Zeitschrift fear P h y s i k B
Z. Physik B 27, 267-271 (1977)
© by Springer-Verlag 1977
Statics and Dynamics of "Soft" Impurities in a Crystal* ** K.-H. H6ck and H. Thomas Institut fiir Physik, Universit~it Basel, Basel, Switzerland Received March 25, 1977 We investigate the effect of an impurity on the local static and dynamic behaviour of a crystal undergoing a displacive phase transition. It is shown that in mean-field approximation there may occur a freezing-out of local order driven by a soft local mode above the phase transition of the host. For a crystal containing a dilute impurity concentration, the impurity contribution to the static structure factor is calculated. The results are discussed in detail for the specific cases of a displacive and an Ising-type impurity.
1. Introduction Recently, growing interest has been devoted to the problem of defects in a crystal undergoing a structural phase transition. Halperin and Varma [-1] pointed out the possibility that the presence of a certain type of defect may account for the extremely narrow width of the central peak in the dynamic structure factor appearing in addition to the soft phonon modes [2]. Furthermore, substitutive impurities are used as probes in EPR, NMR, N Q R and M6ssbauer experiments in order to obtain information on local critical properties [3]. It appears desirable to examine the assumption implicitly made in the interpretation of these experiments that the local critical exponents are not changed by the presence of the impurity. In this paper, we therefore study the influence of a displacive structural phase transition of the host crystal on the local static and dynamic properties near "soft" impurities in mean-field approximation (MFA). We call an impurity "soft" if its local normal coordinate is more susceptible than the local normal coordinates of the soft-phonon branch of the host crystal to which it is coupled. We assume the impurities to be independent (an approximation which will break down close to the bulk transition temperature * Work supported by the Swiss National Science Foundation ** The results of this paper were presented at the Spring Meeting of the German Physical Society, Miinster 1977 (Verhandlungen DPG (VI) 12, 1977)
T~ of the pure system) such that we may consider a single impurity localized at lattice site l = i. In Section 2, we obtain the equations determining the local static and dynamic behaviour, and derive the conditions for the existence of a local mode below the soft optical-phonon band and ,for its condensation. The contribution of a dilute concentration of impurities to the structure factor is calculated. In Section 3, the results are applied to the two specif-. ic cases of a displacive impurity (weakly anharmonic potential) and an Ising-type impurity (double-trough potential). The conditions for local freezing-out* are discussed in detail. 2. Local Properties of a Crystal with Impurities We consider a crystal undergoing a displacive phase transition described by a one-dimensional order parameter. Its behaviour is assumed to be described by a model which takes only the local normal coordinates Q; of the optical soft-mode branch and their conjugate momenta into account, and we assume the usual model with quartic single-particle potential F(Q1 ) =~KQ; 1 2 + z1v Q ~ and bilinear interactions [5]. An impurity at lattice site l = i couples linearly to the coordinate Qi and is described by a single-particle * The possibility of such a local freezing-out was recognized in the one-dimensional case by W. Senn [4]
268
K.-H. H6ck and H. T h o m a s : Statics and Dynamics of "Soft" Impurities
Hamiltonian Its(Qi, Pi). We then have the model Hamiltonian H = ~ [P12/2M + V(Q,)] - l ~ . ' v , r QzQr 14=i If' +/ts(Q,, P3.
(1)
For simplicity, we disregard any change of the bilinear interactions v~r by the impurity. The properties of the host crystal are assumed to be known. For definiteness, we consider the case of a ferrodistortive transition, but the model is easily extended to the antiferrodistortive case. The collective susceptibility of the host crystal has the form
Zq(co) =
[M(co2 (T) - c°z)]
-1
(2)
where coq(T) is the frequency of the soft optical branch, for which we assume a Debye dispersion
M co2 (T) = M c02(T) + A 2 (q/qD)2.
