study of ferroelectric phase transitions by electric field perturbation of nuclear quadrupole...

Physica 142B (1986) 71-79

North-Holland, Amsterdam

STUDY OF FERROELECTRIC PHASE TRANSITIONS BY ELECTRIC FIELD PERTURBATION OF NUCLEAR QUADRUPOLE RESONANCE SPECTRA

A. GORDON Department of Physics and Mathematics, Oranim - School of Education of the Kibbutz Movement, Haifa University, Tivon, 36910, Israel

Received 6 November 1985

Revised manuscript received 18 March 1986

Theoretical aspects of electrically induced shifts of nuclear quadrupole resonance spectra are considered at ferroelectric

phase transitions. The displacive and order-disorder types of phase transitions are analysed. Calculations and experimental

data for the low-temperature phase transition in pyrargyrite (Ag,SbS,) show good agreement. It is confirmed that the

above phase transition is of the order-disorder type. The phase transition is shown to be connected with the ordering of

the silver ions in the double-minimum potential of the ionic bond S-Ag-S.

1. Introduction

Nuclear quadrupole resonance (NQR) spectra are extremely sensitive to changes in symmetry which often occur at the sites of the resonant nuclei as a result of structural phase transitions [l]. However, the information which may be obtained from NQR spectra is mainly a local one. It shows the charge distribution in vicinity of resonant nuclei. Hence the study of NQR provides a microscopic probe and thereby makes it possible to investigate local aspects associated with phase transitions. The information obtained is therefore complementary to that provided by techniques such as light scattering and neutron scattering which are sensitive to the collective excitations in the crystal. Nevertheless, some examples of the NQR study of ordering in crystal as a whole at ferroelectric phase transitions may be mentioned. Firstly, the temperature variation of NQR frequency often coincides with the temperature dependence of the spontaneous polarization (see [2] and references cited therein) yielding information about the cooperative properties and type of ordering in ferroelectric crystals. One can obtain the critical exponent associated with the order parameter. Secondly, the temperature dependence of the NQR frequency may reflect the softening of the lattice mode

which governs the phase transition [3]. Thirdly, the NQR spectra may serve as a tool to discover pretransitional clustering and heterophase fluctuations [4,5]; these are long-lived metastable regions with a high degree of order.

An additional method to investigate the non- local, collective peculiarities of crystals undergo- ing structural, in particular ferroelectric, phase transitions is the study of external electric field effect in NQR spectra. Indeed, the above- mentioned relationship between the spontaneous polarization and nuclear quadrupole coupling constants in ferroelectrics can lead to a connec- tion between their responses to an external electric field, that is, between the dielectric susceptibility and electric field induced perturbations of the nuclear quadrupole interaction. Consequent- ly, a strong temperature dependence of the electrically induced shift of the nuclear quadrupole interaction should be expected at ferroelectric phase transitions. Some initial attempts to detect the temperature dependence of electrically induced shift of NQR frequencies have failed [6]. However, the very sensitive method using pulsed external electric field enabled the detection of the effect in question [7]. Nevertheless, the only available experimental result on this effect has not yet been explained.

In this paper we propose a theory of the

0378-4363 /86/$03.50 0 Elsevier Science Publishers B .V. (North-Holland Physics Publishing Division)

72 A. Gordon / Study of ferroelectric phase transitions

electric field effect in NQR spectra at ferroelectric phase transitions and compare it with the result on the low-temperature phase transition in pyrargyrite (Ag,SbS,) crystals.

2. Theory of the electric field effect in NQR at ferroelectric phase transitions

If a nucleus or a paramagnetic ion occupies a site which lacks inversion symmetry in a crystal lattice, an external homogeneous electric field will cause, in general, linear changes in a spectrum of a magnetic resonance [S]. The linear electric field effect in NQR spectra has been observed for the first time by Bloombergen et al. in ref 9. This effect may be explained as follows. Since an external electric field is homogeneous within a nucleus, there is no direct interaction with a quadrupole moment. However, an in- direct interaction between a nucleus and the external electric field is possible for the field gives rise to the ionic and electronic polarizations yielding changes of the electric field gradient (EFG) at the nuclear site. Interactions of the electrically perturbated EFG with a nuclear quadrupole moment bring about the shift of the NQR frequencies. Since the EFG response to an external electric field reflects the ability of sub- stances to be polarized, hence in such strong polarizable crystals as ferroelectrics the electric field effect in NQR frequencies will be sensitive to peculiarities of polarization.

