Frequency and temperature dependence of the electrical conductivity of KTaO3; Li and PbTiO3; La, Cu: Indication of a low temperature polaron mechanism

ARTICLE IN PRESS Physica B 403 (2008) 3608– 3611

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Frequency and temperature dependence of the electrical conductivity of KTaO3; Li and PbTiO3; La, Cu: Indication of a low temperature polaron mechanism A. Levstik a,, C. Filipicˇ a, O. Bidault b, M. Maglione c a

Jozˇef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia Laboratoire de Physique de l’Universite de Bourgogne, UMR CNRS 5027, Faculte des Sciences Mirande, 9 Avenue Alain Savary, B.P. 47 870, F-21078 Dijon Cedex, France c ICMCB-CNRS, Universite de Bordeaux I, 87 Avenue du Dr. A. Schweitzer, 33608 Pessac, France b

a r t i c l e in f o

a b s t r a c t

Article history: Received 18 April 2008 Received in revised form 30 May 2008 Accepted 2 June 2008

Recently, the concept of polarons has again been at the focus of solid-state research, as it can constitute the basis for understanding the high-temperature superconductivity or the colossal magnetoresistance of materials. More than a decade ago there were some indications that polarons play an important role in explaining low temperature maxima in imaginary part of the dielectric constant 00 ðTÞ in ABO3 perovskites. In the present work we report the ac electrical conductivities of KTaO3; Li and PbTiO3; La, Cu and their frequency and temperature dependence. The real part of the complex ac conductivity was found to follow the universal dielectric response s0 / ns . A detailed theoretical analysis of the temperature dependence of the parameter s revealed that, at low temperatures, the tunnelling of small polarons is the dominating charge transport mechanism in ABO3 perovskites. & 2008 Elsevier B.V. All rights reserved.

PACS: 72.20.Fr 73.40.Gk 71.38.k 77.84.Dy Keywords: Low-field transport Tunnelling Polarons Titanates Tantalates

1. Introduction More than a decade ago dielectric properties at high and low temperatures in ABO3 perovskites (more than 100 samples) were systematically studied [1,2]. In many pure and doped samples the low temperature (below 50 K) dielectric loss anomaly was observed. Dielectric spectra were described by the Arrhenius law with activation energies around 80 meV and attempt relaxation time t0 in the range of 1013 21014 s. It was proposed that polaronic excitations are responsible for such behaviour of dipoles. KTaO3, KTaO3; Nb, and KTaO3; Li, Fe have various properties ranging from quantum paraelectric to orientational glass and ferroelectric. Further, such low temperature dielectric losses with similar activation energies were observed in doped ceramics BaTiO3 and PbTiO3 [1,3]. The manifestation of Jahn-Teller polarons [4] was shown in BaTiO3 crystals with EPR. Recently [5],

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E-mail address: [email protected] (A. Levstik). 0921-4526/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2008.06.005

on the basis of measurements of magnetic susceptibility and electrical conductivity in BaTi1x Nbx O3 , it was shown that immobile spin-singlet small bi-polarons are formed in this system. With one exception [3], the concept of polarons connected with the existence of low-temperature peaks in 00 ðTÞ has not been supported by theoretical analysis. In the present work we analyze the frequency and temperature dependence of ac electrical conductivity in KTaO3; 8% Li and PbTiO3; 20% La, 8% Cu (abbreviated as PLTC(20,8)), for the first time using the theoretical model for small polarons, and determine the energy barriers for polarons, reduced polaron radius, attempt relaxation times and the density of polarons. Very probably it will be possible to analyze the ac conductivity in many other ABO3 perovskites with anomalies in 00 ðTÞ at low temperatures, using the same method, thus supporting the polaronic concept proposed approximately 15 years ago. Besides infrared excitation in optical conductivity [6,7], the fingerprint of polarons is usually connected with the temperature and frequency dependence of ac electrical conductivity [8–10].

