Predicting the Structure of Solids by DFT

Transcription

1 Questions? Hudson Hall 235 or Hudson Hall 1111 Predicting the Structure of Solids by DFT Hands-On Instructions Contents 1 Cohesive Energy for Bulk Phases of Si 11 Setting up the Structures 12 Structure-Dependent Total Energies for a Small 13 Plot the Resulting Total Energies 14 Converging the k-grid 15 Converging the Basis Set Phase Stability and Cohesive Properties 21 Total Energies as a Function of Lattice Parameter 22 Cohesive Energies as a Function of Volume per Atom 23 Birch-Murnaghan Equation of State k-point Grid A Transitions Between Different Phases 10 B Information on the BCC, FCC, and Diamond Lattices 11 1 Cohesive Energy for Bulk Phases of Si In this exercise, we will work on different possible structural phases of bulk silicon The correct description of the phase stability of Si by Yin and Cohen is one of the early success stories of computational materials science[1] To do this, we require a mechanism to compute the total energy of an infinite, periodic solid with certain lattice vectors {ai, i = 1,, 3} and (possibly more than one) atomic positions {bi } in the unit cell In the present exercise, we will use the FHI-aims code and set (in the FHI-aims input file controlin): 1

2 # P h y s i c a l s e t t i n g s xc pw lda s p i n none r e l a t i v i s t i c a t o m i c z o r a s c a l a r # SCF s e t t i n g s s c a c c u r a c y r h o 1E 4 s c a c c u r a c y e e v 1E 2 s c a c c u r a c y e t o t 1E 5 # k g r i d s e t t i n g s ( to be a d j u s t e d ) k g r i d nkx nky nkz Please keep these basic settings as default for this exercise (unless specified otherwise) You should have seen most of these keywords in the previous exercise for free atoms, but one thing is new: the k grid specification As outlined in class, the different Kohn-Sham eigenfunctions of a periodic solid, ϕ i (r) can be classified a little further by refining the index i {k, n} According to the Bloch theorem, the different (inequivalent) eigenfunctions on the (periodic) lattice can be written as ϕ k,n (r) = exp(ikr) u k,n (r), (1) Here, u k,n (r) is a so-called Bloch function, which has the same periodicity as the crystal itself In contrast, the phase factor exp(ikr) need not have any periodicity at all All that the translational symmetry of the crystal dictates is that the Bloch functions are only inequivalent for the set of three-dimensional, real-valued vectors {k} that are located in a unit volume of the so-called reciprocal space This space is spanned by vectors {G i, i = 1,, 3} that are defined by a i G j = 2πδ ij (2) (where a i are the lattice vectors of the crystal) In short, in order to get a good sampling of quantities such as the electron density n(r) = ϕ i (r) 2 ϕ k,n (r) 2, (3) i k n we must sample a sufficiently large number of points k in a practical calculation to get a well converged result You will note that the number of possible points k is in principle infinite (the possible values of k are continuous and therefore infinitely many even within the unit volume of reciprocal space) Thus, the sum over k should formally rather be an integral in practice, however, we can only compute such an integral as a sum over specific points on a computer (and we have, somewhat precariously, left out any integration weights in the expression above) For a practical calculation, it thus remains to specify these k-points This is what the k grid setting does, by explicitly setting up a number of integer grid divisions nkx, nky, nkz along each of the reciprocal lattice vectors G i, as illustrated in Figure 1 for a two-dimensional example In three dimensions, the total number of k-grid points is thus nkx nky nkz The other players in the FHI-aims input sample above should all be well familiar The Perdew- Wang LDA (xc pw-lda) exchange-correlation functional[2] will be used for all calculations Silicon would turn out to be nonmagnetic, so no explicit spin treatment is needed The relativistic treatment triggered by the relativistic atomic zora scalar setting is not strictly necessary for silicon The nuclear charge of silicon (Z = 14) is still small enough to allow for a non-relativistic 2