In strong electric fields quadratic electric shifts in NQR frequencies may be observed. In the general case the influence of an external electric field E on the tensor of an EFG I$ at sites of resonant nuclei is expressed as follows:

(1)

where the subscript 0 pertains to E = 0 and i, m, I denote the components of the electric field vector B. We restrict ourselves to the linear

terms so far only the linear effect has been measured. The only deviation from a linear E-

dependence was observed in single crystals of pyrargyrite (Ag,SbS,) [lo] where a linear dependence was detected up to E, = 3.6 kV I cm, while for a higher value of E there was no further shift of the Sb NQR. The saturation of the E-dependence of the electrically induced

shift in the NQR frequency seems to imply screening of the external electric field by the volume charge accumulated in the samples under the action of this field. Evidently the electret state is formed which is characteristic of pyrargyrite and of its isomorph proustite (Ag,AsS,)

[ll]. The memory of the external electric field influence on a crystal was retained during one month [lo]. The linear E-dependence of the electrically induced shift in Sb NQR frequency in Ag,SbS, was restored after annealing of samples at T- 200°C. Therefore the annealing process led to the liberation of carriers from traps and the destruction of the electret state. The important role of traps has been indeed observed in experiments on the residual photoconductivity in pyrargyrite and proustite. In proustite the residual photoconductivity due to traps [12] disap- peared near the first-order phase transition as a result of heterophase fluctuations [13], which, in turn, were detected by NQR [5].

Following [9], the changes induced in quan- tities I$ by the application of an external electric field of unit strength in the ith direction define a

tensor Riik, where

qk e '9i!i R,, = aE = -

I aE, ’

where e is the electronic charge. Then the most general description of the linear electrically induced perturbations of the jkth component of the EFG is

A yk = eA q,k = C RljkEi ’ (3)

As it has been shown in [9], the third rank tensor R consists of the two parts. The first term describes linear changes of the EFG components which occur at constant strain and exist at sites

A. Gordon I Study of ferroelectric phase transitions 73

without inversion symmetry, whether or not the lattice as a whole has inversion symmetry. The second term, a function of the piezoelectric coef- ficients, describes linear changes in the EFG components which occur because of the strain induced by the application of the electric field.

To generalise this theory to the order-disorder phase transition case, we take the quadrupole Hamiltonian Ho as follows:

m=2

& = c Qms, [a:(t)1 7 (4) m=-2

where Q, are nuclear quadrupole operators and q, are EFG operators,

moment m is the

magnetic spin quantum number, and cr’ is the z-component of the pseudospin operator presented by the matrix 1: _‘f ) . As it is known, taking the time average of the EFG tensor and calculat- ing the ensemble average in the mean field approximation (( cz) = (cf) = (a’) = p, where p

is the normalized spontaneous polarization), one obtains [ 141

( 4 > = 40

in the paraelectric phase and

(5)

(4) =qo+Aqp

in the ferroelectric phase, where

(6)

40 = 1(4, + q2)

and

(7)

Aq = ;(q, - q2). (8)

where q, refers to the “right” and q2 to the “left” equilibrium position in the double-well potential.

Thus the EFG in the order-disorder case should be independent of temperature in the paraelectric phase. One finds that the paraelectric EFG tensor is the average of the two ferroelectric ones required by eq. (7).

One can find the electric field shift of the NQR frequency by considering the electric field as a

perturbation of the quadrupole Hamiltonian. We use the perturbation theory because the external electric field is very small compared with the internal electric field corresponding to the spontaneous polarization.

