Application to Alkali Halides - Lammps - Sandia National Laboratories

Using LAMMPS to DeriveCrystal Structure RuleFebruary 24-26, 2010Xiaowang Zhou, F. Patrick Doty, and Pin YangSandia National Laboratories

Example: Alkali HalidesTo predict alkali halide crystals, our embeddedionmethod interatomic potential database1. contains all nine alkali halide elements (Li, Na, K, Rb, CsF, Cl, Br, I),2. incorporates atomic size, electronegativity, and bondenergy effects,3. uses elemental properties directly as model parameterswithout parameterization, and4. gives good trend of charge, energy, and bond lengthpredictions.2

Role of ElectronegativityDouble Perovskite Cs 2 LiLaBr 6Observations:1. Lattice constant of Cs 2 LiLaBr 6 is a = 2r LiLa= 11.289 Å [1] , i.e., r LaLa = 7.983 Å.2. fcc La has a lattice constant a = 5.307 Å anda cohesive energy E c = -4.446 eV/atom [2] . Infcc La, r LaLa = 3.753 Å.3. It is difficult to for a potential to betransferrable to both Cs 2 LiLaBr 6 and La.[1]. P. Yang, M. A. Rodriguez, F. P. Doty, X. Zhou,M. R. Sanchez, and K. S. Shah, submitted.[2]. X. W. Zhou, and F. P. Doty, Phys. Rev. B, 78,224307 (2008).Solution:The electronegativity differenceinducedionization can increasethe bond length and reduce thebond strength.3

Embedded Ion Method (EIM)E=12N∑iN∑N( rij) + ∑ Ei( q i i)φ ,σiji= 1 j=ii=11system total energyEi12( q σ ) = q ⋅σ*i, i i iembedding energy at ii= N∑j=iji( )q iη rcharge on atom i1ijiNσi= ∑qj⋅ϕijj= i 1( r )electrical potential (in voltage) at iijE=12⎧⎛⎞ ⎡⎛⎞ ⎤⎪⎫⎜⎜⎥ ⎬= ⎪⎩ ⎝⎟ ⎠ ⎢⎣⎝⎟ ⎠ ⎥⎦⎪ ⎭N iNN iNiNjN⎪∑∑φ( ) + ∑⎨⎜∑( ) ⎟⋅∑ ⎢⎜∑( ) ⎟ijrijηjirijηkjrjk⋅ϕ( rij)i= 1 j i1i= 1 j=i1j= i1 k=j1*when ϕ ij (r) ~ r -1 , the embedding energy reduces toCoulomb interactions between i and its neighbors:Ei12i N( qi,σi) = ∑j=i1q ⋅ qirijj4

Fitting-Free Parameters5

Predicted Charge vs. Electronegativity6

Cation-Anion Spacing7

Cohesive Energy Prediction8

Simulated Annealing: NaCl at 300 KOver a 25 ps period9

Crystal Phase Diagram10

Working on Lanthanide11

Conclusions• LAMMPS can now be used to explore crystal rule inthe atomic size-electronegativity-bond energy space.• Preliminary application of LAMMPS in the atomicsize space already results in a new crystal rule beyondthe hard sphere model.• The new rule is mathematically very robust in that inAB binary compounds, larger A-B spacing withrespect to A-A or B-B spacings favors the CsCl type ofcrystals and smaller A-B spacing with respect to A-Aor B-B spacings favors the NaCl type of crystals.12

13Derivative Calculation( ) ( ) ( )( )( )( )∑∑∑∑∑∑∑∑∑∑= = == = === ==⎥⎦⎤⎢⎣⎡∂∂⋅⋅⎟ +⎠⎞⎜⎝⎛⋅∂∂=⎥⎦⎤⎢⎣⎡∂∂⋅⋅⎟ +⎠⎞⎜⎝⎛⋅∂∂⎟ +⎠⎞⎜⎝⎛⋅∂∂=⎥⎦⎤⎢⎣⎡∂∂⋅⋅+⋅∂∂⋅+⋅⋅∂∂=∂∂NiNiiijijjiiiNiNiiijijjiNjjjiiNiiijijjiijjiijjiNiiiiNNNXrqqXqXrqqXqXqXrqqrXqqrqXqXqE1 11 11111112121212121,ϕσϕσσϕϕϕσ

Pair Functionsfc( r,r , r )pc( 2r− r − r )⎡⎛1.64498⎞ ⎤p c= ⎢0.510204⋅erfc⎜⎟ − 0. 010204⎥⎢⎣⎝ rc− rp⎠ ⎥⎦cutoff function (r c : cutoff distance)φφαij⎡ Eb,ij⋅ βij⎛ re, ij ⎞ Eb,ij⋅αij⎛ re, ij ⎞r = ⎢ ⋅⎜⎟ − ⋅⎜⎟⎢ βij−α⎣ij ⎝ij ijpair energy function between cation and anionijijij( ) ⎥ ⋅ f ( r r , r )⎡ E⋅ β,e,ij cφ,Pair Functionsijr ⎠ β −α⎝ r ⎠ ⎥⎦r − rEβb,ij ije,ij b,ij ije,ij( ) = ⎢ ⋅exp⎜−α⋅ ⎟ − ⋅exp⎜−β ⋅ ⎟⎥⋅ f ( r,r , r )rijijc e,ij cφ, ijβij−α⎜ijr ⎟e,ijβij−α⎜ijr ⎟e,ij⎢⎣⎛⎝⎞⎠⎤⋅αpair energy function between cations or between anionsc⎛⎝r − r⎞⎤⎠⎥⎦ηji( ) = A ( χ − χ ) ⋅ f ( r r r )rη j i c,ij sη, ij,cη, ijcharge transfer function (χ: electronegativity)ϕij( ) = A ⋅exp( −ζ⋅ r) ⋅ f ( r,r r )rϕij ij c sϕ, ij,cϕ, ijCharge interaction function14

Cohesive Energy Profiles15

Reactivity of EIM16

Goldschmidt Criteria 1,2PerovskiteHard sphere mode: r AX = R A +R X ; r BX = R B +R X .Basic criterion:RA+ RX0 .77