(3)
Here, A 2 is related to the q = 0 Fourier component v0 of the interaction by A2 =(5/3)%, and the soft-mode frequency coo(T) is given by
The static value ~c(0) is just the reciprocal correlation length of the host crystal• For the determination of the static configuration at a given temperature, we obtain in M F A the set of coupled nonlinear equations
i = X s(Fims) ~= X,(F• f)
(lOa)
(14= i)
(10b)
where < >/denotes the statistical average in the presence of an impurity at site i, & f = ~ ' v, r I
(10c)
l'
is the mean field acting on {2z, and X S and )~, are the static nonlinear single-particle response functions at a normal site and the impurity site, respectively. If the response (10b) of the normal cells is linearized, * can be expressed in terms of <(21>i as i = (Z li/~ii) i
(11)
where f2(T) is an effective single-particle Einstein frequency determined selfconsistently by [5]
where Xu is the static nonlocal susceptibility of the host crystal. This result can be simply understood by the following consideration: Write the response of the crystal to a fixed value of i formally as the response of the pure crystal to a fictitious field F~* at the impurity site such that
Mr22(T) = K + 37(a + 2)
(5 a)
, =)~. F~*
a = KT/Mf22(T)
(5b)
M co2(T) = MQ2 (T) - v o
(4)
whence for T__>Tc
Mf2e(T) =½[K "Jv(g 2 ~- 12~2kT)l/a].
(5c)
The nonlocal susceptibility is obtained as the Fourier transform of Zq(co), 7~zr(co) = N 1 ~ Zq(co) exp ( - i q- R l l').
(6)
q
For I= l', one finds for a three-dimensional crystal for frequencies co
(
42
• arctg \M(coT_(o=)! j.
(71
The asymptotic behaviour at large distances
Z~t(co)=(3~/2qoA2)lRizl-*exp(-~c(oo)lR~l)
and eliminate F/*. Relation (11) might appear to be of very limited validity, both because of the use of M F A and of the linearization of Equation (10b) which can be expected to hold only sufficiently far above the bulk critical temperature. It is therefore interesting to note that for an Ising system the relation (11) is in fact exact if the true nonlocal susceptibility is used [6]. This result is due to the fact that the nonlinearities cancel when F~* is eliminated. We expect therefore that Equation (11) has wider validity than indicated by its derivation. We therefore obtain for the determination of the static expectation value at the impurity site the selfconsistency equation
i = J{s(2 i)
(13 a)
where the factor (8)
shows an exponential decay with reciprocal decay length ~c(co)= [M(cog(T) - oa2)/A 2]~*- qD"
(12)
(9)
2 = ~' vilZu/Zu
(13b)
I
describes the enhancement of the local response at lattice site i caused by the coupling to its environment,
K.-H. HSck and H. Thomas: Statics and Dynamicsof "Soft" Impurities (14)
Zii ~" ZOO = ZJ(1 - A Z,).
For sufficiently high temperatures, Equation (13) will have (Qz}~=0 as only solution. Local freezing-out will occur at a temperature T~1°~ determined by the condition that the function 2~(2(Qi}i) in Equation (13a) has an initial slope of unity, i.e. ).(T~~°¢))~°(T~I°¢)= 1
(15)
if this equation has a solution T~l°~ such that T~1°~> T~. Here, 2°s=O~JOFm:[o is the zero-field single-particle susceptibility of the impurity coordinate. Below T~1°°, the order parameter (Q~}~ at the impurity .site is given by the nontrivial solution of (13). The values (Q~}~ at the other lattice sites I+i are then obtained from (11) which shows that the order decays spatially with the correlation length of the host crystal which diverges for T--, T~. The linear response of the perturbed crystal to external fields 6F~ is calculated as the linear response of the pure crystal to fields cSF~+6F~'6~,i where 6F,.* = # 6 (Qi},
(16)
is the change of the fictitious field induced by the response 6(Qi} i. It can be calculated from the single-particle susceptibilities 2s=a2gjOF: ": and gs =~Xs/~Fi': of the perturbed and the pure crystal, respectively, (17)
,5 ( Q~), = :is,5<": = Zs(aU': +,55")
269
which shows that the response of the perturbed crystal diverges at the local freezing temperature T~1°~ determined by Equation (15). The dynamic response at frequency co can be obtained from the above relations by replacing all static susceptibilities g by their dynamic counterparts Z(co), and the factors # and 2 by the corresponding frequency functions #(co) and 2(o)). If local mode frequencies coloc exist, they can be found from #(coloc) )~ii(coloc) = 1
(24)
or equivalently from 2(colo~) )~(co,o~)= 1.