ARTICLE IN PRESS A. Levstik et al. / Physica B 403 (2008) 3608–3611

For many amorphous and crystal systems it has been shown that, besides the dc contribution of conductivity, sdc , the frequency dependent real part of complex ac conductivity, s0 , follows the so-called universal dielectric response (UDR) [11], thus

T (K) 50

20

10-5

1 MHz 10-7

The real part of the complex ac conductivity s0 of KTaO3; 8% Li at various frequencies is shown as a function of inverse temperature (Fig. 1a). We observe s0 ðTÞ peaks at various temperatures below approximately 30 K, which move with increase in frequency to higher temperatures. Below 20 K, s0 is nearly temperature independent. Strongly frequency-dependent plateaus in the s0 ðTÞ data below 20 K suggest that hopping or tunnelling of localized charge carriers governs the electrical transport at lower temperatures [17]. The relaxation frequency determined from the peaks in s0 ðTÞ vs. reciprocal temperature follows the Arrhenius law f ¼ f 0 expðDU=TÞ (Fig. 2). The activation energy has the value DU ¼ 76:1  0:4 meV and f 0 the value ð2:6  0:3Þ  1011 s1 (t0 ¼ 1=ð2pf 0 Þ ¼ 6:0  1013 s). Such behaviour is the result of the simple model [18,19]. The frequency dependence of s0 at different temperatures is presented in Fig. 3a. It is clearly evident that, at higher frequencies, the conductivity follows the UDR behaviour, s0 / ns , while at lower frequencies the data shift away from this dependence due to the dc conductivity contribution. Thus, in one decade, the frequency range s0 could be described by Eq. (1). The UDR parameter s, determined by linear fits as presented in the Fig. 3a, is shown in Fig. 4a as a function of temperature. It exhibits non-monotonous temperature behaviour, with a minimum at  25 K. Such behaviour is predicted by the tunnelling

30 kHz 10 kHz 3 kHz 1 kHz 300 Hz 100 Hz 30 Hz

0.3 Hz

10-13

0.02

0.04

0.06 1/T (K-1)

0.08

0.10

T (K) 10

300 50 10-5

5

PLTC (20, 8) 1 MHz

σ′ (Ω-1 cm-1)

300 kHz

10-7

100 kHz 30 kHz 10 kHz 3 kHz 1 kHz 300 Hz 100 Hz

10-9

30 Hz 10 Hz

10-11

3 Hz 1 Hz

10-13 0.00

0.05

0.10

0.15

0.20

0.25

1/T (K-1) Fig. 1. The real part of the complex ac conductivity s0 . (a) In KTaO3; 8% Li. Measured along the [1 0 0] crystallographic axis at several frequencies versus inverse temperature. (b) In PLTC ð20; 8Þ. Measured at seven frequencies, as a function of inverse temperature.

55

50

45

T (K) 40

35

10 104 KTaO3; 8% Li

8

103 6 ln [f (Hz)]

3. Results and discussion

100 kHz

10-9

10-11

2. Experiment KTaO3; 8% Li mono-crystals were grown by spontaneous crystallization. The starting components were K2CO3 and Ta2O5 of high purity. PbTiO3; La, Cu samples are labelled PLTCðY; XÞ where Y denotes the amount (in %) of lanthanum ðY ¼ 100yÞ and the second ðX ¼ 100xÞ the amount (in %) of copper. PLTC ð20; 8Þ ceramic samples were prepared from very pure PbO, TiO3, CuO, and La2O3 as starting materials with y=2 excess of PbO. Materials were sintered in an alumina crucible in a PbO rich atmosphere so that loss of Pb during sintering was minimized. Details of the sintering procedure have been described [16]. The colour of PLTC ð20; 8Þ is brown. The composition was determined by chemical analysis and the homogeneity by X-ray powder analysis. The complex dielectric constant  ðn; TÞ ¼ 0  00 was measured between 4.5 and 300 K in the frequency range of 0.3 Hz–1 MHz, using a Novocontrol Alpha High Resolution Dielectric Analyzer. The amplitude of the probing ac electric signal was 1 V/mm. The temperature was stabilized within 0:1 K using an Oxford Instruments continuous flow cryostat. The real part of the complex ac conductivity s ¼ s0 þ s00 was calculated via s0 ¼ 2pn0 00 , with 0 being the permittivity of the vacuum.

