High entropy alloys
High-entropy alloys (HEAs) are substances that are constructed with equal or nearly equal quantities of five or more metals. These alloys are currently the focus of significant attention in materials science and engineering because they have potentially desirable properties. Prior to the existence of these substances, alloys consisted of only one or two base metals. For example, additional elements can be added to iron to improve its properties, thereby creating an iron based alloy. Hence, high entropy alloys are a novel material.
Furthermore, research indicates that some HEAs have considerably better strength-to-weight ratios, with a higher degree of fracture resistance, tensile strength, as well as corrosion and oxidation resistance than conventional alloys. Although HEAs have existed since before 2004, research substantially accelerated in the 2010s.
Although HEAs were described as early as 1981 and 1996, significant research interest did not develop until after independent 2004 papers by Jien-Wei Yeh and Brian Cantor, with Yeh's first paper on the topic published 2 months sooner. Yeh also coined the term "high-entropy alloy" when he attributed the high configurational entropy as the mechanism stabilizing the solid solution phase. Cantor, not knowing of Yeh's work, did not describe his alloy as a "high-entropy" alloy, but the base alloy he developed, equiatomic FeCrMnNiCo, has been the subject of considerable work in the field.
Before the classification of high entropy alloys and multi-component systems, nuclear science had highlighted a system that can now be classified a high entropy alloy: within nuclear fuels Mo-Pd-Rh-Ru-Tc particles form at grain boundaries and at fission gas bubbles. Understanding the behavior of these '5 metal particles' was of specific interest to the medical industry as Tc-99m is an important medical imaging isotope.
There is no universally agreed-upon definition of a HEA. Yeh originally defined HEAs as alloys containing at least 5 elements with concentrations between 5 and 35 atomic percent. Later research however, suggested that this definition could be expanded. Otto et al. suggested that only alloys that form a solid solution with no intermetallic phases should be considered true high-entropy alloys, as the formation of ordered phases decreases the entropy of the system. Some authors have described 4-component alloys as high-entropy alloys while others have suggested that alloys meeting the other requirements of HEAs, but with only 2–4 elements or a mixing entropy between R and 1.5R should be considered "medium-entropy" alloys.
In conventional alloy design, one primary element such as iron, copper, or aluminum is chosen for its properties. Then, small amounts of additional elements are added to improve or add properties. Even among binary alloy systems, there are few common cases of both elements being used in nearly-equal proportions such as Pb-Sn solders. Therefore, much is known from experimental results about phases near the edges of binary phase diagrams and the corners of ternary phase diagrams and much less is known about phases near the centers. In higher-order (4+ components) systems that cannot be easily represented on a 2-dimensional phase diagram, virtually nothing is known.
Gibbs' phase rule, , can be used to determine an upper bound on the number of phases that will form in an equilibrium system. In his 2004 paper, Cantor created a 20-component alloy containing 5 at% of Mn, Cr, Fe, Co, Ni, Cu, Ag, W, Mo, Nb, Al, Cd, Sn, Pb, Bi, Zn, Ge, Si, Sb, and Mg. At constant pressure, the phase rule would allow for up to 21 phases at equilibrium, but far fewer actually formed. The predominant phase was a face-centered cubic solid solution phase, containing mainly Fe, Ni, Cr, Co, and Mn. From that result, the FeCrMnNiCo alloy, which forms only a solid solution phase, was developed.
The Hume-Rothery rules have historically been applied to determine whether a mixture will form a solid solution. Research into high-entropy alloys has found that in multi-component systems, these rules tend to be relaxed slightly. In particular, the rule that solvent and solute elements must have the same crystal structure does not seem to apply, as Fe, Ni, Cr, Co, and Mn have 4 different crystal structures as pure elements (and when the elements are present in equal concentrations, there can be no meaningful distinction between "solvent" and "solute" elements).
