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Dynamical conductivity of boron carbide: heavily damped plasma vibrations
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 425801 (http://iopscience.iop.org/0953-8984/26/42/425801) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 128.138.73.68 This content was downloaded on 19/04/2017 at 05:17 Please note that terms and conditions apply.
You may also be interested in: Are there bipolarons in icosahedral boron-rich solids? H Werheit On the dynamical conductivity in icosahedral boron-rich solids R Schmechel and H Werheit Superconductivity in boron carbide? Clarification by low-temperature MIR/FIR spectra H Werheit and U Kuhlmann Advanced microstructure of boron carbide Helmut Werheit and Sulkhan Shalamberidze Correlation between structural defects and electronic properties of icosahedral boron-rich solids R Schmechel and H Werheit On excitons and other gap states in boron carbide H Werheit High-pressure phase transition makes B4.3C boron carbide a wide-gap semiconductor Anwar Hushur, Murli H Manghnani, Helmut Werheit et al. Isotopic effects on the phonon modes in boron carbide H Werheit, U Kuhlmann, H W Rotter et al. Materials for thermoelectric energy conversion C Wood
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 425801 (8pp)
doi:10.1088/0953-8984/26/42/425801
Dynamical conductivity of boron carbide: heavily damped plasma vibrations Helmut Werheit and Guido Gerlach Institute of Physics, University Duisburg-Essen, Campus Duisburg, D - 47048 Duisburg, Germany E-mail: [email protected] and [email protected] Received 16 June 2014, revised 5 August 2014 Accepted for publication 22 August 2014 Published 2 October 2014 Abstract
The FIR reflectivity spectra of boron carbide, measured down to ω ~10 cm−1 between 100 and 800 K, are essentially determined by heavily damped plasma vibrations. The spectra are fitted applying the classical Drude-Lorentz theory of free carriers. The fitting Parameter Π = ωp/ωτ yields the carrier densities, which are immediately correlated with the concentration of structural defects in the homogeneity range. This correlation is proved for band-type and hopping conductivity. The effective mass of free holes in the valence band is estimated at m*/me ~ 2.5. The mean free path of the free holes has the order of the cell parameters. Keywords: Boron carbide, electronic transport, dynamical conductivity, plasma vibrations, FIR reflectivity, structural defects (Some figures may appear in colour only in the online journal)
1. Introduction
on the reflectivity is significant; but it failed largely at low temperatures. In this paper we choose an opposing method to determine the parameters of the dynamical electronic transport in boron carbide. Using the classical Drude–Lorentz model of free carriers [11, 12] (see [13, 14]), we simulate the reflectivity spectra measured. The highly sensitive fitting parameter in this procedure is Π = ωp/ωτ, the quotient of plasma and relaxation frequency, yielding information on the composition and temperature dependent properties of the electronic transport, including the effective mass m* of free charge carriers and the mean free path of charge carriers.
The homogeneity range of boron carbide comprises the chemical compositions between B4.3C and B~10.4C (see [1] and references therein). Boron carbide belongs to the αrhombohedral boron structure group. However, there is no real unit cell representing the whole structure and consisting of a well defined combination of structural elements. Instead, the structure is formed by rhombohedral elementary cells randomly composed of B12 or B11C icosahedra at the vertices of the cells and CBC, CBB or B□B arrangements (□, vacancy) on the main cell diagonal. Their concentrations depend on the actual chemical composition [2]. Apart from the comprehensive collection of data in [1], reviews are available on the technical and industrial aspects of boron carbide by Lipp [3], on the general properties by Domnich et al [4] and on the electronic properties by Werheit [5]. Structural aspects in relation to the electronic properties were updated in [6]. In previous investigations of the dynamical conductivity [7–10], the dielectric parameters of boron carbide including the optical conductivity were derived from the measured reflectivity spectra, applying Kramers–Kronig transformations and taking into account the different models of electronic transport. This procedure was successful at higher temperatures, where the effect of the dynamical conductivity 0953-8984/14/425801+8$33.00
2. Electronic structure The electronic properties of boron carbide are characterized by the minimum transition energies between the valence and conduction bands 2.09 and 2.41 eV (for the actual band scheme see [6, 15]). The electronic transport is determined by high-density gap states due to the high concentrations of structural defects generating split-off valence states. The valence band is left completely occupied thus bringing out the semiconducting character of boron carbide (see [5, 6, 16]). The band-type and hopping transport of charge carriers determine the electronic properties. Both mechanisms coexist, but 1
© 2014 IOP Publishing Ltd Printed in the UK
H Werheit and G Gerlach
J. Phys.: Condens. Matter 26 (2014) 425801
(a)
(b)
(c)
Figure 1. Attribution of electronic gap states of boron carbide to the structural defects taken from [2] versus carbon content. (a) Charge of gap states, based on (B11C)CBC as a reference. (b) Charge of gap states, based on (B12)CBC as a reference. (c) Density of defects, referred to the particular preference structure ((B12)CBC in the boron-rich region; (B11C)CBC in the carbon-rich region of the homogeneity range).
one of them usually prevails depending, for example, on the temperature. This qualitative relation between structural defects and electronic properties is meanwhile widely accepted, although it is in fundamental contrast to the theoretical band structure calculations performed so far [17–23]; determining boron carbide, represented by the hypothetical, energetically most favorable structure (B12)CBC, to be metallic. Merely Bylander et al [24] obtained completely filled valence bands on the likewise hypothetical structure (B11C)CBC, which is outside the homogeneity range of boron carbide. The quantitative concentrations of the structural elements, determined by Werheit and Shalamberidze [2] on isotopically pure boron carbides, enables us to check the suitability of the hypothetical structure models (B12)CBC and (B11C) CBC by adapting them to the real compounds considering the actual concentrations of donors and acceptors. The results are shown in figures 1(a) and (b) respectively. The chain-free cells (represented by B□B (□, vacancy) are alternatively taken into account with three acceptors each for the missing central B atom and five acceptors considering the exchange of the carbon atoms at the chain ends by boron additionally. The results are qualitatively the same: Figure 1(a) shows the concentration of acceptors monotonously increasing with the carbon content decreasing in contrast to the experimental findings (for examples, see below, figures 7–9) indicating a maximum acceptor concentration near the centre of the homogeneity range. Figure 1(b) indicates a majority of donors near the carbon-rich limit of the homogeneity range, in contrast to the experimentally proved p-type behaviour of boron carbide within the whole homogeneity range. Hence, both hypothetical models are individually unsuitable to describe the electronic properties of boron carbide throughout the whole homogeneity range. Based on the results presented below, we propose to waive such single reference structures for the whole homogeneity range and instead take that structure model for reference, which is prevailing in the respective part of the homogeneity range. These are the structure models mentioned, but they are to be limited separately to the boron-rich and the carbon-rich part of the homogeneity range respectively. We take the structural formula (B11C)CBC prevailing in the carbon-rich and B12CBC prevailing in the boron-rich part of the homogeneity
Figure 2. B4.1C, FIR reflectivity spectra (T = 100–800 K).