We consider the half-integral spin case since the majority of experiments has been carried out on such nuclei. For simplicity we study the zero asymmetry parameter case, the case of our experiment .

Let us write z+, the NQR frequency in the presence of the external electric field as

VE =vo+A+, (9)

where vo is the NQR frequency for E = 0 and Av, is the shift of the NQR frequency. Calculat- ing V~ from the quadrupole Hamiltonian in an external electric field by perturbation approximation we obtain for the order-disorder phase transition

3eQ VQ = 41(21- 1)h (2]m] - I)(q” + p Aq) 3 (10)

AvE = 4Z(23Jel)h (2lml - 1)

RI:3 +p ARi33 + Aq

where (11)

RR, = +$ , I

a Aq AR,,, = - dE, ’

(12)

(13)

and I is the nuclear spin and h is the Planck constant. In eq. (11) we take into account that the R-tensor is traceless [6]. In the case of zero asymmetry parameter we obtain

3eQ 'Q= 41(2Z_l)h (q”+PAdy

3eQ AvE = 41(2Z - 1)h

R;,, + p AR333+ Aqg E .

74 A. Gordon i Study of ferroelectric phase transitions

These results are also applicable for the displacive phase transitions, where there is a linear relationship between the spontaneous polarization and quadrupole constants.

For order-disorder phase transitions the mac- roscopic spontaneous polarization may be writ- ten as follows [14]:

P=Nt-q, (16)

where N is the number of ferroelectric dipoles in unit-volume and TV is the dipole moment of the ferroactive ion. The factor 2 appearing in [14] is caused by the use of Pauli matrices there. Then

dp F--l

dE - 4rrNp ’ (17)

E is the dielectric susceptibility. We suppose that the difference between the

polarizabilities in two equilibrium positions of the ferroactive ion is negligible. Then the second term in eq. (15) equals zero:

3eQ AvE = 41(21- 1)h i 2lml- 1)

x(R:,,+*yg E. i (18)

However, the second term in eq. (15) is not equal to zero for displacive phase transitions. It is seen from (15) and (17) that the electric field shift A+ must exhibit the critical singularity determined by the critical temperature dependence of the dielectric susceptibility. In displacive ferroelectrics the term containing an order parameter also contributes to the electrically induced shift of the NQR in an expression which is analogous to (15). Thus the temperature dependence of A+ does not necessarily exhibit the critical singularity of the e-type. It should be stressed that eq. (15) does not describe the displacive phase transition when the splitting of the NQR spectrum takes place in a ferroelectric phase.

In the paraelectric phase, both for the order- disorder ferroelectrics and for displacive ones the temperature dependence of the electric shift of

the NQR frequency may be also of the s-type. In fact, the expression for the A+ in the paraelectric phase is given by

3eQ Au, = 41(21- 1) (21m1- 1) R:,, (19)

In the general case the expression for R,,, may be presented as follows [15]:

(20)

where superscripts “ion” and “et” denote the contributions of the ionic and electronic polarizabilities to the electric field tensor component. According to [16] Rr3 includes the difference (E - n2), where rz2 is the square of the refractive index. If the contribution of the Rz3 is large compared with R;:, the temperature dependence of R,,, may be determined by the dielectric susceptibility. So far the contribution of RE3 was found to dominate only in some ionic crystals [16] and in a number of layer crystals [15]. However, in majority of crystals the electronic polarizability gives the principal contribution to R,,, In ferroelectrics such microscopic calculations have not been carried out. It should be only mentioned that in the ferroelectric KH,AsO, a considerable R,,,-component has been reported [17].

3. Comparison with experiment

We compare these results with the experimental data on the antimony NQR spectrum in an external electric field in pyrargyrite (Ag,SbS,) [7]. At room temperature pyrargyrite belongs to crystals of trigonal syngony (the space group R3c = Cg,) [18]. The covalently bound complex- es SbS, form trigonal pyramids with an antimony atom at the vertex. The pyramids are connected by ionic bonds S-Ag-S. The ionic bond to the silver is quite weak, as is confirmed by the presence of ionic conduction [19] accompanied by the emergence of silver at the electrodes applied to the samples. The crystal of Ag,SbS, is composed of two interpenetrating sublattices


which are apparently not connected by any bonds at all [18]. The Sb atoms at (0, 0,O) belong to one sublattice and the crystallographi- tally equivalent Sb atoms at (O,O, i) belong to the other.

