Computational Materials Physics

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1 Computational Materials Physics narrated by Hans L. Skriver Center for Atomic-scale Materials Physics CAMP-DTU Lyngby thanks to L. Vitos P. Söderlind S.I. Simak A. Ruban N. Rosengård J. Nørskov A. Niklasson S. Mirbt A. McMahan A.R. Mackintosh W. Lambrecht P.A. Korzhavyi J. Kollar B. Johansson G. Johannesson O. Jepsen J.-P. Jan K. Jacobsen O. Eriksson N.E. Christensen T. Bligård O.K. Andersen M. Aldén I. Abrikosov title

Wigner and Seitz If one had a great calculating machine, one might apply it to the problem of solving the Schrödinger equation for each metal and obtain thereby the interesting physical properties,

Presumably the results would agree with the experimentally determined quantities and nothing vastly new would be learned from the calculation.

2 Wigner and Seitz If one had a great calculating machine, one might apply it to the problem of solving the Schrödinger equation for each metal and obtain thereby the interesting physical properties, such as the cohesive energy, the lattice constant, and similar parameters. It not clear, however, that a great deal would be gained by this. Presumably the results would agree with the experimentally determined quantities and nothing vastly new would be learned from the calculation. It would be preferable instead to have a vivid picture of the behaviour of the wave functions, a simple description of the essence of the factors which determine cohesion and an understanding of the origins of variation in properties from metal to metal. The task before us is not a purely scientific one; it is partly pedagogic. Eugene Wigner and Frederick Seitz (1955) wignerseitz

3 Eriksen's Theorem Physics is is an an experimental science eriksen

4 Measured Cohesive Energies (kj/mole) C. Kittel: Introduction to Solid State physics, 7th edition (1996) Hf Zr Ta W Re Os Mo Tc Ru Lu La 500 Y V Pd Au 400 Ti Sc Cr Fe Co Ni Ag 300 Ba Cu Sr 200 Mn 100 Ca 3d 4d 0 5d d-occupation Ir Rh Pt coh3

5 Cohesive Energy Energy difference between atom in the ground state and the energy of the solid per atom at zero temperature E coh = E bond - E atom E atom = E pro + E metal (d n+1 ) E (d n+1 )* E pro E bond (d n )* atom E coh E : Calculated from local exchange integrals M.S. Brooks and B. Johansson, J. Phys. F13, L197 (1983) E pro : From experiment (Moore 1958) coh2

6 Cohesive Energy M.S. Brooks and B. Johansson, J. Phys. F13, L197 (1983) coh1

7 d-band occupation: n d = E F D(E) de 0 Bonding energy: E bond = - E F 0 _ = - ( E The Friedel Model J. Friedel in The Physics of Metals (ed. J.M. Ziman, 1969) D(E) ( E - E a ) de - E a ) n d E W E F _ E 0 E bond W 10 W D(E) E a W = n 20 d ( 10 - n d ) 0 10 n d friedelm

8 Local Density Theory of Metallic Cohesion Calculations by V.L. Morruzzi, A.R. Williams and J.F. Janak, Phys. Rev. B15, 2854 (1977) using DFT LDA LSD P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964) W. Kohn and L. Sham, Phys. Rev. 140, A1133 (1965) L. Hedin and B.I. Lundqvist, J. Phys. C 4, 2064 (1971) U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972) and fast KKR A.R. Williams, J.F. Janak and V.L. Morruzzi, Phys. Rev. B6, 4509 (1972) Morruzzi

9 Crystal Structures Li Na Be Mg fcc bcc hcp K Ca Sc Ti V Cr Mn Fe Co Ni Cu Rb Sr Y Zr Mo Tc Ru Rh Pd Ag Cs Ba Lu Hf Ta W Re Os Ir Pt Au cstruc

10 Canonical Band Theory O.K. Andersen, J. Madsen, U.K. Poulsen, O. Jepsen and J. Kollar, Physica 86-88B, 249 (1977) canon

11 Calculated Structural Energy H.L. Skriver, Phys. Rev. B31, 1909 (1985) A.R. Miedema and A.K. Niessen, CALPHAD 7, 27 (1983) d metals Theory Ru 30 "Exp." Rh 20 Tc Li Na K Rb Cs Be Mg Ca Sr Ba Sc Ti Y Zr Lu Hf fcc bcc hcp V Ta Cr Mn Mo Tc W Re Fe Co Ru Os Rh Ir Ni Cu Pd Ag Pt Au [mry] Y Zr Sr fcc hcp hcp fcc Pd Ag bcc Mo d-occupation struce-4d

12 Database: Calculated Structural energy differences Li - Pa at experimental volume LMTO-ASA plus Madelung correction Andersen's force theorem Barth-Hedin XC strdb H.L. Skriver, Phys. Rev. B31, 1909 (1985) H.L. Skriver, Phys. Rev. Lett. 49, 1768 (1982) d-occupation [states/atom]