range as a reference in these sections and count the deviations as structural defects. Accordingly, B12 icosahedra are defects in the carbon-rich and B11C icosahedra defects in the boronrich regions respectively. In the centre of the homogeneity range with 50% B12 and B11C icosahedra each, both descriptions are equal. The resulting defect concentrations shown in figure 1(c) are largely compatible with the experimental results as shown below. Figure 1(c) confirms previous conclusions that the chemical compound B13C2 exhibits the mostdistorted structure in the whole homogeneity range. There, the structural defects amount to nearly 80% of elementary cells with some kind of distortion. 3. Samples Most of the boron carbides investigated were provided by ESK Kempten, Kempten, Germany, and H. C. Starck, Goslar, 2
H Werheit and G Gerlach
J. Phys.: Condens. Matter 26 (2014) 425801
5. Results FIR reflectivity spectra of a boron carbide sample with the nominal composition B4.1C, close to B4.3C, the carbon-rich limit of the homogeneity range, were measured between 100 and 800 K in steps of 100 K. As the dynamical conductivity is the object of interest, the spectral range, starting at the equipment-related low frequency limit 10 cm−1, was limited at 600 cm−1 thus including the strong phonon mode close to 400 cm−1 only. The results are shown in figure 2. The whole homogeneity range of boron carbide was included by re-evaluating the 450 K reflectivity spectra published by Schmechel and Werheit in [1, 7, 9] and compiled in figure 3. These spectra are distinguished by their particularly high accuracy in the low frequency FIR range. This low-frequency accuracy was not required in the FIR phonon spectra measured by Kuhlmann [26], which are additionally considered in this study. The variation in the absorption of the phonons near 400 cm−1, obviously split at high temperatures, will be the subject of a separate study. 6. A short summary of the classical theory of dynamical conductivity (Drude–Lorentz model [11–14])
Figure 3. BxC, FIR reflectivity spectra at 450 K, x = 4.3, 6.3, 6.95,
7.91, 8.52, 10.37.
Germany. Samples of about 10 × 10 × 5 mm3 were cut from the compacts and mechanically polished on one side with diamond pastes of systematically decreasing grain sizes down to 1 μm. The reflectivity spectrum of a single crystal B4.3C sample as provided by Leithe-Jasper and Tanaka [25] was included.
The reflectivity of solids is determined by R=
1 with n2 = 2
(
(n − 1)2 + κ 2 (n + 1)2 + κ 2
1 ε12 + ε22 + ε1 and κ 2 = 2
)
(1)
(
)
ε12 + ε22 − ε1
The effect of mobile carriers in the dielectric function is described by
4. Experimental details The reflectivity spectra of boron carbide in the FIR range 10–600 cm−1 were measured with a Bruker Fourier spectrometer IFS 113v established for an external sample compartment. The source of radiation was a mercury high pressure lamp (T ~ 5000 K). Suitable Mylar foils were used as beam splitters in the interferometer (50 μm thickness for 10–100 cm−1 and 3.5 μm for 100–600 cm−1). The detector was a liquid-Hecooled Ge Bolometer. The spectra obtained in the overlapping spectral subsections were combined to the complete spectrum using a special computer program minimizing errors. Low sample temperatures were obtained in a liquid-He cryostat (Oxford instruments) mounted in the internal sample compartment of the spectrometer. For high temperatures, a special sample compartment was constructed allowing a high long-term stability. At high temperatures, the thermal radiation contributed considerably to the signal measured. For the required corrections the corresponding black body radiation was measured separately. The evacuated high temperature sample compartment was separated from the evacuated spectrometer by a FIR transparent polyethylene window, irradiated from the sample compartment. Melting processes in the window limited the range of achievable sample temperatures to ~800 K.
ε1 = εL −
ωp2
ωτ2
⋅
ωτ2 1 = εL − Π2 1 + Ω2 + ω2
ωτ2
(2)
and
ωp2
ωτ2 Π2 1 = ⋅ 2 2 ωτ ω ωτ + ω Ω 1 + Ω2 ωp ω with Π = and Ω = ωτ ωτ
ε2 = εL −
⋅
(3)
εL is the sum of the contributions of the bonded valence electrons and the phonon resonances in the dielectric susceptibility, taken here as a constant value without considering specific frequency dependencies, for example due to specific phonons. In the Drude–Lorentz model, the free charge carriers in the extended electronic states of semiconductors form a ‘gas of free electrons’, whose resonance occurs at the plasma frequency
ωp =
4πNe2 / mε
(4)
with N, the concentration of the free charge carriers, m, their mass and ε, the dielectric constant. The relaxation of the charge carriers and the damping of their vibration respectively is unspecifically considered by the 3
H Werheit and G Gerlach
J. Phys.: Condens. Matter 26 (2014) 425801
Figure 4. Calculated FIR reflectivity R of a solid (ε1 = 10.8; ωp(opt) = 300 cm−1); parameter: Π = ωp/ωτ.
relaxation time τ = 2π/ωτ, whereby τ is attributed for example to the average period between two collisions or between the generation and recombination or retrapping of free charge carriers. Figure 4 demonstrates the plasma edge allocated to 300 cm−1 in the reflectivity spectrum of a semiconductor and the sensitive effect of the parameter Π = ωp/ωτ on R in particular below ωp.
