Polaronic conduction and Anderson localization in reduced strontium barium niobate

Transcription

1 Polaronic conduction and Anderson localization in reduced strontium barium niobate Christopher S. Dandeneau 1, a), YiHsun Yang 1, Marjorie A. Olmstead 2, Rajendra K. Bordia 3, Fumio S. Ohuchi 1 1 Department of Materials Science and Engineering, University of Washington, Seattle, Washington 98195, USA 2 Department of Physics, University of Washington, Seattle, Washington 98195, USA 3 Department of Materials Science and Engineering, Clemson University, Clemson, South Carolina 29634, USA Electron transport mechanisms in reduced Sr 0.5 Ba 0.5 Nb 2 O 6 (SBN50) are investigated from ~100 to 955 K through an analysis of the electrical conductivity (σ) and Seebeck coefficient (S) with respect to temperature (T). Notably, experimental evidence is presented that supports a scenario of Anderson localization below 600 K and carrier excitation across a mobility edge at higher temperature. As a relaxor ferroelectric, stoichiometric SBN has intrinsic disorder associated with both the distribution of Sr/Ba vacancies and the formation of polarized nanoregions. The removal of oxygen through reduction generates conduction electrons in SBN. At the lowest temperatures measured ( K), the electrical conductivity exhibits a temperature dependence characteristic of variable range hopping, followed by a transition to small polaron hopping at intermediate temperatures ( K). In both the variable range and small polaron hopping regimes, a semiconductor-like temperature dependence of the electrical conductivity (dσ/dt > 0) was observed. However, above 615 K, dσ/dt decreases dramatically and eventually becomes metal-like (dσ/dt < 0). Concomitantly, the Seebeck coefficient exhibits a linear dependence on lnt from K with the same slope (~104 μv/k) for both polycrystalline SBN50 and single crystalline SBN61 (both reduced), indicating a similar, constant density of states near the Fermi level for both systems. The application of Seebeck coefficient theory to this inherently disordered material reveals that the excitation of carriers across a mobility edge is likely responsible for the change in dσ/dt at high temperature. Such findings may have a significant impact in the field of conductive ferroelectrics. 1

2 Relaxor ferroelectrics are a class of materials that display peculiar dielectric properties due to their intricate, disordered structure. Unlike conventional ferroelectrics, relaxors exhibit a broad, frequency-dependent transition from a polarized (ferroelectric) to non-polarized (paraelectric) state. 1 Furthermore, polarization in relaxors is locally correlated on the nanoscale within areas known as polarized nanoregions (PNRs). 2 This is in contrast to conventional ferroelectrics, where long-range, homogeneous polarization exists within macroscopic domains. The upper limit of PNR stability for a given relaxor system is defined by the Burns temperature (T B ), and it is above T B that a fully paraelectric state is realized. 3 The presence of PNRs in relaxors not only imparts unique attributes, but also allows for investigations into the influence of polarization and disorder on other material properties. Among the relaxor ferroelectrics, strontium barium niobate (Sr x Ba 1-x Nb 2 O 6, SBN100x) has been extensively researched for its electro-optic 4 and photorefractive 5 characteristics. SBN crystallizes into a tetragonal tungsten bronze (TTB) structure with the unit cell (A1) 2 (A2) 4 (C) 4 (B1) 2 (B2) 8 O Details regarding the SBN structure and cationic site occupancy in the material were provided in our previous report. 7 In the polarized state, all three cationic elements in SBN are displaced from oxygen planes in the same sense. However, as the temperature is raised towards T B, Sr, Ba, and 20% of the Nb cations move into alignment with oxygen planes, while the remaining 80% of Nb cations are equally distributed above and below the oxygen planes. 8,9 It has been determined that T B in the SBN system is approximately 623 K. 1 Recently, it was discovered that reduced SBN is a promising candidate for use as an n- type thermoelectric oxide. In particular, a high power factor of approximately 40 W/cm K was obtained for reduced single crystals of SBN From both our previous work 11 and results obtained by Lee et al., 10 a consistent trend has also been identified in the electrical conductivity 2