The low-temperature structural phase transitions have been detected by As’~ NQR: in proustite at T = 26 K ([5] and references therein) and in pyrargyrite at T = 9.7 K [20]. It was shown that the phase transition in Ag,AsS, is a fer-

roelectric one [21]. In Fig. 1 the antimony NQR frequencies in

the single crystals of pyrargyrite are shown in the temperature range 4.2 K < T < 30 K [20]. The gradual change of the NQR spectrum accompanied by the doublet splitting shows that the phase transition is of second order. Within the framework of Landau theory of phase transitions [22] the temperature dependence of the order parameter in second order phase transitions is given by p - (T, - T)“2. Then the temperature dependence of the quadrupole doublet splitting should follow: Au - (T, - T)1’2. The experimental data of [20] give indeed this dependence A V( T). Consequently, we can use the mean field approximation in this case.

In ref. 20 it has also been shown that the phase transition seems to be an order-disorder one. There are three facts in favour of this as-

sumption: (i) The antimony EFG in the paraelectric

phase is independent of temperature [see

eq. WI.

u,MHz r I 91.5

t o-X

c oo-o~*~----Q~-- o-

90.7 &a@+++ I

Fig. 1. The three NQR frequencies v of the ‘?Sb in Ag,SbS, as function of temperature within the temperature range

4.2 K < T < 35 K [20].

(ii) The paraelectric EFG tensor is the average of the two ferroelectric ones [see eqs. (7) and (14)].

(iii) The temperature dependence of the NQR lineshape g(v) is the same as in the case of chemical exchange [23]:

gtvj = KU + 7/T2)P+ MR

P2+ R2 ’ (21)

where

M=21rr “1+%-v 3 1

(23)

R=2rr [ &,+v*)-V](l+++ (24) 2

where V, and V, are frequencies of the Sb NQR doublet splitting. In this case l/r is the rate of jumps of ferroactive ions between the two equilibrium positions, T, is the transverse relaxation time which does not depend on temperature according to [20], and K is a normalization constant. A comparison of the theory with the experiment makes it possible to find the temperature dependence of l/7. It has been found that l/r- T. An order of a magnitude of the rate: 1 /r - 10-h s-’ within the temperature range 4.2K< T<lOK.

The analysis of the microscopic mechanism of phase transitions in pyrargyrite and proustite made in [24] shows that they may be caused by redistribution of the silver ions among the possible equilibrium positions of the S-Ag-S bond. If we also take into account the weak ionic bond S-Ag-S one can conclude that the ions ordering at the phase transition in Ag,SbS, are silver ones. Thus the phase transition is of an order- disorder type connected with silver ordering in the double-minimum potential of the S-Ag-S ionic bond.

The low rate of silver jumps in Ag,SbS, is also confirmed by the calculation [20] of the Sb nuc-

76 A. Gordon I Study of ferro&ctric phase transitions

lear quadrupole spin-lattice relaxation near the phase transition in Ag,SbS,. From the spin- lattice relaxation analysis we also obtain that l/r-- 1P’sC’.

The slow jumps of silver ions and deviation of (1 /T(T) from an exponential dependence seem to show that the silver ions move between the two equilibrium positions not by hopping over the hill of the double-minimum potential but by phonon-assisted tunnelling through the potential barrier. In fact, at low temperatures direct one- phonon process may dominate in phonon-assisted tunnelling leading to temperature dependence of the rate: 1 /T - T [25]. The phonon- assisted tunnelling in ferroelectrics was considered in [26]. Following [26] we write

1 lr - ,u?-~K, T (25)

Tunnelling, which may have a negligible effect as far as the static properties of ferroelectrics are concerned, plays an important part in their dynamic properties [27].