At temperatures exceeding ~300 K, the adaptation to the experimental reflectivity spectra was so good that the small remaining deviations can easily be attributed to experimental errors or uncertainties in the Fourier transformation of the interferograms. This confirms that the underlying classical Drude–Lorentz model is applicable. Towards lower temperatures the deviations become noticeable. However, in this range the effect of the dynamical conductivity on the reflectivity spectra is small and moreover restricted to a narrow range near the low frequency limit of the spectrometer (see figure 2). Therefore, we cannot decide, whether these deviations are produced by an increasing influence of the hopping processes prevailing at low temperatures, or by experimental or transformation errors. As Π2 yields relative carrier concentrations only, quantifying requires results of a different method. In figure 5, the temperature-dependent Hall density of holes determined by Wood [27] is adapted; the thermal activation energies agree and comply with those of the electrical conductivity in this temperature range (see [5]). Using instead the spin densities determined by Chauvet et al [28], but attributing them, contrary to the authors, to localized electrons in the split-off gap states (for argumentation, see [16]), carrier densities are obtained, which are about four times higher. We estimate the effective mass of holes using the data obtained at 800 K. At this temperature, the contribution of the dynamical conductivity to the reflectivity spectrum is highest, and hence errors should be low. Using the hole density in figure 4 and the dielectric constant ε1 = 10.8, equation (4) yields
7. Evaluation and discussion of the spectra Comparing the spectra in figures 2 and 3 with the model calculations in figure 4 shows that the plasma vibrations in boron carbide are heavily damped. Therefore, the plasma frequency cannot be exactly determined but can only be estimated. However, the obtained fitting parameters Π = ωp/ωτ are rather accurate. Compared with Π, the accuracy of the components ωp and ωτ is much lower, but they may have the correct order of magnitude at least. Indeed, it must be obeyed that in boron carbide band type conductivity and hopping conductivity coexist, and their relative share depends on the temperature. We assume that the dynamical conductivity in the range of FIR frequencies (~1012 Hz) is preferably determined by carriers in extended states only. 7.1. Temperature dependence
The reflectivity spectra shown in figure 2 were obtained from the very same sample, and therefore the structural properties are exactly the same. Hence, the related impact on ωτ is unchanged in all spectra. A temperature-dependent influence of the velocity of carriers and the occupancy of traps in the band gap on ωτ seems possible; but it cannot be quantified reliably at present. Therefore, we assume the fitting parameter Π squared to be largely proportional to N, the density of charge carriers. The results are shown in figure 5.
m * /me = 2 .4
Based on Chauvet’s spin density, which is four times higher, m*/me ~ 10 would be obtained. These values confirm the effective masses of free holes in boron carbide as previously estimated (see [5]). 4
H Werheit and G Gerlach
J. Phys.: Condens. Matter 26 (2014) 425801
Figure 5. Squared fitting parameter Π to the reflectivity spectra in figure 2; quantifying by the Hall density of holes (Wood [27]). Insert, ωp and ωτ, components of Π.
Figure 6. Mean free path of holes l versus T. The lattice parameters a and c are indicated. Dashed line, linear fit.
In spite of the very limited accuracy of ωτ displayed in the insert in figure 5, a rough estimation of the mean free path is made. In the classical Drude–Lorentz model, the velocity of charge carriers forming the ‘gas of free electrons’ is essentially determined by m / 2v 2 = 3 / 2kT . The effective relaxation time is determined by various processes which are represented as 1/τ = 1/τ1 + 1/τ2 + 1/τ3 + …. Examples of such processes are scattering by phonons, scattering by charged or uncharged lattice defects and trapping in unoccupied gap states; all of them depending differently on various parameters, which are largely unknown in the case of boron carbide. Hence the roughly estimated mean free path
l = v ⋅ τ = v / ωτ in figure 6 based on ωτ taken from the insert in figure 5 and m*/me = 2.4 is unspecified, and indeed comparable with the lattice parameters and compatible with the concentration of defects in the order of ~1 cell−1 (figure 1(c)). The increase of l at increasing T can be explained by the trapping probability decreasing, with the carrier velocity increasing. 7.2. Dependence on the chemical composition
Re-evaluating the spectra in 7–10 was confined to those at 450 K, where the influence of the dynamical conductivity on R is most significant. In figure 7; the squares of the obtained 5
H Werheit and G Gerlach
J. Phys.: Condens. Matter 26 (2014) 425801
Figure 7. Square of the Π parameters of the plasma vibrations in BxC at 450 K. Full symbols, derived from the spectra in 7, 10; open symbols, from [26], Dashed line, sum of defects, taken from figure 1.
Figure 8. Selection of electronic transport properties of BxC versus carbon content.
fitting parameters Π are plotted versus carbon content. The FIR phonon spectra [26] with a lower accuracy (see above) yield somewhat more varying results, but agree satisfactorily. Figure 7 shows that the curve shape, consisting of two almost linear branches meeting at a maximum in the middle of the homogeneity range, is correlated with the sum of the structural defects taken from figure 1(c). The results were obtained from different samples of different chemical compositions and separate preparations; in particular containing different defect concentrations. Hence, a certain effect of ωτ2 on the pattern of data in figure 7 cannot be excluded. But they cannot be specified. Nevertheless the correlation between carrier density and defect concentration is obvious. In order to verify this correlation more generally, figure 8 compares a selection of electronic transport properties with the sum of the intrinsic defects: dc and hf (1010 Hz) conductivity
by Samara et al [29], dc conductivity at 200 K by Werheit et al [30, 31], Seebeck coefficient at 500 K by Werheit et al [32] and a selection of Seebeck coefficients at 300 K obtained from [1]. In all these cases the correlation is confirmed (further examples in [1] figure 283). An additional investigation of this correlation is shown in figure 9 comparing the electrical conductivity of BxC at 10 K [1, 32, 33] with the sum of the defects. At low temperatures, the electrical conductivity of boron carbide is nearly entirely determined by variable-range hopping [34]; hence at 10 K a measurable contribution of free carriers is largely excluded. According to Mott’s theory of variable-range hopping [35] advanced by other authors [36–39], the hopping probability varies ~exp (−2αD) (α, localization length; D, distance of localized states, immediately related to their density). The expected exponential correlation with the defect concentration is satisfactorily confirmed in figure 9. 6
H Werheit and G Gerlach
J. Phys.: Condens. Matter 26 (2014) 425801
Figure 9. Electrical conductivity of BxC versus carbon content at 10 K.
into the gap states split-off from the valence band. Their concentration is quantitatively correlated with the intrinsic structural defects in the order of one per rhombohedral cell. These originate essentially from deviations of the actual structures from B12CBC and (B11C)CBC, which are prevailing in the boron-rich and the carbon-rich section of the homogeneity range respectively. The correlation between electronic transport properties and the composition-dependent concentration of structural defects has been proved for the temperature-dependent ranges of band-type conduction and for the hopping processes as well. The results on boron carbide demonstrate that even in cases of effective masses m* clearly exceeding me the classical Drude–Lorentz model is applicable.