(25)
Comparison with (15) shows that local freezing-out is always associated with a soft local mode coloc~ 0 for T-~ T~l°°. In one dimension, the sum in (6) determining the static susceptibility Z00 diverges as coo ~ for T~T~. Therefore, (15) which is equivalent to #(Tc l°c) Zoo(Tc l°c) = 1 always has a solution T~1°°> T~ if only #(Tc)>0, i.e. in one dimension a soft impurity always freezes out. In three dimensions, however, Z00 stays finite, and #(T~) has to exceed a certain positive value before freezing-out occurs. The correlations
~i~, (z) = (Ql(z) Q r (0))i - (Q l)i (Q r )i
(26)
of the perturbed crystal can be found from the fluctuation-dissipation theorem. Of particular interest is the Debye-Waller factor of the impurity coordinate
whence
g2 = k T 2, = S,/(1 - p Zi~) L = Zs/(1 - # z).
(18)
We have a soft impurity if 2~>)~s, i.e. # > 0 . Expressing the response 6 (Q l}i as indicated above,
and the correlation of other coordinates Q z with the impurity coordinate Qz, J~] = k T 2~ = Si j(1 - # )~1.
,5(Qz)~=~i,,,,SFr=~Z~r(,SFr+,SF,.* l'
I'
6,, ~)
(19)
one finds 2l l' =){l l'-{'-# )lvli 2i l'"
(20)
From the equation for l=i, the response 2il, can be calculated 2i z"= )~ii'/( 1 - # Xii).
(21)
We therefore obtain for the susceptibility of the perturbed crystal 2, r = Z,,, + [#/(1 - # Z,,)] Z ,i ,~,,'-
(22)
The factors # and 2 are related by )fii/( 1 -- # ~ii) = 2s/( 1 -- ~ L )
( = 2ii)
(23 )
(27)
(28)
This shows how the local fluctuations are enhanced by the soft impurity. We finally calculate the impurity contribution to the static structure factor of the crystal containing a dilute concentration N = c N of impurities. We assume that the impurities do not interact, and their contributions can be linearly superimposed. This approximation will be valid as long as the distance between impurities is larger than the correlation length, i.e. c 1/a ~tc. This is the opposite limit to the "average crystal approximation" considered by Halperin and Varma [1] which may be expected to hold close to T~ where the correlation clouds of the impurities overlap. Then, writing ~{l,=Slr + A ~ I , , one has S,,, = ~ ,~:~,= S,r + c (A Sq/{,>conf i
(29)
270
K.-H. H6ck and H. Thomas: Statics and Dynamics of "Soft" Impurities
where ( )~o,f=N-1
~
all sites
\
is a configurational average
over the random impurity positions. From (22) one obtains
A, Sq= (c/kT)
[#/(1 - , u Zoo)] Sq S q.
^2 _~e=Kc/3kTc
,////
(31)
where (Qq)i=N-l~,(Q,l)iexp(iqRu) is the Fourier l
transform of the local order parameter. In contrast to zI1Sq which is of dynamic origin, the contribution A2S q is purely static and corresponds to a central peak in the dynamic structure factor of zero frequency width.