13 Database: Calculated Stacking fault energy Transition metals at experimental volume LMTO-ASA Green function approach, semi-infinite geometry LDA: Perdew-Zunger, Ceperley-Alder Instrinsic stacking fault N.M. Rosengaard, H.L. Skriver, Phys. Rev. B47, (1993) stackdb

14 Experimental Surface Tension J.N. Schmit: Ph.D. Thesis (1978), University of Liege; in Zangwill: Physics at Surfaces (1988) Atomic number surfacetension

15 Eötvös Law Surface tension (energy) is linear in temperature. 2.0 Shaler and Wulff, Trans. AIME 185, 186 (1949) Belforti and Lopie, Trans. AIME 227, 20 (1963) Surface tension of Cu Discontinuity at melting is approximately 20 %. (T) = (Tm)+ ( Tm - T) Surface tension [J/m 2 ] T m ~ 20 % Temperature [K] eotvos

16 Surface Energy Energy required to transform a bulk atom into a surface atom with a corresponding increase in surface area Friedel: W E coh = n 20 d ( 10 - n d ) E surf = Wn d ( 10 - n d ) 20 E W E F _ E 0 E surf W W D(E) E W- W E surf 0 10 W- W E F _ E D(E) Tight-Binding: W = 1 - (8/12) 1/2 ~ n d surfaceenergy

17 "Experimental" Surface Energy : de Boer et al., Transition Metals Alloys in Cohesion and Structure, Vol 1 (North-Holland, 1988) : Vitos et al., Surf. Sci. 411, 186 (1998); DFT-LMTO-FCD 3 4 4d metals 4d metals Surface tension (J/m 2 ) 2 1 Surface tension at the melting temperature Y Zr Mo Tc Ru Rh Pd Ag Surface energy (J/m 2 ) Calc. fcc111 De Boer Sr Rb Rb Sr Y Zr Mo Tc Ru Rh Pd Ag Cd Atomic number surft-4d

18 Calculated Surface Energy : de Boer et al., Transition Metals Alloys in Cohesion and Structure, Vol 1 (North-Holland, 1988) : Vitos et al., Surf. Sci. 411, 186 (1998); DFT-LMTO-FCD Surface energy (J/m 2 ) d metals Calculated de Boer Mo Zr Y Sr Tc Ru Rh Pd Ag Surface energy (ev/atom) d metals Sr Y Zr Mo Tc Ru Calculated Friedel theory Rh Pd Ag Rb Number of valence electrons d-band filling surfe-4d

19 Database: Calculated Surface, step and kink energy Li - Pu at calculated volume LMTO-ASA plus full chargedensity correction FCD Green function approach, semi-infinite geometry Surface energy (ev/atom) Transition metals 3d 4d 5d W Mo Cr Re Tc Mn Calculated surface energies for the most close-packed surfaces d-band filling 4 surfedb E surf = E 2D FCD ( N atom N vac + ) - N atom Perdew, Burke, Ernzerhof GGA L. Vitos, A.R. Ruban, H.L. Skriver, J. Kollar, Surf. Sci. 411, 186 (1998) FCD E 3D L. Vitos, H.L. Skriver, J. Kollar, Surf. Sci. 425, 212 (1999) Surface energy (J/m 2 ) fcc (100) Surface energy (100) x (111) (100) x (010) Y Zr Mo Tc Ru Rh Pd Ag Step energy (10 J/m)

20 Surface Segregation Energy Bulk A 1-c B c A B c = de bulk dc c=0 Surface A 1-c B c E surf = ( E - E ' ' bulk ) - c E segr E B -> A segr, Friedel model = de surf dc c =0 E A surf E B surf E B -> A E B segr ~ surf - E A surf 0 c 1 segrth

21 Surface Segregation Energy 2.0 Friedel model Surface Energy (ev/atom) 1,5 1,0 0,5 Y Zr Mo Wf(1-f) Tc Ru Rh Pd Ag Host Mo Tc Ru Rh Zr Solute Mo Tc Ru Rh Pd Ag Pd 0,0 0,0 0,2 0,4 0,6 0,8 1,0 d-band filling Ag segrem

22 Calculated Segregation Energy Solute 3d 4d 5d Ti V Cr Mn Fe Co Ni Cu Zr Mo Tc Ru Rh PdAg Hf Ta W Re Os Ir Pt Au hcp Ti E segr Host [fcc(111), bcc(110), hcp(0001)] 5d 4d 3d bcc bcc bcc bcc hcp fcc fcc hcp bcc bcc hcp hcp fcc fcc fcc hcp bcc bcc hcp hcp fcc V Cr Mn Fe Co Ni Cu Zr Mo Tc Ru Rh Pd Ag Hf Ta W Re Os Ir Zr Mo Tc Ru Rh Pd Ag Zr < ev ev ev ev ev ev > 0.7 ev Friedel model Mo Tc Ru Rh Pd Ag fcc Pt fcc Au A.V. Ruban, H.L. Skriver, J.K. Nørskov, Phys. Rev. B59, (1999) esegr