3 Figure 1: Illustration of the k-grid for the 2D rectangular lattice The reciprocal vectors G 1 and G 2 define a rectangular reciprocal cell A k-space grid of 3 3 (example) divides the reciprocal cell into 9 sub-rectangles (green lines) and evaluates the total energy based on the light green and the grey dots for the reciprocal cell integration treatment Since the correction is computationally inexpensive, it does not hurt to use it, either Just be sure to never compare total energies from different relativistic settings Please use the default light species settings for Si in /programs/fhi-aims/aimsfiles/species defaults/light/14 Si default 11 Setting up the Structures The first step towards studying periodic systems with FHI-aims is to construct periodic geometries in the FHI-aims geometry input format geometryin and visualize them As a next step, we set basic parameters in controlin for periodic calculations Finally, we compare total energies of different Si bulk geometries Thus: Set up geometryin files for the Si fcc, bcc, and diamond structures (see Appendix B) Use the approximate lattice constants a of 38 Å for fcc, 31 Å for bcc, and 54 Å for the diamond structure Visualize the resulting structures (eg, using jmol) To set up a periodic structure in FHI-aims, all three lattice vectors as well as the atomic positions in the unit cell must be specified The lattice vectors are specified by the keyword lattice vector There are two ways to specify the atomic positions You can specify absolute Cartesian positions with the keyword atom Alternatively, you can specify the atomic positions in the basis of the lattice vectors, the so called fractional coordinates, with the keyword atom frac The fractional coordinates s i are dimensionless and the coefficients for the linear combination of the lattice vectors a i Written out as a formula, this linear combination reads as follows R = s 1 a 1 + s 2 a 2 + s 3 a 3 (4) 3

4 where R is the Cartesian position of the specified atom To visualize the resulting files geometryin files in jmol, please type /opt/jmol-1407/jmolsh geometryin To get periodically repeated units of the lattice, open the console in Jmol and type load geometryin {3 3 3} The numbers give the repetition of the structure along the corresponding lattice vector 12 Structure-Dependent Total Energies for a Small k-point Grid In this exercise, we compare total energies of different lattice structures for Si as a function of lattice constant Prepare a controlin file using k-points and the settings given in the introduction Use a shell script (see below) to calculate total energies of the fcc, bcc, and diamond phases of Si as a function of lattice constant a Consider 7 different values of a in steps of 01 Å, centered around the lattice parameters given above, for each structure The basic settings in controlin were given near the beginning of this documents To start out, please use: k g r i d We will later find that, for a small unit cell, this k-point density is by no means enough but let us go ahead anyway, for now Once all input files are set up for a given structure, you can run the parallel version of FHI-aims by mpirun -n 4 aims031214scalapackmpix tee aimsout This command starts a parallel calculation, using four processors simultaneously It is good practice to use a separate directory for every single run of FHI-aims in order to preserve the exact input files along with the output files Here, however, most of the calculations can be started using the shell script described below, which takes care of these things and which only needs slight adjustments The example script below calculates the total energy of fcc Si with different lattice constants 4

5 #! / bin / bash l s e t e # Stop on e r r o r f o r A i n ; do echo P r o c e s s i n g l a t t i c e c o n s t a n t $A Angstrom mkdir $A # Use t h i s c o n s t r u c t f o r simple c a l c u l a t i o n s As v a l u e s # are r e p l a c e d verbatim, always put them i n t o (, ) A2=$ ( python c p r i n t ($A ) / 2 0 ) # Write geometry in cat >$A/ geometry i n <<EOF # f c c s t r u c t u r e with l a t t i c e c o n s t a n t $A Angstrom l a t t i c e v e c t o r 0 0 $A2 $A2 l a t t i c e v e c t o r $A2 0 0 $A2 l a t t i c e v e c t o r $A2 $A2 0 0 a t o m f r a c S i EOF # Write c o n t r o l in cp c o n t r o l i n $A/ c o n t r o l i n # Now run FHI aims with 4 p r o c e s s o r s in d i r e c t o r y $A cd $A mpirun n 4 aims s c a l a p a c k mpi x > aims out cd done For the other two phases, bcc and diamond, you will have to create very similar shell scripts Please copy these scripts to dedicated folders for bcc and diamond Si To make your script executable, type chmod 700 scriptsh if your script is named scriptsh To retrieve the total energies, you could use the following script for post-processing: #! / bin / bash l s e t e # Stop on e r r o r echo # l a t t i c e c o n s t a n t s in Angstrom, e n e r g i e s in ev > e n e r g i e s dat f o r A i n ; do # Check f o r convergence o f c a l c u l a t i o n i f (! grep q u i e t S e l f c o n s i s t e n c y c y c l e converged \ <$A/ aims out ) \ (! grep q u i e t Have a n i c e day \ <$A/ aims out ) ; then echo pwd / $A/ aims out did not converge! f i # Get 6 th column from t h e l i n e with Total energy o f t h e DFT E=$ ( gawk / \ Total energy o f the DFT/ { p r i n t $12 } $A/ aims out ) # Write r e s u l t s to data f i l e echo $A $E >> e n e r g i e s dat done This script extracts the total energies and writes them to a file called energiesdat, along with the lattice constants You will need to adapt this script to the other phases of silicon In particular, adjust the lattice constants 5