An additional evidence on the order-disorder type of the low-temperature phase transition in Ag,SbS, we obtain from the electric field effect in NQR.

In fig. 2 we show the temperature dependence of the electric shift of the “’ Sb NQR spectrum AZ+ in the single crystals of Ag,SbS, for the spin-resonance transition * $ f-$ ? 2 (71. The

AQHZ’

1250 -

1000-s_d

750 - 0 =

500 -

250 ->, I I I 1

5 7 9 II 13 15 T,OK

Fig. 2. The temperature dependence of electrically induced shift of the IL’ Sb NQR frequency A+ in external electric field

E = 2.5 kV/cm for the spin-resonance transition 2 $ ++ k f in Ag,SbS, in the temperature range 4.2 K < T < 17 K [7].

electric field effect in the “‘Sb NQR is detected as a modulation (slow beats) of the “‘Sb spin- echo signals envelope. Upon application of the external electric field between two microwave pulses or between the second pulse and spin- echo signal slow beats of the envelope of spin- echo signals occur [28]:

j = j,, COS~~~TAV~T~,/ , (2fi)

where 7(, is the time between radio frequency pulses and j is the amplitude of the envelope of the spin-echo signals. It follows that

(27)

where n is the number of a null of the cosine in (26). Eq.(27) IS used to obtain the electric field shift in the NQR frequency A+. The data in fig. 2 are given for E = 2.5 kVlcm. We see that in the low-temperature phase the electric field shifts of the two “‘Sb lines of the quadrupole

doublet show a strong temperature dependence, while there is no temperature dependence in the high-temperature phase.

To explain the above experimental results we must take into account the peculiarities of the experimental method used [15,29]. The phenomenon of slow beats in the spin-echo envelope may be observed in crystals subjected to electric field pulses only in cases where NQR frequencies split as a result of an external electric field application [15,29, 301. Firstly, this may occur in centrosymmetric crystals, in which resonant nuclei are at inversion-coupled positions before the application of the field [28,31] and where the local symmetry of the resonant nucleus site is noncentrosymmetric. Secondly, it may take place in noncentrosymmetric crystals when twins are formed during single-crystal growth [15, 29,301. In ferroelectrics and antiferroelec- tries slow beats may be displayed for the same reason, i.e., due to lifting of the spatial degeneracy by an external electric field because of the presence of two domains [29] or because of the presence of two sublattices [30]. Thirdly, slow beats may result from lifting of the spin degener-

A. Gordon I Study of ferroelectric phase transitions

acy by an external electric field for the case of nuclei with integral spins.

As it has been shown [20], the asymmetry parameter of the EFG tensor at a nuclear site in Ag,SbS, equals zero. Consequently, eq. (18) may be applied for pyrargyrite. We can rewrite (18) as follows:

A+ =

(

E-l x RI:,,+&- 4lrN/.L 1

E. (28)

Taking into account the electronic polarization and replacing l-+ E,+ n2 [32], we obtain

3eQ AvE = 41(21- 1)h (2lml- 1)

F - n2 R:,, + Aq ~ 47rN/.~ 1 E, (29)

We consider the resonant transition + 2 cz + 2 for which the data in fig. 2 are presented. Then we obtain for T < T,

2 2

tiv,=0.3~AqE-nE, 47TpN (30)

wheresv, is the electric field shift of the nuclear quadrupole splitting below the phase transition temperature. Using the temperature dependence of the dielectric susceptibility measured along the direction of the z-axis of the EFG tensor at T < T, [33] and taking the Sb NQR spectrum data [20], we calculate &v,(T) according to [30]. We put n = 3 [34]. The results are presented in fig. 3. The solid line is the calculation and the points are the experimental data. We see good agreement between the theory and experiment. Consequently, the order-disorder character of the phase transition is confirmed. From a comparison of the calculation with the experiment we find the value of the electric dipole moment CL: p = 0.35 Debye. This value is close to that of the electric dipole moment p = 0.33 Debye of the O-D- -0 bond as obtained in the deuterated Rochelle salt [35] which is the characteristic order-disorder ferroelectric [ 141.

