Home
Search
Collections
Journals
About
Contact us
My IOPscience
Effect of the variation of the electronic density of states of zirconium and tungsten on their respective thermal conductivity evolution with temperature
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 165501 (http://iopscience.iop.org/0953-8984/27/16/165501) View the table of contents for this issue, or go to the journal homepage for more
Download details: This content was downloaded by: jpcrocombette IP Address: 132.166.68.92 This content was downloaded on 10/04/2015 at 08:27
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 165501 (5pp)
doi:10.1088/0953-8984/27/16/165501
Effect of the variation of the electronic density of states of zirconium and tungsten on their respective thermal conductivity evolution with temperature Jean-Paul Crocombette, Paul Notargiacomo and M-C Marinica CEA, DEN, Service de Recherches de M´etallurgie Physique, F-91191 Gif-sur-Yvette, France E-mail:
[email protected] Received 27 October 2014, revised 3 March 2015 Accepted for publication 6 March 2015 Published 30 March 2015 Abstract
The thermal conductivity of zirconium and tungsten above 500 K is calculated with atomistic simulations using a combination of empirical potentials molecular dynamics and density functional theory calculations. The thermal conductivity is calculated in the framework of Kubo–Greenwood theory. The obtained values are in quantitative agreement with experiments. The fact that the conductivity of Zr increases with temperature while that of tungsten is essentially constant is reproduced by the calculations. The evolution with temperature of the electronic density of states of these two pseudo-gap metals proves to explain the observed variations of the conductivity. Keywords: thermal conductivity, zirconium, tungsten, DFT calculations, density of states (Some figures may appear in colour only in the online journal)
Zirconium and tungsten are two examples of such pseudogap materials. However, though chemically rather close, they behave differently regarding thermal conductivity. It slightly decreases in tungsten [2], while it strongly rises in zirconium [3]. In this paper we calculate with numerical simulations from first principles the thermal conductivities of these two metals. We reproduce quantitatively the experimental data, especially the difference in variation with temperature between zirconium and tungsten. From these simulations we are able to explain the origin of these variations. We especially highlight the role of the variation of the electronic DOS with temperature for the conductivity of metallic materials, an effect vastly overlooked in literature. Section 2 is devoted to the presentation of technicalities of our work; the results are presented in section 3 and discussed in section 4.
1. Introduction
Thermal conductivity is an important property of materials relevant for many technological applications. From a fundamental point of view the mechanisms of heat conduction are known: heat is transported by vibrations in insulators while in metals electrons are the main contributors. Nevertheless, some things remain to be clarified especially for what concerns the variation of thermal conductivity with temperature in metals. Moreover, this quantity proves difficult to calculate for such materials, as complex electronic structure calculations are needed. Qualitative arguments predict that thermal conductivity of metals should be constant or slightly decreasing at high temperature. However, some metals exhibit an increase of thermal conductivity with temperature, especially the ones exhibiting a pseudo-gap, i.e. a low lying minimum in the electronic density of states (DOS) at the Fermi level. The qualitative explanation of this behaviour relates the positive curvature of the DOS close the Fermi level to the increase of conductivity [1]. 0953-8984/15/165501+05$33.00
2. Technicalities
Part of the technicalities of the present work are identical to the ones used in our previous study on zirconium carbide [4], 1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 165501
Jean-Paul Crocombette et al
especially the details of electronic structure calculations. Nevertheless, we briefly repeat them for clarity. We restricted our calculations to temperatures fairly higher than the Debye temperature of both metals (see below). Doing so, we can safely disregard any vibrational quantum effects. We calculate both the vibrational (phonon) and electronic part of the thermal conductivity. As far as electronic heat conduction is concerned, electron transport coefficients are related to the kinetic Onsager coefficients for electrical and heat transport Lij which relate the electrical (je ) and heat (jq ) and a temperature current densities to an electric field (E) gradient according to L12 ∇T 1 eL11 E − , (1) je = e T 1 jq = 2 e
L22 ∇T eL12 E − T
.
