Anderson transition: a novel route to high thermoelectric performance

Discovered exactly 200 years ago in 1821, 1 thermoelectricity is nowadays of global interest as it allows to directly interconvert thermal and electrical energy via the Seebeck/Peltier eﬀect, which could be ex-ploited to enhance energy eﬃciency 2,3 . In their seminal work 4 , Mahan and Sofo mathematically derived the conditions for ’the best thermoelectric’ − a delta-distribution-shaped electronic transport function, where charge carriers contribute to transport only in an inﬁnitely narrow energy interval. So far, however, only approximations to this concept were expected to really exist in nature 4,5 . Here, we propose as a physical realisation of this scenario the Anderson transition in an impurity band, i.e. the transition from Anderson-localised to extended quantum states 6 . We obtained a signiﬁcant enhancement and dramatic change of the thermoelectric properties from p -type to n -type in the stoichiometric Heusler compound Fe 2 VAl, which we assign to a narrow region of delocalised electrons in the energy spectrum near the Fermi energy. We achieved this through an innovative approach of driving the Anderson transition via continuous disorder tuning: variable amounts

Discovered exactly 200 years ago in 1821, 1 thermoelectricity is nowadays of global interest as it allows to directly interconvert thermal and electrical energy via the Seebeck/Peltier effect, which could be exploited to enhance energy efficiency 2,3 . In their seminal work 4 , Mahan and Sofo mathematically derived the conditions for 'the best thermoelectric' − a delta-distribution-shaped electronic transport function, where charge carriers contribute to transport only in an infinitely narrow energy interval. So far, however, only approximations to this concept were expected to really exist in nature 4,5 . Here, we propose as a physical realisation of this scenario the Anderson transition in an impurity band, i.e. the transition from Anderson-localised to extended quantum states 6 . We obtained a significant enhancement and dramatic change of the thermoelectric properties from p-type to n-type in the stoichiometric Heusler compound Fe 2 VAl, which we assign to a narrow region of delocalised electrons in the energy spectrum near the Fermi energy. We achieved this through an innovative approach of driving the Anderson transition via continuous disorder tuning: variable amounts of atomic defects are induced in a controlled fashion by thermal quenching from high temperatures (950 − 1380 • C). Based on our experimental electronic transport and magnetisation results, supported by Monte-Carlo and density functional theory calculations, we demonstrate a universal enhancement strategy towards colossal thermoelectric performance that is applicable to diverse material classes.
Thermoelectric (TE) devices are capable of converting 1 wasted heat into useful electrical energy or act as Peltier 2 coolers. Facing an increasing worldwide demand for effi-3 cient energy utilisation, the immense diversity of potential 4 technologocial applications has sparked great interest 2,3 .

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Still, TE devices are currently restrained in their applica-6 bility due to their limited efficiency. The dimensionless 7 figure of merit ZT = S 2 σT /(κ e + κ ph ), which is closely re-8 lated to the conversion efficiency, comprises three material-9 dependent parameters. These are the thermopower S, the 10 electrical conductivity σ and the thermal conductivity κ, 11 consisting of a contribution from electrons κ e and phonons  part of ZT is a much more formidable, yet necessary task 16 and new exotic concepts for enhancement are required. 17 In 1996, Mahan and Sofo mathematically identified 'the 18 best thermoelectric' as an ideal system, characterised by 19 * Corresponding author an infinitely narrow delta-distribution-shaped transport 20 function Σ(E) 4 .

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Here, we propose that this seemingly unrealisable math-22 ematical concept becomes actually realised in real mate-23 rials at the Anderson transition in an impurity band, as 24 predicted theoretically 9 . As sketched in Fig.1, such a tran-25 sition occurs when the number of randomly distributed 26 impurities increases above a critical value x c , known as 27 quantum percolation. Below x c , all impurity states are 28 Anderson-localised due to disorder 10 . A singularity of the 29 transport function occurs at x c when an infinitesimally 30 small region of states in the density of states (DOS) be-31 comes delocalised. This was explained by Mott in 1967 32 through the concept of 'mobility edges', which are two crit-33 ical energies E c1,2 that appear at the centre of an impurity 34 band, separating localised states in the band tails from 35 delocalised, extended states in the centre 11 . Far above 36 x c , E c1 and E c2 shift towards the band edges, eventually 37 delocalising all impurity states.