300 kHz

102

4

101

2 0

f (Hz)

Here, A is a temperature dependent constant, n is the measuring frequency, and s the frequency exponent, sp1. Such behaviour has been deduced by several theoretical approaches from the microscopic transport properties, including hopping or tunnelling of the charge carriers through the energy barrier separating different localized states [12–15]. Different mechanisms, however, lead to different frequency dependence of the ac conductivity, i.e., to different temperature dependence of the UDR parameter s.

10

KTaO3; 8%Li

(1)

σ′ (Ω-1 cm-1)

s0 ¼ sdc þ Ans .

3609

f = f0 exp(-ΔU/T) ΔU = 0.076 eV τ0 = 1/2πf0 = 6.0×10-13 s

100

-2

10-1 0.020

0.024

0.028

1/T (K-1) Fig. 2. Relaxation frequency determined from the peaks in s0 in Fig. 1a versus reciprocal temperature in KTaO3; 8% Li.

ARTICLE IN PRESS 3610

A. Levstik et al. / Physica B 403 (2008) 3608–3611

1.0

KTaO3; 8%Li

10-9

KTaO3; 8%Li

σ′ (Ω-1 cm-1)

0.9 10-10

0.8 S

0.7

10-11 30 K

10-12 100

25 K

15 K

23 K

10 K

20 K

5K

0.6 0.5

1000

0.4

ν (Hz)

PLTC (20, 8)

10-9

τ0 = 10-13 s W∞ = 0.043 eV r0' = 1.1

0

5

10

15 T (K)

20

25

30

1.00

σ′ (Ω-1 cm-1)

PLTC (20, 8) 0.95 10-10 s = 0.81

55 K 41 K

10-11

S

0.90

30 K

0.99

0.85

16 K

τ0 = 5 × 10-14 s W∞ = 0.146 eV r0' = 5.0

4.4 K 10-12

0.80 1

10

100 ν (Hz)

1000 0.75 0

Fig. 3. Frequency dependence of the real part of the complex ac conductivity s . Solid lines denote the universal dielectric response. (a) In KTaO3; 8% Li at seven temperatures. (b) In PLTC ð20; 8Þ at five temperatures.

10

20

30

0

polaron model, which yields the temperature dependence of s as [12] 02

s¼1

4 þ 6W 1 r 00 =ðkTR Þ 02

R0 ½1 þ W 1 r 00 =ðkTR Þ2

.

(2)

Here, W 1 denotes the energy barrier and r 00 the reduced polaron radius while R0 is the reduced tunneling distance, also being a function of W 1 and r 00 and, additionally, of the inverse attempt frequency t0, as [12] 2R0 ¼ ln

1

ot0



W1 þ kT

!1=2   1 W 1 2 4r 00 W 1 ln  þ , ot0 kT kT

(3)

where o ¼ 2pn. We are aware that the model predicts a frequency dependence of s and therefore Eq. (1) should no longer be valid in a strong sense, i.e. with s ¼ constant. However, in the present data a pure ns contribution is seen only over a relatively small frequency range and it can be assumed that the rather weak lnðnÞ dependence of R0 leads to only small deviations from Eq. (1). The fitting of sðTÞ data to Eq. (2) (denoted by the solid line in Fig. 4a) gives values of W 1 ¼ 0:043  0:001 eV, r 00 ¼ 1:1  0:03, and t0 ¼ 1013 s. From the s0 ðn; TÞ and sðTÞ data we conclude that, below 30 K, tunnelling of polarons governs the charge transport in KTaO3; 8% Li.

40 50 T (K)

60

70

80

Fig. 4. Temperature dependence of the UDR parameter s. The solid line represents the fit of the experimental data to a model for polaron tunnelling (Eq. (2)). (a) In KTaO3; 8% Li. (b) In PLTC ð20; 8Þ.