The multi-component alloys Yeh developed also consisted mostly or entirely of solid solution phases, contrary to what had been expected from earlier work in multi-component systems, primarily in the field of metallic glasses. Yeh attributed this result to the high configurational, or mixing, entropy of a random solid solution containing numerous elements. Because , and the phase with the lowest Gibbs free energy of formation (ΔG) will be the phase formed at equilibrium, increasing ΔS (entropy) will increase the likelihood of a phase being stable. The mixing entropy for a random ideal solid solution can be calculated by:
where R is the ideal gas constant, N is the number of components, and ci is the atomic fraction of component i. From this it can be seen that alloys in which the components are present in equal proportions will have the highest entropy, and adding additional elements will increase the entropy. A 5 component, equiatomic alloy will have a mixing entropy of 1.61R.
|∆Hmix||> -10 and < 5 kJ/mol|
|VEC||≥ 8 for fcc, <6.87 for bcc|
However entropy alone is not sufficient to stabilize the solid solution phase in every system. The enthalpy of mixing (ΔH), must also be taken into account. This can be calculated using:
where is the binary enthalpy of mixing for A and B. Zhang et al. found, empirically, that in order to form a complete solid solution, ΔHmix should be between -10 and 5 kJ/mol. In addition, Otto et al. found that if the alloy contains any pair of elements that tend to form ordered compounds in their binary system, a multi-component alloy containing them is also likely to form ordered compounds.
Both of the thermodynamic parameters can be combined into a single, unitless parameter Ω:
where Tm is the average melting point of the elements in the alloy. Ω should be greater than or equal to 1.1 to promote solid solution development.
The atomic radii of the components must also be similar in order to form a solid solution. Zhang et al. proposed a parameter δ representing the difference in atomic radii:
For those alloys that do form solid solutions, an additional empirical parameter has been proposed to predict the crystal structure that will form. If the average valence electron concentration (VEC) of the alloy is ≥8, the alloy will form a face-centered cubic (fcc) lattice. If the average VEC is <6.87, it will form a body-centered cubic (bcc) lattice. For values in between, it will form a mixture of fcc and bcc. VEC has also been used to predict the formation of σ-phase intermetallics (which are generally brittle and undesirable) in chromium and vanadium-containing HEAs.
High-entropy alloys are mostly produced using distinct methods that depend on the initial phase - starting either from a liquid, solid, or gas state.
- Most HEAs have been produced using liquid-phase methods include arc melting, induction melting, and Bridgman solidification.
- Solid-state processing is generally done by mechanical alloying using a high-energy ball mill. This method produces powders that can then be processed using conventional powder metallurgy methods or spark plasma sintering. This method allows for alloys to be produced that would be difficult or impossible to produce using casting, such as AlLiMgScTi, in which the melting points of the constituent elements has a range of nearly 1500 °C.
- Gas-phase processing includes processes such as sputtering or molecular beam epitaxy (MBE), which can be used to carefully control different elemental compositions to get high-entropy metallic or ceramic films.
Modeling and simulationEdit
The atomic-scale complexity presents additional challenges to computational modelling of high-entropy alloys. Thermodynamic modelling using the CALPHAD method requires extrapolating from binary and ternary systems. Most commercial thermodynamic databases are designed for, and may only be valid for, alloys consisting primarily of a single element. Thus, they require experimental verification or additional ab initio calculations such as density functional theory (DFT). However, DFT modeling of complex, random alloys has its own challenges, as the method requires defining a fixed-size cell, which can introduce non-random periodicity. This is commonly overcome using the method of "special quasirandom structures," designed to most closely approximate the radial distribution function of a random system, combined with the Vienna Ab-initio Simulation Package. Using this method, it has been shown that results of a 4-component equiatomic alloy begins to converge with a cell as small as 24 atoms. The exact muffin-tin orbital method with the coherent potential approximation has also been employed to model HEAs. Other techniques include the 'multiple randomly populated supercell' approach, which better describes the random population of a true solid solution (although is far more computationally demanding). This method has also been used to model glassy/amorphous (including bulk metallic glasses) systems without a crystal lattice.
Further, modeling techniques are being used to suggest new HEAs for targeted applications. The use of modeling techniques in this 'combinatorial explosion' is necessary for targeted and rapid HEA discovery and application.
Simulations have highlighted the preference for local ordering in some high entropy alloys and, when the enthalpies of formation are combined with terms for configurational entropy, transition temperatures between order and disorder can be estimated. - allowing one to understand when effects like age hardening and degradation of an alloy's mechanical properties may be an issue.