3 (σ) of highly reduced SBN with respect to temperature (T). Namely, a change in the sign of dσ/dt from positive (semiconductor-like temperature dependence) to negative (metal-like temperature dependence) has been observed at elevated testing temperatures. It was postulated that the presence of PNRs in reduced SBN may act to sustain a positive dσ/dt up to T B by hindering the mass percolation of electrons. However, in a recent report by Bock et al., 12 the peak σ value of reduced SBN70 was located far above the derived T B of 625 ± 25 K, thereby casting doubt on the notion that PNRs are responsible for the change in sign of dσ/dt. Instead, it was suggested that electrons in reduced SBN are localized at lower T due to disorder-induced Anderson localization. At progressively higher temperatures, carriers in the vicinity of the Fermi energy (E F ) are excited above a mobility edge into delocalized states in the conduction band. These promoted carriers tend to dominate the electrical conductivity behavior due to their higher mobility, resulting in a metal-like temperature dependence of σ. Unfortunately, this scenario could not be validated by Seebeck coefficient (S) measurements, as a larger testing temperature range was required. In this Letter, the underlying mechanisms of electrical conductivity in reduced SBN are examined over a temperature range of approximately K through the application of polaron hopping models and Seebeck coefficient theory to experimental measurements. Using Sr 0.5 Ba 0.5 Nb 2 O 6 (SBN50) as a representative composition, distinct regions of localized carrier transport were identified over different temperature regimes. The results provide strong evidence that carrier excitation across a mobility edge drives the change in sign of dσ/dt at elevated temperatures in highly reduced SBN. SBN50 powder was fabricated by a solution combustion synthesis (SCS) method with aqueous Sr(NO 3 ) 2, Ba(NO 3 ) 2, and NbCl 5 precursors; urea and NH 4 NO 3 were employed as a fuel 3

4 and additional oxidizer, respectively. Details regarding the SCS procedure are provided in our previous work. 11 The as-processed SBN50 powder was sintered in air at 1523 K and then reduced at 1273 K for 25 h in N 2 /H 2 (5%); the partial pressure of oxygen during reduction was ~10 22 atm. Low-temperature σ data were obtained from ~100 to 300 K by a four-point probe method using a liquid nitrogen-cooled cryostat. Current was supplied with a Keithley 6220 DC current source, while voltage readings were obtained with a Keithley 2182A nanovoltmeter. Over a temperature range of K, both S and σ were measured with a ZEM-3 thermoelectric property measurement system (ULVAC-RIKO, Japan) in an inert, low-pressure (0.1 atm) environment. Shown in Fig. 1(a) is the electrical conductivity of reduced polycrystalline SBN50 with respect to temperature. The plot can be roughly divided into two regions based on the trends in dσ/dt. In region I, conductivity increases with temperature in a manner consistent with thermally-activated conduction (semiconductor-like behavior). This regime of relatively constant, positive dσ/dt persists up to ~615 K, at which point the gradient begins to show a noticeable decrease. In region II, dσ/dt is significantly reduced and even becomes negative above 800 K, indicating a metal-like temperature dependence of σ. To examine the possible correlation between T B and the change in sign of dσ/dt, the electrical conductivity of an SBN61 single crystal reduced for 25 and 50 h was examined; the results are displayed in the inset of Fig. 1(a). The temperature at which dσ/dt became negative was lowered by ~100 K as the reduction time was doubled. It has been shown that T B is rather invariant to the reduction annealing conditions 12 and thus, the point at which the temperature dependence of σ becomes negative may be related to the defect density (and thus, the carrier concentration); this will be explained later when addressing the phenomenon of Anderson localization. Furthermore, in σ tests conducted 4

5 FIG. 1. (a) Electrical conductivity and (b) magnitude of the Seebeck coefficient for reduced SBN50; the electrical conductivity behavior can be divided into two regions (labeled as I and II) based on the trends in dσ/dt. Electrical conductivity data obtained for an SBN61 single crystal reduced for 25 and 50 h are shown in the inset of (a); arrows denote the temperature of maximum σ. 5

6 for Ca 0.18 Ba 0.82 Nb 2 O 6 (CBN18), which is isostructural to SBN, the sign of dσ/dt changed from positive to negative at ~563 K, more than 600 K below the T B of ~1200 K 13 reported for CBN18 (see Fig. S1 of the supplementary material). 32 In our previous research on SBN50, the onset of metal-like dσ/dt values occurred around 650 K for samples reduced at 1273 to 1423 K for 2 h. 11 However, in the same report, a sample reduced at 1173 K for 2 h showed a positive dσ/dt up to a higher temperature of ~740 K. The magnitude of the Seebeck coefficient ( S ) with respect to T for reduced SBN50 is displayed in Fig. 1(b); all S values were negative, indicating n-type conductivity. Interestingly, S exhibits a nearly constant increase from K. The parallel increase in both S and σ in region I can yield insight into the electrical conductivity mechanism. For a non-degenerate, n- type semiconductor with a thermally-activated carrier concentration, the Seebeck coefficient may be expressed as: 14,15 S = k q (E c E F + C) = k kt q (ln N c n + C) (1) where E c is the conduction band energy, k is the Boltzmann constant, N c is the effective density of states in the conduction band, n is the carrier concentration, and C is a constant. Under these assumptions, an increase in the carrier concentration with temperature would reduce S while increasing the electrical conductivity. However, if the carrier concentration did not change with temperature, S would be proportional to the natural logarithm of T due to the dependence of N c on T 3/2 (this will be addressed later). In addition, a rise in σ with temperature can be observed if electron transport occurs via thermally-activated polaron hopping, as charge carriers can more easily overcome the energy barrier associated with hopping to an adjacent site. To identify the nature of electron transport in Fig. 1(a), the electrical conductivity data were fit according to the small polaron hopping (SPH) model, 16 expressed as 6