The method [12], which is used for the determination of polarons in ABO3 perovskites, is widely used for the analysis of the ac conductivity in high-T c superconductors [20], glasses [21], semiconductors [22], and other systems [9,23,24]. The ac conductivity in tunnelling polaron model used above is given by

s0 ðnÞ ¼

p5 6

e2 ðkTÞ2 N2f

nR4 2akT þ W 1 r 0 =R2

.

(4)

Here, N f is the density of states at the Fermi level and a the decay parameter for the wave function of a carrier localized at a given site. The reduced polaron radius and tunnelling distance in Eqs. (2) and (3) are defined via R0 ¼ 2aR and r 00 ¼ 2ar 0 [12]. By introducing these reduced quantities it can be seen that only the ratio N2f =a5 can be determined from Eq. (4), which prevents determination of the number of centres N ¼ kTN f . However, by using the value of a ¼ 108 cm1 [21] and values of W 1 , r 00 , and t0 determined from our sðTÞ fit, Eq. (4) yields Nf ¼ 7  1020 eV1 cm3 , when applying the value of s0 ¼ 1:27  1011 O1 cm1 determined experimentally at T ¼ 10 K and n ¼ 1 kHz. This value of Nf further justifies the application of the polaron tunnelling model to the ac conductivity data in KTaO3; 8%

ARTICLE IN PRESS A. Levstik et al. / Physica B 403 (2008) 3608–3611

Li. The possible existence of bipolarons in KTaO3; 1.1% Li was supposed from the analysis of the characteristic frequency [25] as a function of temperature. To check whether the polaron concept is valid for other ABO3 perovskites, as proposed earlier [1–3], the ac conductivity s0 of PLTC (20,8) was determined at different frequencies and plotted versus inverse temperature in Fig. 1b. It is evident that, below approximately 20 K, the ac conductivity is almost independent of temperature, while at higher temperatures it becomes temperature dependent. As in KTaO3; 8% Li, the strongly frequency dependent plateaus in s0 ðTÞ below 20 K suggest that hopping or tunnelling of localized charge carriers governs the electrical transport at lower temperatures. The frequency dependence of s0 at five different temperatures is presented in Fig. 3b. It can be seen that, between 1 and 100 Hz, the conductivity follows the UDR behaviour s0 / ns. The UDR parameter s, determined by linear fit (Fig. 3b), is shown in Fig. 4b as a function of temperature. It exhibits temperature dependence with a minimum at approximately 55 K. Such behaviour can be described with the tunnelling polaron model which gives the temperature dependence determined by Eqs. (2) and (3). The solid line in Fig. 4b represents the fit to experimental data with values of W 1 ¼ 0:146  0:016 eV, r 00 ¼ 5:0  1:2, and t0 ¼ 5  1014 s. From s0 ðn; TÞ and sðTÞ data we conclude that, below 50 K, the tunnelling of polarons governs the charge transport in PLTC ð20; 8Þ. As the activation energies involved [1] (0.043 eV for KTaO3; 8% Li and 0.146 eV for PLTC ð20; 8Þ compared to 0.2–1 eV for ion jumping) are much smaller, the possibility that such a dielectric relaxation is due to ion jumping is ruled out. As for KTaO3; 8% Li, the conductivity in PLTC ð20; 8Þ can be described by Eq. (4), from which the density of polaron states at the Fermi level can be determined. This leads to a value for PLTC ð20; 8Þ of Nf ¼ 8  1020 eV1 cm3 , using the value of s0 ¼ 8:7  1011 O1 cm1 determined experimentally at T ¼ 14 K and n ¼ 100 Hz. This value of Nf further justifies the application of the polaron tunnelling model to data for ac conductivity in PLTC ð20; 8Þ.