Properties and potential usesEdit
The crystal structure of HEAs has been found to be the dominant factor in determining the mechanical properties. bcc HEAs typically have high yield strength and low ductility and vice versa for fcc HEAs. Some alloys have been particularly noted for their exceptional mechanical properties. A refractory alloy, VNbMoTaW maintains a high yield strength (>600 MPa (87 ksi)) even at a temperature of 1,400 °C (2,550 °F), significantly outperforming conventional superalloys such as Inconel 718. However, room temperature ductility is poor, less is known about other important high temperature properties such as creep resistance, and the density of the alloy is higher than conventional nickel-based superalloys.
CoCrFeMnNi has been found to have exceptional low-temperature mechanical properties and high fracture toughness, with both ductility and yield strength increasing as the test temperature was reduced from room temperature to 77 K (−321.1 °F). This was attributed to the onset of nanoscale twin boundary formation, an additional deformation mechanism that was not in effect at higher temperatures. As such, it may have applications as a structural material in low-temperature applications or, because of its high toughness, as an energy-absorbing material. However, later research showed that lower-entropy alloys with fewer elements or non-equiatomic compositions may have higher strength or higher toughness. No ductile to brittle transition was observed in the bcc AlCoCrFeNi alloy in tests as low as 77 K.
Al0.5CoCrCuFeNi was found to have a high fatigue life and endurance limit, possibly exceeding some conventional steel and titanium alloys. But there was significant variability in the results, suggesting the material is very sensitive to defects introduced during manufacturing such as aluminum oxide particles and microcracks.
A single-phase nanocrystalline Al20Li20Mg10Sc20Ti30 alloy was developed with a density of 2.67 g cm−3 and microhardness of 4.9 – 5.8 GPa, which would give it an estimated strength-to-weight ratio comparable to ceramic materials such as silicon carbide, though the high cost of scandium limits the possible uses.
Rather than bulk HEAs, small-scale HEA samples (e.g. NbTaMoW micro-pillars) exhibit extraordinarily high yield strengths of 4-10 GPa —one order of magnitude higher than that of its bulk form—and their ductility is considerably improved. Additionally, such HEA films show substantially enhanced stability for high-temperature, long-duration conditions (at 1,100 °C for 3 days). Small-scale HEAs combining these properties represent a new class of materials in small-dimension devices potentially for high-stress and high-temperature applications.
Electrical and magneticEdit
CoCrCuFeNi is an fcc alloy that was found to be paramagnetic. But upon adding titanium, it forms a complex microstructure consisting of fcc solid solution, amorphous regions and nanoparticles of Laves phase, resulting in superparamagnetic behavior. High magnetic coercivity has been measured in a BiFeCoNiMn alloy. Superconductivity was observed in TaNbHfZrTi alloys, with transition temperatures between 5.0 and 7.3 K.
The high concentrations of multiple elements leads to slow diffusion. The activation energy for diffusion was found to be higher for several elements in CoCrFeMnNi than in pure metals and stainless steels, leading to lower diffusion coefficients. Some equiatomic multicomponent alloys have also been reported to show good resistance to damage by energetic radiation .
- Wang, Shaoqing (13 December 2013). "Atomic Structure Modeling of Multi-Principal-Element Alloys by the Principle of Maximum Entropy". Entropy. 15 (12): 5536–5548. Bibcode:2013Entrp..15.5536W. doi:10.3390/e15125536.
- Tsai, Ming-Hung; Yeh, Jien-Wei (30 April 2014). "High-Entropy Alloys: A Critical Review". Materials Research Letters. 2 (3): 107–123. doi:10.1080/21663831.2014.912690.
- Lavine, Marc S. (2014). "A metal alloy that is stronger when cold". Science. 345: 1131. Bibcode:2014Sci...345Q1131L. doi:10.1126/science.345.6201.1131-b. Retrieved 9 January 2015.
- Shipman, Matt (10 December 2014). "New 'high-entropy' alloy is as light as aluminum, as strong as titanium alloys". Phys.org. Retrieved 9 January 2015.
- Youssef, Khaled M.; Zaddach, Alexander J.; Niu, Changning; Irving, Douglas L.; Koch, Carl C. (9 December 2014). "A Novel Low-Density, High-Hardness, High-entropy Alloy..." Materials Research Letters. 3: 1. doi:10.1080/21663831.2014.985855.
- Yarris, Lyn (4 September 2014). "A metallic alloy that is tough and ductile at cryogenic temperatures". Berkeley Lab News Center. University of California, Berkeley. Retrieved 9 January 2015.