600 -

500- /

I I I I 1 5 6 7 T,‘K

Fig. 3. The temperature dependence of the electric field shift

of the NQR doublet splitting 6v, in external electric field

E = 2.5 kV/cm for the spin-resonance transition + $ - + 2

in Ag,SbS, for T < TC; the experimental data are shown by

points and the theoretical results are demonstrated by the

solid line.

As it is stressed in the preceding paragraph, the temperature dependence of the electrically induced shift of the Sb NQR frequency may also appear in the high-temperature phase of the Ag,SbS,. However, this is not observed. This may be explained as follows. Within the temperature range 11 K < T < 17 K dielectric susceptibility changes by 2-3% [33], whereas the accura- cy of the measurement of the electric field shifts of the antimony NQR frequency is only 5%. Consequently, within this temperature range this method is not sufficiently sensitive to be able to detect any temperature dependence of the electric field shift of the 121Sb NQR frequency. Near the phase transition temperature (9.7 K < T < 11 K) we cannot observe the electric field effect at all because of the strong broadening of the NQR lines and sharp decrease of their intensity.

The critical singularity in the dielectric susceptibility as a function of temperature at the phase transition and the close agreement between the ferroelectric model proposed indicate that the low-temperature structural phase transition in pyrargyrite is a ferroelectric one.

The appearance of slow beats in the paraelectric phase of Ag,SbS, may be due to either one of two reasons: twins or sublattices which are characteristic of the pyrargyrite structure. The experiment [30] shows that the slow beats in the

78 A. Gordon I Study of ferroelectric phase transitions

envelope of signals in the antimony spin-echo are due to the twins. In the ferroelectric phase the presence of slow beats seems to be connected with two ferroelectric domains. In uniaxial Ag,SbS, two MY’-domains must exist. The Sb resonant nucleus does not distinguish between the two domains. In each domain the quadrupole doublet is given by the dynamic order-disorder

model [eqs. (5)-(S)]. It should be noted that eqs. (5), (6) do not describe the quadrupole splitting in the static order-disorder model. To explain the splitting of the latter type one must use for the EFG the two following equations [29,30]:

qi= 40 + Ap,

qz = qo - Ap

(31)

(32)

Good agreement is obtained between the calculations and experimental data on the electric field shift of the “‘Sb NQR in the ferroelectric

phase of Ag,SbS,. It is found that the electric dipole moment of S-Ag-S equals: I_L = 0.35 Debye, close to the value of the electric dipole moment of the O-D- -0 bond in the order- disorder ferroelectric deuterated Rochelle salt. Thus the order-disorder nature of the phase transition is proved. An absence of a temperature dependence of the electrically induced shift of the Sb NQR frequency in the paraelectric phase of Ag,SbS, is explained. Reasons of slow beats in the antimony spin-echo signals envelope are discussed. It is shown that the low-temperature structural phase transition is a ferroelectric one. Thus the electric field effect in NQR is a useful method for the study of the ferroelectric ordering in crystals.

In the present case the external electric field lifts the spatial degeneracy related to the two ferroelectric domains. This leads to slow beats in the envelope of signals in the antimony spin echo.

Acknowledgements

The strong electret effect appearing as a result The author is indebted to Drs. D. Baisa and of the external electric field influence on fer- A. Bondar for the fruitful collaboration in the roelectric proustite samples prevents the observ- ation of the electric field shift of 75As NQR

works [7] and [20]. It is a pleasure to thank Professor J.Genossar for his interest in this work

frequency. and a critical reading of the manuscript.

4. Summary References

In this paper we derive expressions for NQR frequencies in presence of an external electric field for order-disorder and displacive ferroelec- tries (in which there is a linear relationship between the spontaneous polarization and quadrupole constants). Perturbation theory is em- ployed to derive expressions for a case of nuclei with half-integral spins.

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study of ferroelectric phase transitions by electric field perturbation of nuclear quadrupole...

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