23 Surface Core Level Shift Thermodynamical model of a fully screened core-hole B. Johansson and N. Mårtensson, Phys. Rev. B21, 4427 (1980) SB = E bulk (Z*) ~ sol Z+1 -> Z E segr - E surf sol (Z*) Z Z* ~ Z+1 XPS surface Z Friedel model, d -band filling f Z+1 -> Z E segr = Z+1 E surf - E Z surf initial state XPS bulk Z final states de surf ~ ~ W df 10 (1-2f) scls-th

24 Calculated Surface Core Level Shift Thermodynamical model B. Johansson and N. Mårtensson, Phys. Rev. B21, 4427 (1980) SB = Z+1 -> Z E segr Friedel model, d -band filling f Z+1 -> Z E segr ~ W 10 (1-2f) Surface core-level shift (ev) A.V. Ruban and H.L. Skriver, Comp. Mat. Sci. 15, 119 (1999) 5d metals E segr (Z+1) Hf Ta W Re Exp. SCLS Friedel Os Ir hcp bcc bcc hcp hcp fcc fcc Pt d-band filling scls-5d

25 Database: Calculated Surface segregation energy Transition - transition metal at calculated volume LMTO-ASA plus multipole correction Green function approach, semi-infinite geometry Coherent Potential Approximation CPA LDA: Perdew-Zunger Ceperley-Alder LSD: Vosko-Wilk-Nusair c Pt Surface segregation profile (111) Rh Pt T = 1300 K Calculated Experiment (100) (110) A.V. Ruban, H.L. Skriver, J.K. Nørskov, Phys. Rev. B59, (1999) A.V. Ruban, H.L. Skriver, Comp.Mat. Sci. 15, 119 (1999) Surface layer segrdb

26 Database: Calculated Vacancy formation energy Transition metals Supercell approach: Order-N Locally Self-consistent Greens Funtion LSGF method F ~ H 1V (P=0) = E(1,V) - N-1 N s,p,d,f LMTO-ASA plus multipole correction E(0,V) Perdew, Burke, Ernzerhof GGA Vacancy formation enthalpy (ev) 4 Calculated 3d Experimental 3 Recommended values Sc Ti V Cr Mn Fe Co Ni Cu Y 4d Zr Mo Tc Ru Rh Pd Ag 5d P.A. Korzhavyi, I.A. Abrikosov, B. Johansson, A.V. Ruban, H.L. Skriver, Phys. Rev. B59, (1999) 1 Lu Hf Ta W Re Os Ir Pt Au vacdb

27 Empirically Vacancy Formation Energy F 1 E 1V ~ 3 E coh 2 P.A. Korzhavyi, I.A. Abrikosov, B. Johansson, A.V. Ruban, H.L. Skriver, Phys. Rev. B59, (1999) Calculated LSGF Structural contribution, i.e., beyond Friedel, to the vacancy formation energy E coh E vac ~ z Friedel = E coh + E str ( 1 - [ ] 1/2 ) z-1 z E coh Energy difference (ev) E vac bcc fcc bcc - z [ ] 1/2 z-1 z E str -3 Y Zr Mo Tc Ru Rh Pd Ag vacstr

28 Database: Calculated impurity solution energy 4d transition metals at calculated volume LMTO-ASA Zr Mo Tc Ru Rh Pd Ag Coherent Potential Approximation Perdew-Zunger Ceperley-Alder XC E sol (ev) 1 0 Solution of B -> A -1 E sol = de A 1-c B c dc c=0 + E A - E B -2 A.V. Ruban, H.L. Skriver, J.K. Nørskov, Phys. Rev. Lett. 80, 1240 (1998) -3 Zr Mo Tc Ru Rh Pd Ag Host imp4ddb

29 Impurity Solution Energy Crystal structure contribution to the solid solubility in transition metal alloys - E sol - = 2 E str N d C ( ) - E str - A ( N ) d (ev) bcc-hcp E str A.V. Ruban, H.L. Skriver, J.K. Nørskov, Phys. Rev. Lett. 80, 1240 (1998) Y Host: Zr Mo Tc Ru Tc Ru Rh Ag 0.0 C 1 A B N d = 2 ( N d + N ) d Zr Pd -0.5 Mo N d impstr

Surface segregation energies in transition-metal alloys

Downloaded from orbit.dtu.dk on: Feb 10, 2018 Surface segregation energies in transition-metal alloys Ruban, Andrei; Skriver, Hans Lomholt; Nørskov, Jens Kehlet Published in: Physical Review B Condensed