Electronic thermal conductivity is then L212 1 κel. = 2 . L22 − e T L11
Figure 1. Calculated phonon DOS in zirconium using EAM potential number 2 by Mendelev and Ackland [11].
(2)
a perfectly periodic crystal. Thermal conductivity is the limit at zero frequency of equation (3). To obtain the sets of atomic positions, we use constant temperature molecular dynamics (MD) simulations with empirical potentials. Indeed, using ab initio MD would require an enormous computational power, while good empirical potentials can provide accurate atomic positions for a negligible cost. For both materials we use embedded atom model (EAM) potentials. Mendelev and Ackland [11] have designed three different EAM potentials for zirconium. To choose between these different versions we have calculated the cell parameters and thermal expansions of zirconium for various temperatures. Indeed thermal expansion depends on anharmonic phonon– phonon interactions. The potential with the best thermal expansion is therefore the most suited to obtain accurate atomic positions at high temperatures. We eventually chose Mendelev’s empirical potential no. 2 as it gives the cell parameters of zirconium with an accuracy better than 0.1% at any temperature between 500 and 2000 K. The ability of this potential to reproduce vibrations in zirconium has been further checked by the calculation of the phonon DOS (see figure 1). This DOS is quite close to the ones obtained with second moment potentials [12] or force constants extracted from experiments [13]. Mendelev potential performs however better than these previous models as the high frequency limit of optical modes is only slightly larger than 5 THz as in experiments, while it reaches 6 THz in the previous models. Finally the Debye temperature of this potential, extracted from the DOS is 268 K in very good agreement with the 265 K experimental value [14]. Note that a phase transition exists at 1130 K in zirconium. All calculations below this temperature were thus performed in the low temperature close packed hexagonal structure, while high temperature calculations are done for the body centred cubic (bcc) structure. Tungsten has a bcc structure for all temperatures. Fewer potentials exist to describe this material. We use the recently published potential by Marinica et al [15]. The cell
(3)
Various electron scattering mechanisms exist which may contribute to the thermal or electrical resistance: electron– electron and electron–phonon interactions, as well as scattering by point or extended defects. As we are interested in perfect materials we disregard the latter, while we also neglect the electron–electron interaction to focus on electron–phonon scattering. The accuracy of the obtained results justifies this assumption (see below). Onsager coefficients are evaluated using Kubo–Greenwood theory [5, 6]. They are calculated with an electronic structure code for various atomic positions to obtain an average value. In the present work we use Abinit [7]. The electronic calculation is done in the generalized gradient approximation (GGA) of the density functional theory (DFT) with the exchange-correlation energy functional of Perdew– Burke–Ernzerhof [8] using the projector augmented wave formalism [9]. The Onsager coefficients are extracted from the ground state wave functions using the following frequency dependent formula [10]: 2π Lij (ω) = (−1)i+j wk (|n,k |∇α |m,k |2 3ω n,m,k,α ×(εm,k − εF )i−1 (εn,k − εF )j −1 ×(f (εm,k ) − f (εn,k ))δ(εn,k − εm,k − ω))
(4)
In this expression (written in atomic units), ω is the frequency; k runs over the k-points in the Brillouin zone with weights wk ; is the volume of the simulation cell; α runs over the three directions of space; n and m run over electronic states of wave functions n,k and m,k and energies εn,k and εm,k ; and f is the Fermi–Dirac occupation factor. Electron phonon coupling enters in the values of the velocity operators present in the bra–ket term. The highest the temperature, the more the atomic positions deviate from perfect lattice sites and the smaller these terms are due to the change in wavefunctions from the ones of 2
J. Phys.: Condens. Matter 27 (2015) 165501
Jean-Paul Crocombette et al
values for the conductivity are within 2% of the ones obtained for the grid restricted to the irreducible Brillouin zone which fully justifies this procedure. The thermal conductivity of each position is therefore calculated with 3% accuracy. For some selected configurations we calculated the DOS using the tetrahedron method [19, 20]. This method allows direct comparisons of different temperatures as it does not rely on temperature dependent smearing. The phonon part of the thermal conductivity is estimated within MD with the empirical potentials described above, using the direct method: a heat flux is imposed in a MD simulation box by heating (respectively cooling) one side (the other side) of the box. For tungsten the box is cubic with a size ranging from 7.91 to 8.02 nm depending on temperature. For hcp zirconium the heat flux is applied along the (1 0 1¯ 0) direction in a tetragonal box of 11.2 nm in the flux direction and about 10 nm in the perpendicular directions. Finally, for bcc Zr the size of the box is 9.1 nm. After a few tens picosecond a steady state temperature gradient is achieved. The thermal conductivity is then deduced directly from the ratio of the heat flux and the temperature gradient within the box. The heat flux was such that the difference of temperature between the two edges of the box was of a few tens kelvin.