- Gludovatz, B.; Hohenwarter, A.; Catoor, D.; Chang, E. H.; George, E. P.; Ritchie, R. O. (5 September 2014). "A fracture-resistant high-entropy alloy for cryogenic applications" (PDF). Science. AAAS. 345 (6201): 1153–1158. Bibcode:2014Sci...345.1153G. doi:10.1126/science.1254581. PMID 25190791. Retrieved 9 January 2015.
- Vincent AJB; Cantor B: part II thesis, University of Sussex (1981).
- Huang KH, Yeh JW. A study on multicomponent alloy systems containing equal-mole elements [M.S. thesis]. Hsinchu: National Tsing Hua University; 1996.
- Yeh, J.-W.; Chen, S.-K.; Lin, S.-J.; Gan, J.-Y.; Chin, T.-S.; Shun, T.-T.; Tsau, C.-H.; Chang, S.-Y. (May 2004). "Nanostructured High-Entropy Alloys with Multiple Principal Elements: Novel Alloy Design Concepts and Outcomes". Advanced Engineering Materials. 6 (5): 299–303. doi:10.1002/adem.200300567.
- Cantor, B.; Chang, I.T.H.; Knight, P.; Vincent, A.J.B. (July 2004). "Microstructural development in equiatomic multicomponent alloys". Materials Science and Engineering: A. 375–377: 213–218. doi:10.1016/j.msea.2003.10.257.
- Middleburgh, S.C.; King, D.M.; Lumpkin, G.R. (April 2015). "Atomic scale modelling of hexagonal structured metallic fission product alloys". Royal Society Open Science. 2 (1): 140292. Bibcode:2015RSOS....2n0292M. doi:10.1098/rsos.140292.
- Otto, F.; Yang, Y.; Bei, H.; George, E.P. (April 2013). "Relative effects of enthalpy and entropy on the phase stability of equiatomic high-entropy alloys". Acta Materialia. 61 (7): 2628–2638. doi:10.1016/j.actamat.2013.01.042.
- Zou, Yu; Maiti, Soumyadipta; Steurer, Walter; Spolenak, Ralph (February 2014). "Size-dependent plasticity in an Nb25Mo25Ta25W25 refractory high-entropy alloy". Acta Materialia. 65: 85–97. doi:10.1016/j.actamat.2013.11.049.
- Gali, A.; George, E.P. (August 2013). "Tensile properties of high- and medium-entropy alloys". Intermetallics. 39: 74–78. doi:10.1016/j.intermet.2013.03.018.
- Miracle, Daniel; Miller, Jonathan; Senkov, Oleg; Woodward, Christopher; Uchic, Michael; Tiley, Jaimie (10 January 2014). "Exploration and Development of High-Entropy Alloys for Structural Applications". Entropy. 16 (1): 494–525. Bibcode:2014Entrp..16..494M. doi:10.3390/e16010494.
- Gali, A.; George, E.P. (August 2013). "Tensile properties of high- and medium-entropy alloys". Intermetallics. 39: 74–78. doi:10.1016/j.intermet.2013.03.018.
- Greer, A. Lindsay (25 December 1993). "Confusion by design". Nature. 366 (6453): 303–304. Bibcode:1993Natur.366..303G. doi:10.1038/366303a0.
- Zhang, Y.; Zhou, Y. J.; Lin, J. P.; Chen, G. L.; Liaw, P. K. (June 2008). "Solid-Solution Phase Formation Rules for Multi-component Alloys". Advanced Engineering Materials. 10 (6): 534–538. doi:10.1002/adem.200700240.
- Takeuchi, Akira; Inoue, Akihisa (2005). "Classification of Bulk Metallic Glasses by Atomic Size Difference, Heat of Mixing and Period of Constituent Elements and Its Application to Characterization of the Main Alloying Element". Materials Transactions. 46 (12): 2817–2829. doi:10.2320/matertrans.46.2817.
- Zhang, Yong; Zuo, Ting Ting; Tang, Zhi; Gao, Michael C.; Dahmen, Karin A.; Liaw, Peter K.; Lu, Zhao Ping (April 2014). "Microstructures and properties of high-entropy alloys". Progress in Materials Science. 61: 1–93. doi:10.1016/j.pmatsci.2013.10.001.