B, a local mode freezes out at a temperature T~l°e > To, i.e. 3~kT~+(Vo-½A2)I~<(Vo-½A2) 2. This is accompanied by a divergence of the static local susceptibility at the impurity site, Zu = [ ( a - c/)(T- Tcl°e)~ -1
a) Displacive Impurity
where
We consider an impurity which gives rise to an effective mass M of the coordinate Q~ and to a quartic single-particle potential,
a=37(K2 +127kTl°e) -~
(32)
with
M=M+AM>O I ( = K + AK>O The zero-feld single-particle susceptibility of the impurity coordinate is of the same form as that of the pure crystal [5] Zs(co) = [2~(f]2 (T) __'CO2)]
- 1
(34)
where ~/f22(T)is determined as in (5). From the dynamic equivalent of (18), we obtain for the factor
#(co) # (co) = M(,.Q 2 (T) - co2) _ ,~r (~2 (T) - c02).
(35)
The impurity is "soft" if #(0)>0, i.e. near T~
3~kTc+vOI(
~/= 3~(/( + 12~?k T~l°e)-~.
The conditions (24, 15) for the existence of a local mode below the band and for local freezing-out are presented in Figure 1. We find that the "soft" part of the (/£, ~) parameter plane is divided into four regions A-D, separated by two lines L1,2. In regions A and
(37b)
+ 3 ~^a~ + ~ X A ^s2) = F ~
(38a)
= k T/[I(+ 3 ~(~)+ )~2)].
(38 b)
For temperatures close to T1°°, one finds (Qi)2 = [(a - ~/)/3?]k(T~l°° - T).
(39)
According to (11), the local order extends over a region determined by the bulk correlation length, and spreads for T--*Tc over the whole crystal. For T < T~, one finds local order superimposed on a bulk order parameter. The temperature variation of the localmode frequency O)1oc for T>~ T~1°c is characterized by an initial slope c~, CO2oc= 8 ( T - T~1°c)
(36)
(37 a)
In the range T~< T < Tc1°e there exists local order in a region around the impurity. The order parameter at the impurity site is determined by (13), where the nonlinear single-particle response J(s(F/) is given implicitly by [5] 27s(£
(33)
K
Fig. 1. Local freezing condition in the parameter plane of a displacive impurity. The lines L 1 (TI°~=T~) and L 3 (#(T~)=0) are independent of the impurity mass ~t, the line L 2 is shifted upward with increasing /~. The value of ~0 is given by Yo = (~/.~t) 12 72 k Tj(2 v0 - K)
3. Specific Cases
~?=~+AT>0.
~---" I-t (Tc) =0 ~vL3
~0 ~, : V0 - + A2
d2Sq = c N - 2 Z (Ql)i ( Q r ) i exp(i q(R z-Rr) ) 1,i
1 ^ 2 H^ s = ~1P ~2 +gKQi +¼~Q~
L2
(30)
The dynamic structure factor can be obtained in an analogous way. Below the local freezing temperature T~l°~, there appears an additional contribution to the static structure factor from the local order parameters. One finds
=c(Qq) (Q_q)
D 1~2~
(40)
which is smaller or larger than the slope ct of the softmode frequency co2 _- cffT- To)
(41)
for parameter values in regions A and B, respectively. Thus, in region A, the local mode will persist to
K.-H. H6ck and H. Thomas: Statics and Dynamics of "Soft" Impurities
higher temperatures, whereas in region B it will disappear into the band. In regions C and D, no local freezing-out occurs. In region C, a local mode emerges from the band at a temperature T > To, whereas in region D no local modes exist below the band for temperatures near T~. b) Ising-Type Impurity ( " H a l p e r i n - V a r m a
Center")
We consider an impurity which can assume only two positions Qi=+_pi, with relaxation dynamics [1]. The zero-field single-particle susceptibility is given by 2s(0)) = (~2/k T) I~/(F - io))
(42)
where f is the single-particle relaxation frequency. Thus, #(aO = M ( t 2 2 ( T ) - a) 2) - (kT//32)(1 - ico/F).