Figure 2. Calculated phonon DOS in tungsten using the EAM potential by Marinica et al [15] in tungsten. Inset: corresponding dispersion curves compared to experiments [16].
parameter of tungsten is reproduced within less than 0.6% at all temperatures between 500 and 2500 K. The phonon DOS and dispersion curves are given in figure 2 together with experimental values [16]. For low frequencies, the agreement with experiments is very good, as can be seen in the Debye temperature: the calculated value (399 K) being essentially equal to the experimental one (400 K [17]). The higher frequencies are less perfectly described but the agreement with experiments remains reasonable. For hcp zirconium a 384 atom box, built from a 4 × 4 × 6 repetition of the tetragonal 4 atom unit cell is considered. For bcc Zr and W the box contains 250 atoms and is a 5 × 5 × 5 repetition of the 2 atom conventional unit cell. Boxes are equilibrated for 100 000 MD time steps with the empirical potentials and 5 atomic configurations regularly sampled from the MD run are transferred to Abinit for tungsten. The thermal conductivity is calculated for each box and the overall conductivity is obtained as the average over the 5 positions for each temperature. In view of the small spread of the conductivity values we only calculated three positions for the calculations on zirconium (see the error bars on figure 3). Convergence of the Abinit results must then be achieved with regards the k point sampling. We eventually found that it was necessary to sample rather finely the IBZ of the supercell. Indeed a 5 × 5 × 5 grid is needed for tungsten and 3 × 3 × 3 grids are needed for both phases of zirconium to achieve a convergence of the order of 1%. Moreover, we use the following trick by Alfe et al [18]: as atomic positions are taken from non-zero temperature MD runs, they have no symmetry. In principle, one should then consider all the k-points in the Brillouin zone with identical weights. In practice the summation is performed on k-points of the irreducible Brillouin zone built according to the symmetries of the perfect cubic positions with the corresponding variable weights. This allows decreasing the number of k-points for a given grid. As a check we performed for some configurations the calculations using the complete k point grid. The obtained
3. Results
The calculated electronic and phononic part of the thermal conductivity of Zr and W are given in figure 3. The experimental values of the thermal conductivity of the two materials are plotted for comparison. These values come from analytical recommended values derived from experiments [2, 3]. The first thing to note is that the thermal conductivity of zirconium and tungsten are quantitatively reproduced (note that W conductivity is about five times higher than the one of Zr). Second the difference in behaviour between the two materials is clearly reproduced. Indeed, both in simulations and experiments, the conductivity of tungsten tends to very slowly decrease at high temperature. At the opposite the conductivity of Zr exhibits a steady rise with increasing temperature. The agreement between experiments and simulations is however not perfect. For tungsten, calculations underestimate the conductivity at low temperature. Quantum effects may be responsible for this discrepancy. The Debye temperature of tungsten (400 K) is indeed not too far from the lowest temperature we consider (500 K). At high temperature, the calculated conductivities slightly overestimate the experiments in the two materials. This overestimation should come from the neglect of the additional sources of thermal resistances, presumably mainly the resistance induced by defects, impurities or polycrystallinity. As expected in a metal, the vibrational thermal conductivity proves negligible compared to the electronic component, except for the zirconium at the lowest temperature where it amounts to 17% of the total conductivity. The variations of thermal conductivity with temperatures for Zr and W come completely from their electronic parts. We 3
J. Phys.: Condens. Matter 27 (2015) 165501
Jean-Paul Crocombette et al
Figure 3. Calculated thermal conductivity (circles) of tungsten (left) and zirconium (right) as a function of temperature, compared to experiments (dotted line [2, 3]). The phonon and electronic parts of the thermal conductivity are indicated with rectangles and diamonds respectively.