We note that the preceding correlations are based on the implicit assumption that one structural defect generates one gap state each time. This certainly applies for one B11C icosahedron generating one donor in structures based on B12 icosahedra, and the opposite is the case for B12 acceptors in B11C structures. In the same way, CBB creates one acceptor compared with the reference CBC. However, somewhat questionable is the attribution of one defect to B□B, where the central vacancy is proved beyond doubt (see [2]). Compared with CBC the missing central B atom implies three missing electrons (acceptors) and the exchange of the C atoms at the chain ends would cause a further deficit of two electrons in the electronic structure. Accordingly, there is a certain uncertainty with respect to the one-to-one assignment of gap states to these defects. As the content of B□B is relatively small and depends weakly on the carbon content, the uncertainty might remain in narrow limits, at least concerning the dependence on the carbon content. The correlations between the sum of the intrinsic defects and electronic transport properties shown above suggest that the energetic position of the concerning gap states ΔE = 150 meV is independent of their individual origin. This is not necessarily the case. Indeed, their energetical variation is sufficiently small, so that in the case of band-type conductivity the thermal excitation of electrons is not essentially different. At low temperatures suitable phonons are required for allowing phonon-assisted variable-range hopping. The individual identification of the gap states in the band scheme [6, 15] remains open.
Acknowledgements The authors thank Dr K A Schwetz, ESK Kempten, Germany, and Dr Krismer, Dr Fister and Mr Neisius, H C Starck, Goslar and Laufenburg, Germany, for providing boron carbide samples. References [1] Werheit H 2000 Boron compounds Numerical Data and Functional Relationships in Science and Technology (Landolt–Börnstein New Series Group III, vol 41D) ed O Madelung (Berlin: Springer) pp 1–491 [2] Werheit H and Shalamberidze S 2012 J. Phys.: Condens. Matter 24 385406 [3] Lipp A 1965 Technische Rundschau 57 nos. 14, 28 and 33 Lipp A 1966 Technische Rundschau 58 no. 7 [4] Domnich V, Reynaud S, Haber R A and Chhowalla M 2011 J. Am. Ceram. Soc. 94 3605 [5] Werheit H 2007 J. Phys.: Condens. Matter 19 186207 [6] Werheit H 2014 On microstructure and electronic properties of boron carbide Proc. of the ICACC’2014, ‘Ceramic Engineering and Science Proc.’ (CESP) Advances in Ceramic Armor X ed J C LaSalvia, vol 35 (Westerville, OH: American Ceramic Society) at press
8. Conclusion The plasma vibrations determining the low frequency region of the FIR reflectivity spectra of boron carbide are heavily damped. Apart from the large effective mass of the free holes m*/me = 2.5, this is due to the high concentration of traps formed by intrinsic gap states in the band gap above the valence band edge. The concentration of free holes in the valence band of boron carbide is due to the thermal excitation of valence electrons 7
H Werheit and G Gerlach
J. Phys.: Condens. Matter 26 (2014) 425801
[22] Calandra M, Vast N and Mauri F 2004 Phys. Rev. B 69 224505 [23] Vast N, Sjakste J and Betranhandy E 2009 J. Phys.: Conf. Ser. 176 012012 [24] Bylander D M, Leinmann L and Lee S 1991 Phys. Rev. B 43 1487 [25] Werheit H, Leithe-Jasper A, Tanaka T, Rotter H W and Schwetz K A 2004 J. Solid State Chem. 177 575 [26] Kuhlmann U 1994 PhD Thesis University of Duisburg, Germany [27] Wood C 1985 AIP Conf. Proc. 140 206 [28] Chauvet O, Emin D, Forro L, Aselage T L and Zuppiroli L 1996 Phys. Rev. B 53 14450 [29] Samara G A, Tardy, H L, Venturini E L, Aselage T L and Emin D 1993 Phys. Rev. B 48 1468 [30] Kuhlmann U and Werheit H 1994 Jpn. J. Appl. Phys. Series 10 84 [31] Werheit H 1999 Boron and Boron-rich Compounds in: Electric Refractory Materials ed Y Kumashiro (New York: Marcel Dekker) p 589 [32] Werheit H, Kuhlmann U, Franz R, Winkelbauer W, Herstell B, Fister D and Neisius H 1990 AIP Conf. Proc. 231 104 [33] Werheit H and Winkelbauer W 2014 unpublished [34] Werheit H 2007 J. Phys.: Condens. Matter 19 186207 [35] Mott N F 1969 Phil. Mag. 19 835 [36] Pollak M J 1972 Non-crystalline Solids 11 1 [37] Seager C H and Pike G E 1974 Phys. Rev. B 10 1435 [38] Apsley N and Hughes H P 1974 Phil Mag. 30 963 [39] Overhof H 1976 Hopping conductivity in disordered solids Festkörperprobleme/Advances in Solid State Physics vol. XVI ed J Treusch (Berlin: Springer)
[7] Schmechel R 1998 PhD thesis University of Duisburg, Germany [8] Schmechel R and Werheit H 1996 J. Phys.: Condens. Matter 8 7263 [9] Schmechel R and Werheit H 1998 J. Mat. Proc. and Manufact. 6 329 [10] Schmechel R and Werheit H 1997 J. Solid State Chem. 133 335 [11] Drude P 1900 Z. Phys. 1 161 [12] Lorentz H A 1909 The Theory of Electrons (Leipzig: Teubner) reprinted 1952 (New York: Dover Publications) [13] Gerlach E and Grosse P 1977 Scattering of free electrons and dynamical conductivity Festkörperprobleme XVII/Advances in Solid State Physics ed J Treusch (Braunschweig: Vieweg) p 157 [14] Grosse P 1979 Freie Elektronen in Festköpern (Berlin: Springer) [15] Werheit H and Kuhlmann U 2011 J. Phys.: Condens. Matter 23 435501 [16] Schmechel R and Werheit H 1999 J. Phys.: Condens. Matter 11 6803 [17] Switendick A C 1990 The electronic structure of crystalline boron carbide I: B12 icosahedra and C-B-C chains The Physics and Chemistry of Carbides, Nitrides and Borides ed R Freer (Dordrecht, The Netherlands: Kluwer) p 525 [18] Bylander D M, Kleinman L and Lee S 1990 Phys. Rev. B 42 1394 [19] Kim H and Kaviany M 2013 Phys. Rev. B 87 155133 [20] Armstrong D R, Bolland J, Perkins P G, Will G and Kirfel A 1983 Acta Crystallogr. B 39 324 [21] Kleinman L 1991 AIP Conf. Proc. 231 13
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Dynamical conductivity of boron carbide: heavily damped plasma vibrations
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 425801 (http://iopscience.iop.org/0953-8984/26/42/425801) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 128.138.73.68 This content was downloaded on 19/04/2017 at 05:17 Please note that terms and conditions apply.
