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THE CONDUCTIVITY OF LiD AND LiH
P. Varotsos
To cite this version:
JOURNAL D E PHYSIQUE Colloque C7, suppltment au no 12, Tome 37, Dtcembre 1976, page C7-327
THE CONDUCTIVITY OF
LiD
AND LiH
P. VAROTSOS
Department of Physics, University of Athens, Solonos Str. 104, Athens, 144 Greece
RBsumB. - La conductivite de LiD polycristalline et de LiH monocrystalline a 6t6 mesurke a la temperature de fusion. Leurs courbes de conductivite In 6T = f ( l / T ) ont 6t6 analyskes en accord avec la th6orie habituelle et de cette manikre les paramktres correspondant a la formation et la migration des lacunes ont kt6 determinees. Un comportement non usuel a CtB observe prks de leurs temperatures de fusion.
La conductivite de LiD, irradiee par des rayons X, a Bte Ctudi6e dans la region extrinskque. On a trouve qu'apr6s irradiation la conductivite en courant continu a BtB rkduite, tandis que le nombre des lacunes likes (obtenu par la methode de relaxation dielectrique) n'a pas 6tC change.
Abstract. - The conductivity of polycrystalline LiD and LiH single crystals has been studied up to their melting points. Their conductivity curves In (6 T) = f ( l / T ) were analysed according to the usual theory and thus the corresponding parameters for the formation and the migration of vacancies were determined. An unusual behaviour near their melting point was observed.
Furthermore the conductivity of X-ray irradiated LiD and its extrinsic region has been studied. It was found that after X-ray irradiation the d. c. conductivity was lowered whereas the number of contained bound vacancies (obtained by dielectric loss method) was unchanged.
1 . Introduction.
-
The study of the conductivity 1.1 T>
0,.-
(i) All experimental workers ana- and diffusion of alkali halides is based on the fact that lyse their results by putting arbitrarily [3]. V = VD. defect jump rates are well represented by an Arrhenius Then a combination of (1) and (2) gives :formula
where Vis a frequency factor and Ag, is the free energy barrier between the neighbouring sites available to the defect ; Ag, is usually analysed into enthalpy Ah, and
entropy As, terms as
Ag, = Ah, - T As,,
.
(2) The Arrhenius formula is obtained by using a gene- ralised rate theory based on classical statistical mecha- nics and thus is expected to hold only at high tempe- ratures [I] i. e. for T p 8,. Most alkali halides havetheir Debye temperatures in the region 0, = 150
-
300 K which is lower than the temperatures at which the usual conductivity and diffusion experiments are made ; for them the validity of the Arrhenius formula is rather assured.In the last years a large strong interest has been expressed for investigation in the region T
<
8,. Inthis respect LiH and LiD are suitable materials because of their high Debye temperatures (851 K and 638 K respectively) [2]. The present paper describes measure- ments of the conductivity of these materials extending over a region between 0.5 8, - 1.1 8,. In order to
discuss the results attention has to be drawn to some points concerning the use of the Arrhenius relation for temperatures both above and lower 8, :
We note that there is no strict justification for putting V = VD especially if the lattice contains points with two different masses. Thus we see that if the frequency factor was incorrect (which has been anticipated [4]) a large error will result in the As, value. An error of a factor of 2 in V would change As, value by an amount about 70
%.
It is therefore possible that the published values of As,, which have been determined by the arbitrary assumption V = V,, do not represent the real variation in the phonon spectra during the migrational process.(ii) Even if the case that the assumption V = VD
is correct, we must recall that VD depends on tempera-
ture ; for example in the case of NaCl VD decreases
(with the temperature) by an amount of 20
%
between R. T. and the melting point. This correction, however, will be less important than the previous one.(iii) In the analysis of experimental results a third arbitrarity is made : both Ah, and As, are, a priori,
supposed to be temperature independent. The only justification towards this arbitrarity is that Ag, decreases - approximately - linearly with the tempe- rature. There is however another possibility for them ; at high temperatures anharmonic effects appear and thus both Ah, and AS, may depend on temperature.
C7-328 P. VAROTSOS
Even in the simple case of a linear variation of Ah,
with temperature i. e. Ah, = Ah,
-
aT we easily obtain from eq. (3) that the published values AS,,,,.do not represent the true entropy As,. In the case of
alkali halides with NaC1 structure we have indicated [5] that the values Asexpe,. represent approximatly the
anharmonic variation of the Ag with the tempera-
ture.
1.2 T
<
8,. - In this temperature it seems pro- bable that the pre-exponential frequency V must be replaced by a temperature dependent frequency V(T).Unfortunately the function V(T) is still unknown although many attempts have been made in the last years towards its calculation.