thermal conductivity as this number of states directly enters equation (4). Figure 4 gives the variation of the DOS obtained with the tetrahedron method for the atomic configuration for which the conductivity is closest to the average value. The DOS are averaged over an interval of 5 × 10−3 Ha. One can see that the shape of the DOS varies with temperature especially for zirconium. In this material the DOS rises steadily which can be directly associated with the rise of the conductivity. In tungsten apart from a low temperature kick, which can be tracked in the variation of κ between 500 and 1000 K, the DOS remains roughly constant, and so does κ. The thermal conductivity therefore varies as the average value of the DOS at the Fermi level. This directly relates to the fact that the number of heat conducting states is proportional to the value of the DOS at the Fermi level (see equation (4)). This study confirms the close relation between the variation of the DOS with temperature and the one of thermal conductivity which was first observed in our previous paper on zirconium carbide [4]. The effect of the pseudo-gap is therefore not the expected one. The ambipolar contribution to the rise of the conductivity, expected if the DOS were constant, appears negligible in all cases we considered. Conversely, when the pseudo-gap that exists at low temperature is filled at high temperature, it induces an increase in conductivity. Finally, it is worth pointing out that these arguments on DOS change can only be used to discuss the variation of the thermal conductivity for a given material. The DOS values can, naturally, not be used to compare directly different materials. Indeed the DOS of W is 2 to 3 times lower than the one of Zr while its thermal conductivity is about five times larger (see figure 5). The main factor for the absolute value of the conductivity remains the velocity terms appearing in the brackets of equation (5) which express the intensity of the electron–phonon interactions.
shall therefore focus on the electronic part of the thermal conductivity in the following discussion. 4. Discussion
The increase of the thermal conductivity of semi-metals or metals exhibiting a pseudo-gap is commonly explained by the curvature of the DOS at the Fermi level [1]. This is the so-called semi-metallic effect or ambipolar contribution to thermal conductivity [21, 22]. It relies on the increase of the Lorenz number with temperature. This explanation assumes that the DOS is constant with temperature. In the present case we found that this effect plays little role. First for tungsten, the curvature of the DOS is too spread in energy. Indeed to have an effect the width of the well in the DOS should be of the order of magnitude of the interval in energy sampled by equation (4), i.e. a few kB T (kB T = 3 × 10−3 Ha at 1000 K). Thus, even if tungsten is indeed a pseudo-gap material, the curvature of the DOS at the Fermi level is too smooth to induce an ambipolar contribution to the thermal conductivity. In Zr the pseudo-gap exists only in the hcp phase (see the inset of figure 4). In this phase at 0 K, the well in the DOS is very deep and shallow close to Fermi level. Would the DOS be constant with temperature, its shape would induce some rise in thermal conductivity. The part of the rise of the thermal conductivity due to the curvature of the DOS is equal to the variation of the Lorenz number. We found that this number increases by 4% between 500 and 1000 K. However, the increase of the conductivity between these two temperatures is much larger (+42%). Therefore, only 10% of the increase in conductivity of Zr between 500 and 1000 K can be attributed to the ambipolar contribution. Moreover, the pseudo-gap completely disappears in Zr bcc phase while the thermal conductivity continues to increase. Both points indicate that the main source of increase of the conductivity is not the curvature of the DOS. The rest of the increase comes from a different source: namely the variation of the DOS itself with temperature. Indeed if the number of electronic states close to the Fermi level vary with temperature, so will the
5. Conclusion
Using a combination of empirical potential molecular dynamics and DFT electronic structure calculations we 4
J. Phys.: Condens. Matter 27 (2015) 165501
Jean-Paul Crocombette et al
Figure 4. Electronic DOS close to the Fermi level at various temperatures: tungsten (left) and zirconium (right); kB T = 3 × 10−3 Ha at 1000 K. Insets: complete valence DOS.