7 σ = (A/T s )exp(-e a /kt), where A is a pre-exponential factor, s = 1 or 3/2 for adiabatic or nonadiabatic hopping, respectively, and E a is the activation energy for hopping. Plots of ln(σt s ) vs. (10 3 /T) from 160 to 955 K are displayed in Fig. 2. The electrical conductivity data in the temperature range of K can be closely modeled according to the adiabatic or nonadiabatic SPH expressions with activation energies of 122 and 136 mev, respectively. For adiabatic hopping, electron motion is sufficiently rapid such that a charge can adjust to the instantaneous positions of atoms in the material. 17 In contrast, an electron is unable to follow lattice fluctuations in the non-adiabatic case and thus, the probability of hopping is lower. 18 Further research is currently being conducted to more clearly determine the electrical conductivity mechanism, but regardless, the overall temperature dependence shows that carrier transport in reduced SBN50 likely proceeds via thermally-activated small polaron hopping at intermediate T ( K). The formation of Nb 4+ polarons in reduced SBN61 single crystals was demonstrated in a previous report via optical absorption measurements. 19 For systems in which SPH is observed, strong electron-phonon coupling results in a thermally-activated jump rate above approximately Θ D /2 (Θ D = Debye temperature). 20 However, at temperatures below ~Θ D /2, the number of available phonon modes is reduced and the temperature dependence of the electrical conductivity gradually changes. This is evident in Fig. 2, where the SPH plots deviate from linearity in the vicinity of Θ D /2 (~230 K for SBN). 21 Charge transport at sufficiently low temperatures may be modeled according to Mott s 3D variable range hopping (VRH) expression, 22 given as σ = σ 0 exp[-(t 0 /T)] 1/4, where σ 0 is a pre-exponential term and T 0 (=18α 3 /kn(e F ); α is the inverse of the localization length and N(E F ) is the density of localized states near the Fermi level) is the characteristic temperature coefficient. As shown in the inset of Fig. 2, the electrical conductivity of reduced SBN50 can be represented by the VRH 7

8 FIG. 2. Plots of ln(σt s ) vs /T (SPH model) generated with the electrical conductivity data in Fig. 1(a); s = 1 for adiabatic hopping and 3/2 for non-adiabatic hopping. While close linear fits were achieved from K, deviations from linearity are evident below Θ D /2 = 230 K (Θ D = Debye temperature). The inset shows lnσ vs. T -1/4 data (VRH model) from ~100 to 300 K; good linear fitting was observed from ~100 to 155 K. model in the temperature range of ~100 to 155 K. From valence band X-ray photoelectron spectroscopy data obtained after a 25 h reduction, N(E F ) was estimated to be ~ ev -1 m (see Figure S2 of the supplementary material), 32 or 1 state per 12 Nb atoms per ev. From this estimated N(E F ) and the slope of the VRH plot, the localization length, i.e., the electron wave decay length for a localized state in the disordered system, was calculated to be 5.5 Å. This is slightly larger than the Nb-O-Nb distance (~3.9 Å) reported for SBN48 single crystals. 23 In previous research, Emin reported that the VRH model, which is based on singlephonon emission or absorption, may only be valid for hopping between impurity states at very low (~4 K) temperatures. 24 To explain localized transport from liquid helium temperatures to 8