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To illustrate the nature of localised Fe/V antisite elec-75 tronic states near the Fermi energy E F , we calculated the 76 spin-polarised DOS by making use of the exact muffin-tin 77 orbital coherent potential approximation method (EMTO-78 CPA). This method allows to calculate the DOS of a single-79 impurity embedded in an infinitely large and ordered ef-80 fective medium, mimicking the electronic and structural 81 properties of an alloy in the dilute limit of antisite concen-82 tration x AS → 0. Figs.2c,d show the occurence of sharp, 83 hydrogen-like impurity states near E F for both Fe V and 84 V Fe defects, as compared to the fully ordered compound. 85 Similar results are obtained for Fe Al impurity states (see 86 Extended Data Fig.2). Furthermore, the spin degeneracy 87   due to the delocalisation of antisite electrons with increas-141 ing T quench . Indeed, the residual resistivity ρ 0 decreases 142 by an order of magnitude with increasing T quench , which 143 also manifests itself by a substantial increase of the Hall 144 carrier concentration (see Extended Data Fig.8a), over-145 compensating the increased number of scattering centres 146 due to disorder. Furthermore, the appearance of metallic 147 transport goes hand in hand with the development of a 148 local maximum in ρ(T ) at a temperature T ρ,max , which 149 shifts to higher temperatures as T quench increases.    residual resistivity ρ 0 at 4 K and T ρ,max as a function of 200 T quench . With the spontaneous appearance of T ρ,max > 0 201 around T * quench = 1000 − 1050 • C, ρ 0 simultaneously shows 202 a pronounced kink. In Fig.5b, we show the peak values of 203 the thermopower S max as well as the residual conductivity 204 σ 0 . Again around T * quench , S(T ) displays a sign reversal 205 and σ 0 deviates from a linear scaling behaviour. These 206 anomalies, together with the monotonous increase of T ρ,max 207 and decrease of ρ 0 , clearly indicate the continous delocali-208 sation of the impurity band (IB), as sketched in the insets 209 of Fig.5b. Fe 2 VAl is already 7.6 mW/mK 2 , which is an enhancement 218 by an order of magnitude compared to the as-cast sample 219 and 40% higher than the best PF for optimised rigid-band 220 doping in this system 32 . Considering that the total ther-221 mopower S tot and total conductivity σ tot in a material with 222 multiple electronic bands can be written as with S i , σ i being the respective single-band contributions, 224 we can estimate the contribution of the impurity band 225 which led to the dramatic change in TE transport. Here, 226 the index (i = {pristine, impurity}) refers to contributions 227 from the pristine band structure and the delocalised impu-228 rity band. Bearing in mind that the transport properties 229 of the as-cast sample with purely localised impurity states 230 is mainly dominated by the pristine band structure, we can 231 calculate the additional delocalised impurity contribution 232 to the high-temperature-quenched samples S imp , σ imp from 233 our measured data by solving the system of Eqs.2,3. The 234 contribution of the impurity band to the total measured 235 power factor of the 1380 • C-quenched sample is plotted by 236 red squares in Fig.5c. An extremely large PF of more than 237 18 mW/mK 2 at 400 K is found for a stoichiometric sample 238 not yet optimally doped, exceeding that of the pristine 239 compound by a factor of 30. It is noteworthy to mention 240 that κ ph was also reduced by a factor of 2−3 due to the dis-241 order introduced by quenching (see Extended Data Fig.8b). 242 Consequently, this means that the disorder induced by 243 thermal quenching could be a strategy that can enhance 244 all thermoelectric properties at the same time, which is not 245 achievable by conventional doping strategies. We expect 246 that ZT should be further greatly enhanced by optimis-247 ing the position of E F and by reducing the background 248 DOS of Fe 2 VAl, which can be achieved by appropriate 249 co-substitutions with e.g. Si and Ta 30,32 .