(43)
The impurity is "soft" if/~(0)>0, i.e. near T~ ~2 > kT~/vo"
(44)
The condition T~l°~>T~ for local freezing-out takes the form /32 > kTc/(Vo - ½ A2),
(45)
and the local freezing temperature is given by
271
occur in a crystal with short-range interactions only. We give the following interpretation of the freezing condition: Consider a finite cluster containing the impurity, decoupled from the rest of the crystal. Then, as long as #2s<1 there will exist a finite restoring force giving rise to oscillation or relaxation of the local-mode coordinate about the undisplaced configuration Qloo=0, and this motion will not change drastically by coupling the cluster to the rest of the crystal. For/~)~s > 1, on the other hand, the free energy of the cluster will have minima at displaced configurations _+Qloc, separated by an activation energy which in the limit of large clusters will in reality be independent of cluster size. Whereas in M F A the cluster will be frozen into one of these minima, in a real crystal coupling of the cluster to the rest of the crystal will cause occasional activation processes across the barrier*. We therefore expect a change from local oscillation or ordinary relaxation dynamics of the local mode coordinate at high temperatures to activated jump dynamics at a fairly well-defined temperature T~~°c. One of us (H.T.) thanks K.A. Miiller and E. Courtens for encouragement and clarifying discussions.
References
which freezes out at Tc~°L Recent results of E N D O R measurements on Cr 5 + in KH2AsO 4 seem to indicate that the local dynamics of this center freezes out at T > T~ [7].
1. Halperin, B.I., Varma, C.M.: Phys. Rev. B14, 4030 (1976) 2. Riste, T., Samuelsen, E.J., Otnes, K.: In "Structural Phase Transitions and Soft Modes", (edited by E.J. Samuelsen, E.J. Andersen, J. Feder) Oslo: Universitetsforlaget 1971, p: 395 3. See articles by Miiller, K.A., von Waldkirch, T., Borsa, F., Blinc, R., Rigamonti, A. In: Proceedings of the International School of Physics Enrico Fermi, Course LIX 1976 on "Local Properties at Phase Transitions" (edited by K.A. Miiller and A. Rigamonti) Amsterdam: North-Holland 1976 4. Senn, W.: Diplomarbeit (Basel 1974) 5. Thomas, H.: In "Structural Phase Transitions and Soft Modes", edited by E.J. Samuelsen, E.J. Andersen, J. Feder (Universitetsforlaget, Oslo, 1971), p. 15. See also page 3-42 in the Proceedings mentioned in [3] 6. Thomas, H., Brout, R.: Journal of Applied Physics, Vol. 39, No. 2, 624 (1968) 7. Gaillard, J., Gloux, P., Miiller, K.A.: To be published 8. Heeger, A.J.: Solid State Physics, Vol.23, p. 293 (edited by F. Seitz and D. Turnbull) New York: Academic Press 1969
4. Concluding Remarks
* The freezing:out of a local order parameter bears a certain resemblance to the appearance of a localized moment at a paramagnetic impurity [8]
We close with some remarks concerning the significance of the above results which were obtained in MFA. Although the M F A can be expected to give an acceptable description of the properties of the host crystal for temperatures outside the critical region, a local freezing-out in the strict sense can certainly not
K.-H. H6ck H. Thomas Institut fiir Physik Universit~it Basel Klingelbergstrasse 82 CH-4056 Basel Switzerland
k T ~ ° ° = f i Z ( v o - ½ A2 ).
(46)
The divergence of the local susceptibility at Tc1°° and the appearance of a local order parameter for T < T~1°* are similar as in case a. We are specifically interested in the case of a slowly relaxing impurity, ff~f2(Tc). We then find a local relaxation mode with frequency O~1oo = - iff(1 - T~°C/T~c , ,,
(47)