You may also be interested in: Are there bipolarons in icosahedral boron-rich solids? H Werheit On the dynamical conductivity in icosahedral boron-rich solids R Schmechel and H Werheit Superconductivity in boron carbide? Clarification by low-temperature MIR/FIR spectra H Werheit and U Kuhlmann Advanced microstructure of boron carbide Helmut Werheit and Sulkhan Shalamberidze Correlation between structural defects and electronic properties of icosahedral boron-rich solids R Schmechel and H Werheit On excitons and other gap states in boron carbide H Werheit High-pressure phase transition makes B4.3C boron carbide a wide-gap semiconductor Anwar Hushur, Murli H Manghnani, Helmut Werheit et al. Isotopic effects on the phonon modes in boron carbide H Werheit, U Kuhlmann, H W Rotter et al. Materials for thermoelectric energy conversion C Wood
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 425801 (8pp)
doi:10.1088/0953-8984/26/42/425801
Dynamical conductivity of boron carbide: heavily damped plasma vibrations Helmut Werheit and Guido Gerlach Institute of Physics, University Duisburg-Essen, Campus Duisburg, D - 47048 Duisburg, Germany E-mail: [email protected] and [email protected] Received 16 June 2014, revised 5 August 2014 Accepted for publication 22 August 2014 Published 2 October 2014 Abstract
The FIR reflectivity spectra of boron carbide, measured down to ω ~10 cm−1 between 100 and 800 K, are essentially determined by heavily damped plasma vibrations. The spectra are fitted applying the classical Drude-Lorentz theory of free carriers. The fitting Parameter Π = ωp/ωτ yields the carrier densities, which are immediately correlated with the concentration of structural defects in the homogeneity range. This correlation is proved for band-type and hopping conductivity. The effective mass of free holes in the valence band is estimated at m*/me ~ 2.5. The mean free path of the free holes has the order of the cell parameters. Keywords: Boron carbide, electronic transport, dynamical conductivity, plasma vibrations, FIR reflectivity, structural defects (Some figures may appear in colour only in the online journal)
1. Introduction
on the reflectivity is significant; but it failed largely at low temperatures. In this paper we choose an opposing method to determine the parameters of the dynamical electronic transport in boron carbide. Using the classical Drude–Lorentz model of free carriers [11, 12] (see [13, 14]), we simulate the reflectivity spectra measured. The highly sensitive fitting parameter in this procedure is Π = ωp/ωτ, the quotient of plasma and relaxation frequency, yielding information on the composition and temperature dependent properties of the electronic transport, including the effective mass m* of free charge carriers and the mean free path of charge carriers.
The homogeneity range of boron carbide comprises the chemical compositions between B4.3C and B~10.4C (see [1] and references therein). Boron carbide belongs to the αrhombohedral boron structure group. However, there is no real unit cell representing the whole structure and consisting of a well defined combination of structural elements. Instead, the structure is formed by rhombohedral elementary cells randomly composed of B12 or B11C icosahedra at the vertices of the cells and CBC, CBB or B□B arrangements (□, vacancy) on the main cell diagonal. Their concentrations depend on the actual chemical composition [2]. Apart from the comprehensive collection of data in [1], reviews are available on the technical and industrial aspects of boron carbide by Lipp [3], on the general properties by Domnich et al [4] and on the electronic properties by Werheit [5]. Structural aspects in relation to the electronic properties were updated in [6]. In previous investigations of the dynamical conductivity [7–10], the dielectric parameters of boron carbide including the optical conductivity were derived from the measured reflectivity spectra, applying Kramers–Kronig transformations and taking into account the different models of electronic transport. This procedure was successful at higher temperatures, where the effect of the dynamical conductivity 0953-8984/14/425801+8$33.00
2. Electronic structure The electronic properties of boron carbide are characterized by the minimum transition energies between the valence and conduction bands 2.09 and 2.41 eV (for the actual band scheme see [6, 15]). The electronic transport is determined by high-density gap states due to the high concentrations of structural defects generating split-off valence states. The valence band is left completely occupied thus bringing out the semiconducting character of boron carbide (see [5, 6, 16]). The band-type and hopping transport of charge carriers determine the electronic properties. Both mechanisms coexist, but 1
© 2014 IOP Publishing Ltd Printed in the UK
H Werheit and G Gerlach
J. Phys.: Condens. Matter 26 (2014) 425801
(a)
(b)
(c)
Figure 1. Attribution of electronic gap states of boron carbide to the structural defects taken from [2] versus carbon content. (a) Charge of gap states, based on (B11C)CBC as a reference. (b) Charge of gap states, based on (B12)CBC as a reference. (c) Density of defects, referred to the particular preference structure ((B12)CBC in the boron-rich region; (B11C)CBC in the carbon-rich region of the homogeneity range).