[6] Greenwood D A 1958 The Boltzmann equation in the theory of electrical conduction in metals Proc. Phys. Soc. Lond. 71 585–96 [7] Gonze X et al 2009 ABINIT: first-principles approach to material and nanosystem properties Comput. Phys. Commun. 180 2582–615 [8] Perdew J P, Burke K and Ernzerhof M 1996 Generalized gradient approximation made simple Phys. Rev. Lett. 77 3865 [9] Torrent M et al 2008 Implementation of the projector augmented-wave method in the ABINIT code: application to the study of iron under pressure Comput. Mater. Sci. 42 337–51 [10] Mazevet S et al 2010 Calculations of the transport properties within the PAW formalism High Energy Density Phys. 6 84–8 [11] Mendelev M I and Ackland G J 2007 Development of an interatomic potential for the simulation of phase transformations in zirconium Phil. Mag. Lett. 87 349–59 [12] Willaime F and Massobrio C 1991 Development of an N-body interatomic potential for hcp and bcc zirconium Phys. Rev. B 43 11653–65 [13] Stassis C et al 1978 Temperature dependence of the normal vibrational modes of hcp Zr Phys. Rev. B 18 2632–42 [14] Masuda K, Amada N and Terakura K 1984 J. Phys. F: Met. Phys. 14 47 [15] Marinica M C et al 2013 Interatomic potentials for modelling radiation defects and dislocations in tungsten J. Phys.: Condens. Matter 25 395502 [16] Larose A and Brockhouse B N 1976 Lattice-vibrations in tungsten at 22 ◦ C studied by neutron-scattering Can. J. Phys. 54 1819–23 [17] Kittel C 2004 Introduction to Solid State Physics (New York: Wiley) [18] Alfe D, Pozzo M and Desjarlais M P 2012 Lattice electrical resistivity of magnetic bcc iron from first-principles calculations Phys. Rev. B 85 024102 [19] Jepsen O and Andersen O K 1971 Electronic structure of hcp ytterbium Solid State Commun. 9 1763 [20] Bl¨ochl P, Jepsen O and Andersen O K 1994 Improved tetrahedron method for Brillouin zone integrations Phys. Rev. B 49 16223 [21] Gallo C F et al 1962 Bipolar electronic thermal conductivity in semimetals J. Appl. Phys. 33 3144 [22] Bethin J and Williams W 1977 Ambipolar diffusion contribution to high-temperature thermal conductivity of titanium carbide J. Am. Ceram. Soc. 60 424
Figure 5. Electronic DOS close to the Fermi level at 1000 K, comparison of Zr and W; kB T = 3 × 10−3 Ha at 1000 K.
have reproduced quantitatively the evolution of the thermal conductivity of zirconium and tungsten. Specifically, the rise of the conductivity observed for Zr and not for W is reproduced. These various variations have proven to be directly related to the variation of the electronic DOS of the materials with temperature, an effect which is commonly overlooked in the discussion of the evolution of the conductivities of metals with temperature. References [1] Tye R 1969 Thermal Conductivity vol 1 (London: Academic) [2] Hust J G 1976 Thermal conductivity and electrical resistivity standard reference materials: tungsten (4 to 3000 K) High Temp.-High Press. 8 377–90 [3] Fink J K and Leibowitz L 1995 Thermal conductivity of zirconium J. Nucl. Mater. 226 44–50 [4] Crocombette J P 2013 Origins of the high temperature increase of the thermal conductivity of transition metal carbides from atomistic simulations J. Phys.: Condens. Matter 25 505501 [5] Kubo R 1957 Statistical–mechanical theory of irreversible processes: I. General theory and simple applications to magnetic and conduction problems J. Phys. Soc. Japan 12 570–86 5