It is however expected [6] that quantum effects must lead to a curvature of the Arrhenuis plot leading to lower frequencies. The encouraging fact is that in this region ( T
<
0,) anharmonicity is of less importance. As we mentioned [5] the value As,,,,,. is almost theanharmonic variation of the Ag, vsT; thus in the
region T
<
8, an extremely small value ofas,,,,,^
is expected.
2. Experiment. - Curve a of fig.ure 1 shows the conductivity curve In (6 T) = f (l/T) for a LiH simple crystal containing 30 ppm Cat' (small amounts A1 and Cu have also been detected). The results were reproducible to 1
%
if heated for 2 hours at a tempera- ture-
350 OC. The extrinsic conductivity part shows higher values ( N by a factor of 5) than those reported by Pretzel [2] for a LiHpure monocrystal.The present values also in the extrinsic region are smaller than those reported [7] for a LiH crystal doped with 100 ppm Mgt2. Taking into account that the pure LIH of Pretzel contained 0.5-6 ppm Mgf we see that the conductivity of LiH in its extrinsic region increases with the concentration of divalent cations which proves that the conductivity takes place mainly through the migration of cation vacancies [6].
In the initial part of the intrinsic region (which has been inserted in figure I), we see that the conductivity curve shows a good linearity, whereas at temperatures nearing the melting point, we observed a strong curva- ture towards higher values.
By using the following parameters a = 2.042
A,
n = 30ppm and V = V, = 1.77 x loi3 s-I (8, = 851 K) the application of the usual proce- dure [3, 81 used for alkali halides leads to the following results :We notice here that in fitting of our results we have used only the T-region inserted in figure 1 ; as for the values nearing to the melting point we believe that they
FIG. 1. - Curve a : The conductivity of a LiH crystal doped with 30 ppm Cafz. Curve b : The conductivity of pure poly- crystalline LiD. I n both crystals the conductivity curves nearing to their melting point showed a strong curvature toward higher conductivities ; these last results have not been included in
the figure.
could be described by taking into account anharmonic effects (see Appendix).
Curve b of figure 1 represents the conductivity of a pure polycrystalline LiD which practically contains the same amount of divalent cations as LiHpure crystal of Pretzel. We see that in both regions, extrinsic and intrin- sic, LiD shows smaller conductivity than LiH.
By using for LiD the values a = 2.03
A,
V, = 1.33 x loi3 s-I (8, = 638 K), n-
5 ppm we obtain :Ah: = 2.54 eV As;
-
14.68 k.
THE CONDUCTIVITY OF LiD AND LiH C7-329
vacancy to the imaginary part of the dielectric constant
remains unchanged. This behaviour is different than that appeared in alkali halides ; in them, X-irradiation not oilly decreases the d. c. conductivity but also
disappears the dipolar contribution.
3. Discussion. - 3.1. - At first glance the present measurements would mean that in the region of T < OD, As
,,,,,,,,
for both crystals is practically zero. This would mean that the total change in the vibratio- nal spectra during the migration process is extremely small. However there is an alternative to this argument considering facts already mentioned in the intro- duction :(i) The arbitrary selection of V = V, might not be suitable.
(ii) Even if the selection V = VD is a good appro- ximation for T
>
, %, the materials of the present paper have high 8, values and hence VD should be replaced by a function V ( T ) which has possibly lower values thanVD ; in this case one could obtain a value of AS: higher than zero.
The idea expressed above i. e. that in the region T
<
OD,
V decreases with falling temperature is strengthened by comparing dielectric losses with ITC experiments made on the same LiD+
M~~~ crystal. The low temperature side of the observed ITC peak when analysed by the usual method of Bucci (which is based on the Arrhenius formula z = z0 exp(E/kT)) leads to a pre-exponential frequency-factor--
lo6 Hz which is by a factor of 10' smaller than that resulted from dielectric loss technique. Furthermore another curious fact has to be noticed ; the function om us 1/T, where W, is the peak frequency for each temperature inthe dielectric loss experiment, is linear [I I] leading to a pre-exponential frequency-factor smaller than V, by a factor 10'.
However the ITC curve i(T) (temperature region
<
8,/2) cannot be described by the usual equa- tion proposed by Bucci. This last possibly shows that in this low temperature region an equation of the form z(T) = zo(T) exp(E/kT) must be more appropriate. Although the above discussion concerns a bound- vacancy motion, similar effects in the case of a free cation motion might exist.3.2. - The second unexpected effect is the large value of the formation entropy
-
15 k (in the case of alkali halides Ass-
10 k). At first glance this large value ofAss shows that the formation of a Schottky defect
creates a large variation of frequencies of the ions neighbouring to the vacancies. However there is an alternative of this fact. From the analysis of the intrin-
As;
sic part of the conductivity curve the sum
-
+
AS:2
where T
>
8,) is not unexpected. In the last case the value of Ass is reduced to Ass--
11-
13 k.Acbnowledgments. - The author would like to express his indebteness to Prof. K. Alexopoulos for very usefull discussions. The crystal used in the present paper were furnished us by (1) the late F. Pretzel (polycrystalline 'LiD) and (2), Lawrence Radiation Laboratory, University of California Livermore, Calif. 94550, USA (monocrystal LiH).