one of them usually prevails depending, for example, on the temperature. This qualitative relation between structural defects and electronic properties is meanwhile widely accepted, although it is in fundamental contrast to the theoretical band structure calculations performed so far [17–23]; determining boron carbide, represented by the hypothetical, energetically most favorable structure (B12)CBC, to be metallic. Merely Bylander et al [24] obtained completely filled valence bands on the likewise hypothetical structure (B11C)CBC, which is outside the homogeneity range of boron carbide. The quantitative concentrations of the structural elements, determined by Werheit and Shalamberidze [2] on isotopically pure boron carbides, enables us to check the suitability of the hypothetical structure models (B12)CBC and (B11C) CBC by adapting them to the real compounds considering the actual concentrations of donors and acceptors. The results are shown in figures 1(a) and (b) respectively. The chain-free cells (represented by B□B (□, vacancy) are alternatively taken into account with three acceptors each for the missing central B atom and five acceptors considering the exchange of the carbon atoms at the chain ends by boron additionally. The results are qualitatively the same: Figure 1(a) shows the concentration of acceptors monotonously increasing with the carbon content decreasing in contrast to the experimental findings (for examples, see below, figures 7–9) indicating a maximum acceptor concentration near the centre of the homogeneity range. Figure 1(b) indicates a majority of donors near the carbon-rich limit of the homogeneity range, in contrast to the experimentally proved p-type behaviour of boron carbide within the whole homogeneity range. Hence, both hypothetical models are individually unsuitable to describe the electronic properties of boron carbide throughout the whole homogeneity range. Based on the results presented below, we propose to waive such single reference structures for the whole homogeneity range and instead take that structure model for reference, which is prevailing in the respective part of the homogeneity range. These are the structure models mentioned, but they are to be limited separately to the boron-rich and the carbon-rich part of the homogeneity range respectively. We take the structural formula (B11C)CBC prevailing in the carbon-rich and B12CBC prevailing in the boron-rich part of the homogeneity
Figure 2. B4.1C, FIR reflectivity spectra (T = 100–800 K).
range as a reference in these sections and count the deviations as structural defects. Accordingly, B12 icosahedra are defects in the carbon-rich and B11C icosahedra defects in the boronrich regions respectively. In the centre of the homogeneity range with 50% B12 and B11C icosahedra each, both descriptions are equal. The resulting defect concentrations shown in figure 1(c) are largely compatible with the experimental results as shown below. Figure 1(c) confirms previous conclusions that the chemical compound B13C2 exhibits the mostdistorted structure in the whole homogeneity range. There, the structural defects amount to nearly 80% of elementary cells with some kind of distortion. 3. Samples Most of the boron carbides investigated were provided by ESK Kempten, Kempten, Germany, and H. C. Starck, Goslar, 2
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5. Results FIR reflectivity spectra of a boron carbide sample with the nominal composition B4.1C, close to B4.3C, the carbon-rich limit of the homogeneity range, were measured between 100 and 800 K in steps of 100 K. As the dynamical conductivity is the object of interest, the spectral range, starting at the equipment-related low frequency limit 10 cm−1, was limited at 600 cm−1 thus including the strong phonon mode close to 400 cm−1 only. The results are shown in figure 2. The whole homogeneity range of boron carbide was included by re-evaluating the 450 K reflectivity spectra published by Schmechel and Werheit in [1, 7, 9] and compiled in figure 3. These spectra are distinguished by their particularly high accuracy in the low frequency FIR range. This low-frequency accuracy was not required in the FIR phonon spectra measured by Kuhlmann [26], which are additionally considered in this study. The variation in the absorption of the phonons near 400 cm−1, obviously split at high temperatures, will be the subject of a separate study. 6. A short summary of the classical theory of dynamical conductivity (Drude–Lorentz model [11–14])
Figure 3. BxC, FIR reflectivity spectra at 450 K, x = 4.3, 6.3, 6.95,
7.91, 8.52, 10.37.
Germany. Samples of about 10 × 10 × 5 mm3 were cut from the compacts and mechanically polished on one side with diamond pastes of systematically decreasing grain sizes down to 1 μm. The reflectivity spectrum of a single crystal B4.3C sample as provided by Leithe-Jasper and Tanaka [25] was included.
The reflectivity of solids is determined by R=
1 with n2 = 2
(
(n − 1)2 + κ 2 (n + 1)2 + κ 2
1 ε12 + ε22 + ε1 and κ 2 = 2
)
(1)
(
)
ε12 + ε22 − ε1
The effect of mobile carriers in the dielectric function is described by
4. Experimental details The reflectivity spectra of boron carbide in the FIR range 10–600 cm−1 were measured with a Bruker Fourier spectrometer IFS 113v established for an external sample compartment. The source of radiation was a mercury high pressure lamp (T ~ 5000 K). Suitable Mylar foils were used as beam splitters in the interferometer (50 μm thickness for 10–100 cm−1 and 3.5 μm for 100–600 cm−1). The detector was a liquid-Hecooled Ge Bolometer. The spectra obtained in the overlapping spectral subsections were combined to the complete spectrum using a special computer program minimizing errors. Low sample temperatures were obtained in a liquid-He cryostat (Oxford instruments) mounted in the internal sample compartment of the spectrometer. For high temperatures, a special sample compartment was constructed allowing a high long-term stability. At high temperatures, the thermal radiation contributed considerably to the signal measured. For the required corrections the corresponding black body radiation was measured separately. The evacuated high temperature sample compartment was separated from the evacuated spectrometer by a FIR transparent polyethylene window, irradiated from the sample compartment. Melting processes in the window limited the range of achievable sample temperatures to ~800 K.
ε1 = εL −
ωp2
ωτ2
⋅
ωτ2 1 = εL − Π2 1 + Ω2 + ω2
ωτ2
(2)
and
ωp2
ωτ2 Π2 1 = ⋅ 2 2 ωτ ω ωτ + ω Ω 1 + Ω2 ωp ω with Π = and Ω = ωτ ωτ
ε2 = εL −
⋅
(3)
εL is the sum of the contributions of the bonded valence electrons and the phonon resonances in the dielectric susceptibility, taken here as a constant value without considering specific frequency dependencies, for example due to specific phonons. In the Drude–Lorentz model, the free charge carriers in the extended electronic states of semiconductors form a ‘gas of free electrons’, whose resonance occurs at the plasma frequency
ωp =
4πNe2 / mε
(4)
with N, the concentration of the free charge carriers, m, their mass and ε, the dielectric constant. The relaxation of the charge carriers and the damping of their vibration respectively is unspecifically considered by the 3
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J. Phys.: Condens. Matter 26 (2014) 425801
Figure 4. Calculated FIR reflectivity R of a solid (ε1 = 10.8; ωp(opt) = 300 cm−1); parameter: Π = ωp/ωτ.
relaxation time τ = 2π/ωτ, whereby τ is attributed for example to the average period between two collisions or between the generation and recombination or retrapping of free charge carriers. Figure 4 demonstrates the plasma edge allocated to 300 cm−1 in the reflectivity spectrum of a semiconductor and the sensitive effect of the parameter Π = ωp/ωτ on R in particular below ωp.