6 Note that, in order to compare total energies for different phases of Si, it is advantageous to write out the total energy per atom, not per unit cell This makes a difference for the diamond structure For example, use the expression Eatom=$(python -c print ($E)/20 ) for the diamond structure and puts it into the bash variable $Eatom You can then write this variable instead of $E to the data file 13 Plot the Resulting Total Energies Plot the resulting total energies per atom(!) for fcc, bcc, and diamond silicon as a function of the lattice constant (eg, using xmgrace) What is the most stable bulk phase of Si according to your results? Thus, plot your data (given in fcc/energiesdat, bcc/energiesdat, and diamond/energiesdat) by typing: xmgrace -legend load fcc/energiesdat bcc/energiesdat diamond/energiesdat You might find that, with the current computational settings, the diamond Si phase is unfavorable compared to the other two phases However, the experimentally most stable phase is the diamond structure We will next show that the too coarse k-grid is the reason for this disagreement 14 Converging the k-grid Next, we will explicitly check total energy convergence with respect to the k-grid and to the basis set In principle, each phase needs to be checked separately Within our exercise, however, we can split the effort Everyone should only check one phase of their choice Calculate the total energies for only one of the Si phases as a function of the lattice constant for k-grids of size 8 8 8, , and Otherwise, use the same computational settings and the same lattice constants as before Prepare a plot with all total energies drawn against lattice constant Add the previously calculated results, too Which k-grid would you use to achieve convergence within 10 mev? You should dedicate a separate directory to every series of these calculations The calculations should be done exactly as in the last problem but with the appropriate changes to controlin Discuss the resulting curves and decide which k-space grid would have been good enough for your results 6

7 15 Converging the Basis Set In the following, just use a k-grid for all three phases Calculate the total energies for your chosen phase of Si as a function of the lattice constant for the minimal basis and for the full tier1 basis sets Use the same lattice constants and computational settings as before together with the k-grid Again, prepare a plot with the total energies Add the results for the minimal+spd basis set (the default for the light species settings) from the k-point convergence test above In order to change the basis size settings, look into the species dependent settings within controlin There, you will find a line starting with # F i r s t t i e r Each line after this defines a group of basis functions (radial function type and angular momentum) which is added to the minimal basis In the light defaults for Si, there is one additional radial function for each valence channel (s and p) as well as a d function To run FHI-aims with a minimal basis instead, simply comment out these three lines by prepending a # character To run FHI-aims with a full tier1 basis set, uncomment all four lines following # First tier by removing the initial # character Can you make a statement about the accuracy of the total energy (how strongly does it change and in which direction), as well as about the computational effort? You may also want to look at the Si species defaults for tight settings, found in /programs/fhi-aims/aimsfiles/species defaults/tight/14 Si default Here, you will see that a number of other parameters change (grids, Hartree potential, extent of basis functions, and number of basis functions) in addition to just the basis set 2 Phase Stability and Cohesive Properties After finding converged computational settings, we can now revisit the phase stability of bulk silicon 21 Total Energies as a Function of Lattice Parameter Calculate the total energy of fcc, bcc, and diamond Si as a function of lattice constant a Use a k-grid of points, the minimal+spd basis set (light defaults) and the same lattice constants as before Plot the curves E(a) as done before The resulting binding curves should show that the experimentally observed diamond structure of silicon is most stable in LDA among the crystal structures studied here 7

