole relaxation could not be

treated in this brief survc\,, like the pronouncecl efTects from electric a n d el:~stic dipole interaction

0 5 I 5 I? 5 0 110' dipole systems [I]. [43]. [41]. A n important extension

(-9 of the work described hcrc is the one into the ~~ltrnlovi temperature range. \\here 1, T becomes compar:tblc FIG. 14. - Dielectric loss of O H in KBr and KC1 at 1.22 K

as a function or frecluency, for d i ~ ~ c r c l , t of to the tunneling hplitting A . Quantuln clli'cts I'rom '4.

uniaxial stress Sloe applied I to the electric held strong influences f~.orn hachgrounci striti~i a n d lieltis