At temperatures exceeding ~300 K, the adaptation to the experimental reflectivity spectra was so good that the small remaining deviations can easily be attributed to experimental errors or uncertainties in the Fourier transformation of the interferograms. This confirms that the underlying classical Drude–Lorentz model is applicable. Towards lower temperatures the deviations become noticeable. However, in this range the effect of the dynamical conductivity on the reflectivity spectra is small and moreover restricted to a narrow range near the low frequency limit of the spectrometer (see figure 2). Therefore, we cannot decide, whether these deviations are produced by an increasing influence of the hopping processes prevailing at low temperatures, or by experimental or transformation errors. As Π2 yields relative carrier concentrations only, quantifying requires results of a different method. In figure 5, the temperature-dependent Hall density of holes determined by Wood [27] is adapted; the thermal activation energies agree and comply with those of the electrical conductivity in this temperature range (see [5]). Using instead the spin densities determined by Chauvet et al [28], but attributing them, contrary to the authors, to localized electrons in the split-off gap states (for argumentation, see [16]), carrier densities are obtained, which are about four times higher. We estimate the effective mass of holes using the data obtained at 800 K. At this temperature, the contribution of the dynamical conductivity to the reflectivity spectrum is highest, and hence errors should be low. Using the hole density in figure 4 and the dielectric constant ε1 = 10.8, equation (4) yields
7. Evaluation and discussion of the spectra Comparing the spectra in figures 2 and 3 with the model calculations in figure 4 shows that the plasma vibrations in boron carbide are heavily damped. Therefore, the plasma frequency cannot be exactly determined but can only be estimated. However, the obtained fitting parameters Π = ωp/ωτ are rather accurate. Compared with Π, the accuracy of the components ωp and ωτ is much lower, but they may have the correct order of magnitude at least. Indeed, it must be obeyed that in boron carbide band type conductivity and hopping conductivity coexist, and their relative share depends on the temperature. We assume that the dynamical conductivity in the range of FIR frequencies (~1012 Hz) is preferably determined by carriers in extended states only. 7.1. Temperature dependence
The reflectivity spectra shown in figure 2 were obtained from the very same sample, and therefore the structural properties are exactly the same. Hence, the related impact on ωτ is unchanged in all spectra. A temperature-dependent influence of the velocity of carriers and the occupancy of traps in the band gap on ωτ seems possible; but it cannot be quantified reliably at present. Therefore, we assume the fitting parameter Π squared to be largely proportional to N, the density of charge carriers. The results are shown in figure 5.
m * /me = 2 .4
Based on Chauvet’s spin density, which is four times higher, m*/me ~ 10 would be obtained. These values confirm the effective masses of free holes in boron carbide as previously estimated (see [5]). 4
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J. Phys.: Condens. Matter 26 (2014) 425801
Figure 5. Squared fitting parameter Π to the reflectivity spectra in figure 2; quantifying by the Hall density of holes (Wood [27]). Insert, ωp and ωτ, components of Π.
Figure 6. Mean free path of holes l versus T. The lattice parameters a and c are indicated. Dashed line, linear fit.
In spite of the very limited accuracy of ωτ displayed in the insert in figure 5, a rough estimation of the mean free path is made. In the classical Drude–Lorentz model, the velocity of charge carriers forming the ‘gas of free electrons’ is essentially determined by m / 2v 2 = 3 / 2kT . The effective relaxation time is determined by various processes which are represented as 1/τ = 1/τ1 + 1/τ2 + 1/τ3 + …. Examples of such processes are scattering by phonons, scattering by charged or uncharged lattice defects and trapping in unoccupied gap states; all of them depending differently on various parameters, which are largely unknown in the case of boron carbide. Hence the roughly estimated mean free path
l = v ⋅ τ = v / ωτ in figure 6 based on ωτ taken from the insert in figure 5 and m*/me = 2.4 is unspecified, and indeed comparable with the lattice parameters and compatible with the concentration of defects in the order of ~1 cell−1 (figure 1(c)). The increase of l at increasing T can be explained by the trapping probability decreasing, with the carrier velocity increasing. 7.2. Dependence on the chemical composition
Re-evaluating the spectra in 7–10 was confined to those at 450 K, where the influence of the dynamical conductivity on R is most significant. In figure 7; the squares of the obtained 5
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J. Phys.: Condens. Matter 26 (2014) 425801
Figure 7. Square of the Π parameters of the plasma vibrations in BxC at 450 K. Full symbols, derived from the spectra in 7, 10; open symbols, from [26], Dashed line, sum of defects, taken from figure 1.
Figure 8. Selection of electronic transport properties of BxC versus carbon content.
fitting parameters Π are plotted versus carbon content. The FIR phonon spectra [26] with a lower accuracy (see above) yield somewhat more varying results, but agree satisfactorily. Figure 7 shows that the curve shape, consisting of two almost linear branches meeting at a maximum in the middle of the homogeneity range, is correlated with the sum of the structural defects taken from figure 1(c). The results were obtained from different samples of different chemical compositions and separate preparations; in particular containing different defect concentrations. Hence, a certain effect of ωτ2 on the pattern of data in figure 7 cannot be excluded. But they cannot be specified. Nevertheless the correlation between carrier density and defect concentration is obvious. In order to verify this correlation more generally, figure 8 compares a selection of electronic transport properties with the sum of the intrinsic defects: dc and hf (1010 Hz) conductivity
by Samara et al [29], dc conductivity at 200 K by Werheit et al [30, 31], Seebeck coefficient at 500 K by Werheit et al [32] and a selection of Seebeck coefficients at 300 K obtained from [1]. In all these cases the correlation is confirmed (further examples in [1] figure 283). An additional investigation of this correlation is shown in figure 9 comparing the electrical conductivity of BxC at 10 K [1, 32, 33] with the sum of the defects. At low temperatures, the electrical conductivity of boron carbide is nearly entirely determined by variable-range hopping [34]; hence at 10 K a measurable contribution of free carriers is largely excluded. According to Mott’s theory of variable-range hopping [35] advanced by other authors [36–39], the hopping probability varies ~exp (−2αD) (α, localization length; D, distance of localized states, immediately related to their density). The expected exponential correlation with the defect concentration is satisfactorily confirmed in figure 9. 6
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J. Phys.: Condens. Matter 26 (2014) 425801
Figure 9. Electrical conductivity of BxC versus carbon content at 10 K.