Appendix
I. ANHARMONIC VARIATIONS OF THE FORMATION
ENTHALPY AND ENTROPY IN A SOLID. - Let's suppose that the creation of a vacancy creates a uniform local deformation ; then the corresponding free energy of formation can be expressed as [12]
where B is the bulk modulus, c a dilatation factor which is taken as temperature independent and V the volume that undergoes a strain. At T = 0, eq. (A. 1) gives :
ho = cB0 Vo (A. 2)
by differentiating eq. (A.2) in respect to T and by using the value of c obtained from (A.2) we finally
conclude :
where
p
= volume expansion coef. If we suppose that only the nearest neighbours of a vacancy change their frequency from WE to (Einstein model) we must addW E
to (A. 1) a term nkT In
7
; then in the entropy W Eexpressed by eq. (A.3) or (A.4) we must add a term W E
nk In
7.
This term has been estimated by Beniere to be W E4.16 K in this case of a Schottky defect in NaCl- structure.
A
combination of eq. (A. I), (A. 2) and (A. 3) gives :. ,
is obtained ; although from the extrinsic region
( T < 8,) a value As,
--
0 is obtained, however a value Eq. (A. 5) and (A .4) clearly show that both h and sP. VAROTSOS
References
[I] LIDIARD, A. B., J. Physique Colloq. 34 (1973) C 9-1, C 9-5, C 9-6.
[2] PRETZEL, F. E., RUPERT, G. N., MADER, C . L., STORMS, E. K., GRITTON, G. V. and RUSHING, C . C., J. Phys. & Chem. Solids 16 (1960) 10.
[3] BENIBRE, M., CHEMLA, M. and BENIBRE, F., J. Physique 37 (1976) 525.
[4] FRANKLIN, W. M., Diffusion in Solids, Recent developments, (Academic Press) 1975 p. 53.
[5] VAROTSOS, P. and ALEXOPOULOS, K., Phys. Rev. B (1976) to be published, janv. 1977.
[61 See reference [4] pp. 69, 60.
[7] VAROTSOS, P. and MouRIKrs, S., Phys. Rev. B 10 (1974) 5220.
[8] In this analysis of experimental re! ults we neglect the contri- bution from anion-vacancies motion.
[9] JAIN, S. C., SAI, K. S. K. and LAL, K., J. Phys. C 4 (1971) 1958.
[lo] VAROTSOS, P., Associate Professor's thesis ; University of Athens 1976.
[Ill VAROTSOS, P. and KATSAROS, W., Phys. Status Solidi (a) 25 (1974) 457.
[12] The Appendix has been derived by co-working with prof. K. Alexopoulos. The symbols h, s represent
the corresponding parameters for the formation of a vacancy.
DISCUSSION R.
1.
FRIAUF. - It is a matter of great importanceto determine whether cations or anions make the greater contribution to the ionic conductivity. How good is the experimental evidence on which you claim that cations are predominant ?
P. VAROTSOS. - We observed that the conductivity in its extrinsic region increases with the concentration of divalent cations. This possibly shows that the conduc- tivity takes place mainly through the migration of cations.
2) Experimental results are available only for diva- lent cation doping. The present experiment shows an increase of the extrinsic conductivity with the concentration of divalent cations ; this fact, in the case of a Schottky disorder, shows that the cation vacancies are more mobile.
M. J. NORGETT. - It is very interesting to see measurements on systems carried out well below the Debye temperature. It is worth emphazising that in this case you do not expect Arrhenius behaviour. This is not because any temperature dependence in the defect A. L. LASKAR. - 1) It is difficult to decide whether
formation and activation energies (this may be a real the conductivity Arrhenius plot shows a continuous
but smaller effect) but because of the fact that the curvature in the intrinsic range (will lead to tempera-
phonon distribution cannot be averaged in the classi- ture activation enthal~ies) Or a break cal way applicable at high temperature. The theory
(will mean two mechanisms in the intrinsic range. Are
applicable at low temperatures is the hopping theory there supporting evidence from diffusion etc. to con-
used for polarons and the application I know best is clude that activation enthalpy of migration is tempera-
to the V, centre in fluorites (NORGETT M. .I., STONE- ture dependent ?
HAM M., J. Phys. C. 6 (1973), 238. The effective activa-
2) Are there studies of conductivity of LiH and LiD tion energy deduced at low temperature decreases with cation or anion doping to decide whethe ther very substantially below the classical high temperature defect structure is Frenkel or Schottky type and to value and this is just the behaviour observed by Varot- decide whether cations or anions are more mobile ? sos.