9 Θ D /2, a multi-phonon process was proposed where, due to the non-activated temperature dependence of the hopping rate, the conductivity trends could resemble VRH. While the data here can be closely fit with the VRH expression, the possibility of multi-phonon emission or absorption cannot be excluded from consideration at this time. To formulate a more complete scenario of charge transport in reduced SBN, the mechanism responsible for the transition to a metal-like temperature dependence (dσ/dt < 0) in region II of Fig. 1(a) must be elucidated. In its unreduced state, the structure of SBN50 is highly disordered due to the existence of polarized nanoregions and randomly distributed vacancies at A1 and A2 sites. Reduction annealing further enhances the disorder by introducing oxygen vacancy donor states, with the net result being a distribution of donor energies (due to variations in the local environment) with an associated mean disorder energy. 25 This broad distribution is often referred to as an impurity band. According to the well-known phenomenon of Anderson localization, 26 a sufficient degree of disorder leads to an extended density of states function with characteristic tails at its edges. As such, a degree of overlap can exist between the tails of the impurity and conduction bands. Within this region of overlap, electron states are localized and separated from extended states by mobility edges. To explain why an increase in the reduction time lowered the temperature at which dσ/dt became negative [see inset of Fig. 1(a)], it should be noted that longer reduction times may not only increase the degree of overlap between the tails of the impurity and conduction bands, but also push E F closer to the mobility edge through the addition of carriers. Consequently, a change in sign of dσ/dt from positive (semiconductorlike) to negative (metal-like) could occur at a lower temperature as SBN is subjected to more extensive reduction treatments. 9

10 Referring to Eq. (1), if electrons in the valence band of reduced SBN50 attained the necessary energy to be excited into the conduction band, a decrease in S would be observed at elevated temperatures due to an increase in the carrier concentration. However, the slope of S with respect to T is positive over the entire range of testing temperatures. Alternatively, if a constant carrier concentration is maintained and localized electrons in the vicinity of E F are promoted above the mobility edge, then S should be proportional to the natural logarithm of T since the density of states would have a T 3/2 dependence. More specifically, a parabolic density of states function should yield a linear relationship between S and lnt with a slope of 129 µv/k. 27,28 It should be noted that the Mott relation with S proportional to T is only applicable at lower temperatures, where carrier transport is dominated by hopping processes. 29 To investigate the above scenario, the S data in Fig. 1(b) were plotted against lnt; the results are displayed in Fig. 3. As evident in the graph, a close linear fit to the data was observed over a temperature range of K. Inspection of the electrical conductivity findings in Fig. 1(a) reveals that it is over this same temperature region that the value of dσ/dt significantly decreases and even becomes negative. Excellent linear fits were also obtained above ~615 K for an SBN61 single crystal reduced for 25 h. Furthermore, the slopes of both linear fits were remarkably similar (SBN50: ± 1.7 µv/k, SBN61: ± 1.3 µv/k), implying a close resemblance in the density of states functions for the two compositions. The deviation in the values of the slopes from 129 µv/k is likely the result of reduced SBN possessing a more complex electronic structure with density of states functions that are not perfectly parabolic. Regardless, the linear nature of the S vs. lnt plots at elevated temperatures supports the notion that a change in dσ/dt arises due to the excitation of localized electrons across a mobility edge into extended states. Once promoted into delocalized states, these electrons eventually dominate 10

11 FIG. 3. Plots of S vs. lnt for polycrystalline SBN50 and single crystalline SBN61; both samples were reduced for 25 h. Excellent linear fits with remarkably similar slopes were obtained from K. the electrical conductivity behavior due to their higher mobility, resulting in a metal-like temperature dependence of σ. 15 It should be noted here that, in addition to disorder-induced localization, electron self-trapping is also possible. However, if electron transport occurred via SPH with no energy distribution among the hopping sites, the Seebeck coefficient could be represented by the temperature-independent Heikes formula. 30,31 In summary, the underlying mechanisms of electron transport in reduced polycrystalline SBN50 were investigated over a broad temperature range. The electrical conductivity was closely modeled according to the 3D VRH expression at low temperatures (~100 to 155 K). As the temperature was raised towards Θ D /2 (~230 K), a gradual change in the charge transport mechanism occurred, and SPH appeared to be dominant up to approximately 545 K. While reduced SBN50 showed a semiconductor-like temperature dependence of σ in both the VRH and 11

12 SPH regimes (i.e., dσ/dt > 0), a significant decrease in dσ/dt was noted above 615 K, and the temperature dependence of σ eventually became metal-like (i.e., dσ/dt < 0). High-temperature Seebeck coefficient data were obtained for both polycrystalline SBN50 and single crystalline SBN61 reduced under identical conditions. Linear fits to S vs. lnt plots indicated that (1) the density of states functions for both compositions are quite similar, and (2) an Anderson transition is likely responsible for the reversal in sign of dσ/dt. The work described could be the strongest evidence yet that the previously observed change in the electrical conductivity of reduced SBN at elevated temperature is due to the excitation of localized carriers across a mobility edge into extended states. This study was supported by the Department of Energy under grant no. DE-FE

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