into the gap states split-off from the valence band. Their concentration is quantitatively correlated with the intrinsic structural defects in the order of one per rhombohedral cell. These originate essentially from deviations of the actual structures from B12CBC and (B11C)CBC, which are prevailing in the boron-rich and the carbon-rich section of the homogeneity range respectively. The correlation between electronic transport properties and the composition-dependent concentration of structural defects has been proved for the temperature-dependent ranges of band-type conduction and for the hopping processes as well. The results on boron carbide demonstrate that even in cases of effective masses m* clearly exceeding me the classical Drude–Lorentz model is applicable.
We note that the preceding correlations are based on the implicit assumption that one structural defect generates one gap state each time. This certainly applies for one B11C icosahedron generating one donor in structures based on B12 icosahedra, and the opposite is the case for B12 acceptors in B11C structures. In the same way, CBB creates one acceptor compared with the reference CBC. However, somewhat questionable is the attribution of one defect to B□B, where the central vacancy is proved beyond doubt (see [2]). Compared with CBC the missing central B atom implies three missing electrons (acceptors) and the exchange of the C atoms at the chain ends would cause a further deficit of two electrons in the electronic structure. Accordingly, there is a certain uncertainty with respect to the one-to-one assignment of gap states to these defects. As the content of B□B is relatively small and depends weakly on the carbon content, the uncertainty might remain in narrow limits, at least concerning the dependence on the carbon content. The correlations between the sum of the intrinsic defects and electronic transport properties shown above suggest that the energetic position of the concerning gap states ΔE = 150 meV is independent of their individual origin. This is not necessarily the case. Indeed, their energetical variation is sufficiently small, so that in the case of band-type conductivity the thermal excitation of electrons is not essentially different. At low temperatures suitable phonons are required for allowing phonon-assisted variable-range hopping. The individual identification of the gap states in the band scheme [6, 15] remains open.
Acknowledgements The authors thank Dr K A Schwetz, ESK Kempten, Germany, and Dr Krismer, Dr Fister and Mr Neisius, H C Starck, Goslar and Laufenburg, Germany, for providing boron carbide samples. References [1] Werheit H 2000 Boron compounds Numerical Data and Functional Relationships in Science and Technology (Landolt–Börnstein New Series Group III, vol 41D) ed O Madelung (Berlin: Springer) pp 1–491 [2] Werheit H and Shalamberidze S 2012 J. Phys.: Condens. Matter 24 385406 [3] Lipp A 1965 Technische Rundschau 57 nos. 14, 28 and 33 Lipp A 1966 Technische Rundschau 58 no. 7 [4] Domnich V, Reynaud S, Haber R A and Chhowalla M 2011 J. Am. Ceram. Soc. 94 3605 [5] Werheit H 2007 J. Phys.: Condens. Matter 19 186207 [6] Werheit H 2014 On microstructure and electronic properties of boron carbide Proc. of the ICACC’2014, ‘Ceramic Engineering and Science Proc.’ (CESP) Advances in Ceramic Armor X ed J C LaSalvia, vol 35 (Westerville, OH: American Ceramic Society) at press
8. Conclusion The plasma vibrations determining the low frequency region of the FIR reflectivity spectra of boron carbide are heavily damped. Apart from the large effective mass of the free holes m*/me = 2.5, this is due to the high concentration of traps formed by intrinsic gap states in the band gap above the valence band edge. The concentration of free holes in the valence band of boron carbide is due to the thermal excitation of valence electrons 7
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[22] Calandra M, Vast N and Mauri F 2004 Phys. Rev. B 69 224505 [23] Vast N, Sjakste J and Betranhandy E 2009 J. Phys.: Conf. Ser. 176 012012 [24] Bylander D M, Leinmann L and Lee S 1991 Phys. Rev. B 43 1487 [25] Werheit H, Leithe-Jasper A, Tanaka T, Rotter H W and Schwetz K A 2004 J. Solid State Chem. 177 575 [26] Kuhlmann U 1994 PhD Thesis University of Duisburg, Germany [27] Wood C 1985 AIP Conf. Proc. 140 206 [28] Chauvet O, Emin D, Forro L, Aselage T L and Zuppiroli L 1996 Phys. Rev. B 53 14450 [29] Samara G A, Tardy, H L, Venturini E L, Aselage T L and Emin D 1993 Phys. Rev. B 48 1468 [30] Kuhlmann U and Werheit H 1994 Jpn. J. Appl. Phys. Series 10 84 [31] Werheit H 1999 Boron and Boron-rich Compounds in: Electric Refractory Materials ed Y Kumashiro (New York: Marcel Dekker) p 589 [32] Werheit H, Kuhlmann U, Franz R, Winkelbauer W, Herstell B, Fister D and Neisius H 1990 AIP Conf. Proc. 231 104 [33] Werheit H and Winkelbauer W 2014 unpublished [34] Werheit H 2007 J. Phys.: Condens. Matter 19 186207 [35] Mott N F 1969 Phil. Mag. 19 835 [36] Pollak M J 1972 Non-crystalline Solids 11 1 [37] Seager C H and Pike G E 1974 Phys. Rev. B 10 1435 [38] Apsley N and Hughes H P 1974 Phil Mag. 30 963 [39] Overhof H 1976 Hopping conductivity in disordered solids Festkörperprobleme/Advances in Solid State Physics vol. XVI ed J Treusch (Berlin: Springer)
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