- Guo, Sheng; Ng, Chun; Lu, Jian; Liu, C. T. (2011). "Effect of valence electron concentration on stability of fcc or bcc phase in high-entropy alloys". Journal of Applied Physics. 109 (10): 103505. Bibcode:2011JAP...109j3505G. doi:10.1063/1.3587228.
- Tsai, Ming-Hung; Tsai, Kun-Yo; Tsai, Che-Wei; Lee, Chi; Juan, Chien-Chang; Yeh, Jien-Wei (20 August 2013). "Criterion for Sigma Phase Formation in Cr- and V-Containing High-Entropy Alloys". Materials Research Letters. 1 (4): 207–212. doi:10.1080/21663831.2013.831382.
- Youssef, Khaled M.; Zaddach, Alexander J.; Niu, Changning; Irving, Douglas L.; Koch, Carl C. (9 December 2014). "A Novel Low-Density, High-Hardness, High-entropy Alloy with Close-packed Single-phase Nanocrystalline Structures". Materials Research Letters. 3 (2): 95–99. doi:10.1080/21663831.2014.985855.
- Ji, Wei; Wang, Weimin; Wang, Hao; Zhang, Jinyong; Wang, Yucheng; Zhang, Fan; Fu, Zhengyi (January 2015). "Alloying behavior and novel properties of CoCrFeNiMn high-entropy alloy fabricated by mechanical alloying and spark plasma sintering". Intermetallics. 56: 24–27. doi:10.1016/j.intermet.2014.08.008.
- Zou, Yu; Ma, Huan; Spolenak, Ralph (2015-07-10). "Ultrastrong ductile and stable high-entropy alloys at small scales". Nature Communications. 6: 7748. Bibcode:2015NatCo...6E7748Z. doi:10.1038/ncomms8748. PMC 4510962. PMID 26159936.
- Yao, Chen-Zhong; Zhang, Peng; Liu, Meng; Li, Gao-Ren; Ye, Jian-Qing; Liu, Peng; Tong, Ye-Xiang (November 2008). "Electrochemical preparation and magnetic study of Bi–Fe–Co–Ni–Mn high-entropy alloy". Electrochimica Acta. 53 (28): 8359–8365. doi:10.1016/j.electacta.2008.06.036.
- Zhang, Chuan; Zhang, Fan; Chen, Shuanglin; Cao, Weisheng (29 June 2012). "Computational Thermodynamics Aided High-Entropy Alloy Design". JOM. 64 (7): 839–845. Bibcode:2012JOM....64g.839Z. doi:10.1007/s11837-012-0365-6.
- Gao, Michael; Alman, David (18 October 2013). "Searching for Next Single-Phase High-Entropy Alloy Compositions". Entropy. 15 (10): 4504–4519. Bibcode:2013Entrp..15.4504G. doi:10.3390/e15104504.
- Zunger, Alex; Wei, S.-H.; Ferreira, L. G.; Bernard, James E. (16 July 1990). "Special quasirandom structures". Physical Review Letters. 65 (3): 353–356. Bibcode:1990PhRvL..65..353Z. doi:10.1103/PhysRevLett.65.353.
- Niu, C.; Zaddach, A. J.; Oni, A. A.; Sang, X.; Hurt, J. W.; LeBeau, J. M.; Koch, C. C.; Irving, D. L. (20 April 2015). "Spin-driven ordering of Cr in the equiatomic high-entropy alloy NiFeCrCo". Applied Physics Letters. 106 (16): 161906. Bibcode:2015ApPhL.106p1906N. doi:10.1063/1.4918996.
- Huhn, William Paul; Widom, Michael (19 October 2013). "Prediction of A2 to B2 Phase Transition in the High-Entropy Alloy Mo-Nb-Ta-W". JOM. 65 (12): 1772–1779. arXiv:1306.5043. Bibcode:2013JOM....65l1772H. doi:10.1007/s11837-013-0772-3.
- Tian, Fuyang; Delczeg, Lorand; Chen, Nanxian; Varga, Lajos Karoly; Shen, Jiang; Vitos, Levente (30 August 2013). "Structural stability of NiCoFeCrAl high-entropy alloy from ab initio theory". Physical Review B. 88 (8). Bibcode:2013PhRvB..88h5128T. doi:10.1103/PhysRevB.88.085128.