4. Conclusions In summary, the ac electrical conductivity of KTaO3; 8% Li and PLTC ð20; 8Þ has been investigated as a function of temperature and frequency. The temperature dependence of the ac conductivity indicates the existence of a hopping type mechanism at lower temperatures. The real part of the complex ac electric conductivity

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was found to follow the universal dielectric response s0 / ns , which is typical for hopping or tunnelling of localized charge carriers. For the first time a detailed analysis of the temperature dependence of the UDR parameter s revealed that, at low temperature (To50 K), the tunnelling of small polarons is the dominating charge transport mechanism in two ABO3 perovskites. We propose that tunnelling of polarons in ABO3 perovskites constitutes the general behaviour in these systems. Acknowledgments We thank J.F. Scott and D. Mihailovicˇ for helpful discussion. This research was supported by the Slovenian Research Agency (P1-0125). References [1] O. Bidault, M. Maglione, M. Actis, M. Kchikech, B. Salce, Phys. Rev. B 52 (1995) 4191. [2] M. Maglione, Ferroelectrics 176 (1996) 1. [3] E. Iguchi, N. Kubota, N. Nakamori, N. Yamamoto, K.J. Lee, Phys. Rev. B 43 (1991) 8646. [4] S. Lenjer, O.F. Schirmer, H. Hesse, T.W. Kool, Phys. Rev. B 66 (2002) 165106. [5] T. Kolodiazhnyi, S.C. Wimbush, Phys. Rev. Lett. 96 (2006) 246404. [6] D. Emin, Phys. Rev. B 48 (1993) 13691. [7] C. Hartinger, F. Mayr, A. Loidl, T. Kopp, Phys. Rev. B 73 (2006) 024408. [8] N.F. Mott, E.A. Davis, Electronic Processes in Non-Crystalline Materials, Oxford University Press, New York, 1979. [9] M. Dumm, P. Lunkenheimer, A. Loidl, B. Assmann, H. Homborg, P. Fulde, J. Chem. Phys. 104 (1996) 5048. [10] B.I. Shklovskii, A.L. Efros, Electronic Properties of Doped Semiconductors, Springer, Berlin, 1984. [11] A.K. Jonscher, Dielectric Relaxations in Solids, Chelsea Dielectrics Press, London, 1983. [12] S.R. Elliott, Adv. Phys. 36 (1987) 135. [13] M. Pollak, T.H. Geballe, Phys. Rev. 122 (1961) 1742. [14] A.R. Long, Adv. Phys. 31 (1982) 553. [15] D. Emin, Phys. Rev. B 46 (1992) 9419. [16] O. Bidoult, M. Maglione, Ferroelectric Lett. Section 18 (1994) 157. [17] R.D. Gould, A.K. Hasan, Thin Solid Films 223 (1993) 334. [18] H. Fro¨hlich, Theory of Dielectrics, Clarendon, Oxford, 1958, p. 90. [19] L.A.K. Dominik, R.K. McCrone, Phys. Rev. 163 (1967) 757. [20] S.V. Varyukhin, O.E. Parfenov, J. Exp. Theor. Phys. Lett. 58 (1993) 830. [21] A. Ghosh, Phys. Rev. B 41 (2003) 1479; A. Ghosh, Phys. Rev. B 42 (1990) 5665; A. Ghosh, J. Chem. Phys. 102 (1995) 1385; S. Bhattacharya, A. Ghosh, Phys. Rev. B 68 (2003) 224202. [22] P. Lunkenheimer, G. Knebel, A. Pimenov, G.A. Emelchenko, A. Loidl, Z. Phys. B 99 (1996) 507. [23] A. Seeger, P. Lunkenheimer, J. Hemberger, A.A. Mukhin, V.Y. Ivanov, A.M. Balbashov, A. Loidl, J. Phys.: Condens. Matter 11 (1999) 3273. [24] V. Bobnar, A. Levstik, C. Huang, Q.M. Zhang, Phys. Rev. Lett. 92 (2004) 047604. [25] A. Levstik, C. Filipicˇ, V. Bobnar, R. Pirc, Appl. Phys. Lett. 81 (2002) 4046; W. Kleemann, V. Scho¨nknecht, D. Sommer, D. Rytz, Phys. Rev. Lett. 66 (1991) 762.