- Middleburgh, S.C.; King, D.M.; Lumpkin, G.R.; Cortie, M.; Edwards, L. (June 2014). "Segregation and migration of species in the CrCoFeNi high entropy alloy". Journal of Alloys and Compounds. 599: 179–182. doi:10.1016/j.jallcom.2014.01.135.
- King, D.M.; Middleburgh, S.C.; Liu, A.C.Y.; Tahini, H.A.; Lumpkin, G.R.; Cortie, M. (January 2014). "Formation and structure of V–Zr amorphous alloy thin films". Acta Materialia. 83: 269–275. doi:10.1016/j.actamat.2014.10.016.
- Middleburgh, S.C.; Burr, P.A.; King, D.J.M.; Edwards, L.; Lumpkin, G.R.; Grimes, R.W. (November 2015). "Structural stability and fission product behaviour in U3Si". Journal of Nuclear Materials. 466: 739–744. Bibcode:2015JNuM..466..739M. doi:10.1016/j.jnucmat.2015.04.052.
- King, D.M.; Middleburgh, S.C.; Edwards, L.; Lumpkin, G.R.; Cortie, M. (October 2015). "Predicting the Crystal Structure and Phase Transitions in High-Entropy Alloys" (PDF). JOM. 67 (10): 2375–2380. Bibcode:2015JOM...tmp..273K. doi:10.1007/s11837-015-1495-4.
- Otto, F.; Dlouhý, A.; Somsen, Ch.; Bei, H.; Eggeler, G.; George, E.P. (September 2013). "The influences of temperature and microstructure on the tensile properties of a CoCrFeMnNi high-entropy alloy". Acta Materialia. 61 (15): 5743–5755. doi:10.1016/j.actamat.2013.06.018.
- Wu, Z.; Bei, H.; Otto, F.; Pharr, G.M.; George, E.P. (March 2014). "Recovery, recrystallization, grain growth and phase stability of a family of FCC-structured multi-component equiatomic solid solution alloys". Intermetallics. 46: 131–140. doi:10.1016/j.intermet.2013.10.024.
- Zaddach, A.J.; Scattergood, R.O.; Koch, C.C. (June 2015). "Tensile properties of low-stacking fault energy high-entropy alloys". Materials Science and Engineering: A. 636: 373–378. doi:10.1016/j.msea.2015.03.109.
- Hemphill, M.A.; Yuan, T.; Wang, G.Y.; Yeh, J.W.; Tsai, C.W.; Chuang, A.; Liaw, P.K. (September 2012). "Fatigue behavior of Al0.5CoCrCuFeNi high-entropy alloys". Acta Materialia. 60 (16): 5723–5734. doi:10.1016/j.actamat.2012.06.046.
- Shipman, Matt. "New 'high-entropy' alloy is as light as aluminum, as strong as titanium alloys". Phys.org. Retrieved 29 May 2015.
- Zou, Yu; Maiti, Soumyadipta; Steurer, Walter; Spolenak, Ralph (2014-02-15). "Size-dependent plasticity in an Nb25Mo25Ta25W25 refractory high-entropy alloy". Acta Materialia. 65: 85–97. doi:10.1016/j.actamat.2013.11.049.
- Wang, X.F.; Zhang, Y.; Qiao, Y.; Chen, G.L. (March 2007). "Novel microstructure and properties of multicomponent CoCrCuFeNiTix alloys". Intermetallics. 15 (3): 357–362. doi:10.1016/j.intermet.2006.08.005.
- Vrtnik, S.; Koželj, P.; Meden, A.; Maiti, S.; Steurer, W.; Feuerbacher, M.; Dolinšek, J. (February 2017). "Superconductivity in thermally annealed Ta-Nb-Hf-Zr-Ti high-entropy alloys". Journal of Alloys and Compounds. 695: 3530–3540. doi:10.1016/j.jallcom.2016.11.417.
- Tsai, K.-Y.; Tsai, M.-H.; Yeh, J.-W. (August 2013). "Sluggish diffusion in Co–Cr–Fe–Mn–Ni high-entropy alloys". Acta Materialia. 61 (13): 4887–4897. doi:10.1016/j.actamat.2013.04.058.
- Granberg, Fredric (2016). "Mechanism of radiation damage reduction in equiatomic multicomponent single phase alloys". Physical Review Letters. 116: 135504. Bibcode:2016PhRvL.116m5504G. doi:10.1103/PhysRevLett.116.135504. PMID 27081990.