8 22 Cohesive Energies as a Function of Volume per Atom The cohesive energy (E coh ) of a crystal is the energy per atom needed to separate it into its constituent neutral atoms E coh is defined as E coh = E bulk E atom, (5) where E bulk is the crystal s total energy per atom(!) E atom is the energy of an isolated atom We thus need to recompute the appropriate energy of the isolated Si atom Perform a total energy calculation of the free silicon atom Calculate the cohesive energies and the volume of the crystal per atom for all the structures treated in this section so far Plot the cohesive energies of all three phases into one plot, using the atomic volume as the x axis For the free atom calculation in the LDA, simply use the following settings: (this will lead to a spherically symmetric atomic state, but we do not need to worry about this here) # P h y s i c a l s e t t i n g s xc pw lda s p i n c o l l i n e a r d e f a u l t i n i t i a l m o m e n t hund r e l a t i v i s t i c a t o m i c z o r a s c a l a r In the species defaults, adjust the following keywords c u t p o t b a s i s d e p c u t o f f 0 0 and uncomment all basis functions In order to compare the pressure dependence of phase stabilities, we need to express the lattice constant behavior of all phases on equal footing One possibility to do so is to express the lattice constant in terms of the volume per atom This atomic volume can be calculated quite easily from the lattice constant a The simple cubic (super-)cell has the volume V sc = a 3 This number has to be divided by the number of atoms N sc in this cell V atom = a3 N sc Please verify that there are two, four, and eight atoms in the simple cubic supercell in the case of the bcc, fcc, and the diamond structure, respectively After plotting E(V ) (where V is the volume per atom) using xmgrace, the diamond structure is indeed the lowest-energy phase Yet, it is considerably more space consuming than the two closepacked phases At high pressure, the lower-volume phases might become favorable according to the Gibbs free energies of the different phases, G = E T S + pv (6) 8

9 23 Birch-Murnaghan Equation of State The lowest-energy lattice parameter a 0 is an important quantity which we can calculate from our data In principle, this could be done with a quadratic fit for E(a) or E(V ) Here, we will discuss and use a thermodynamically motivated and more accurate fitting function, the Birch-Murnaghan Equation of State[3, 4] The energy per atom (E = E coh ) is expressed as a function of the atomic volume (V = V atom ) E(V ) = E 0 + B 0 V B ( ) (V 0 /V ) B B B 0V 0 B 1 V 0 and E 0 are the lowest-energy atomic volume and energy per atom, respectively B 0 is the so-called bulk modulus and B 0 its derivative with respect to pressure Equation (7) can be derived by assuming a constant pressure derivative B 0 Fit the cohesive energy data for the three phases to the Birch- Murnaghan Equation of State using the program murnpy Determine the lattice constant a 0, the bulk modulus B 0, and the cohesive energy per atom E coh at minimum energy for each phase Compare the above quantities for the diamond phase with the experimental values of a = 5430 Å, B 0 = 988 GPa, and Ecoh = 463 ev[5] Plot the cohesive energies E(V) with respect to the atomic volume for all three phases The fitting program murnpy is part of the FHI-aims distribution You can get some documentation by typing (7) murn py help The script takes an input file with two columns, the first containing the volume and the second total energies Using the file name murnin as an example, one can simply use the script with murn py murn i n o f i t dat The program then writes the parameters V 0, E 0, B 0, and B 0 for the given data set to the output file, here called fitdat As a quick plausibility check of the fit, you can use the option -p to get a visual impression The script performs no unit conversions, so the bulk modulus B0 is given in units of evå 3 because the cohesive energies and atomic volumes were provided in ev and Å 3, respectively You can use GNU units to convert to SI units For example, use u n i t s v 0 5 ev/ angstrom ˆ3 GPa 9

10 to convert 05 evå 3 to about 80 GPa The optimal lattice constant can be calculated from the equilibrium atomic volume V 0 by a 0 = 3 N sc V 0 (8) with N sc the number of atoms in the cubic unit cell Compare the calculated results with experimental reference values given above Note: Exact agreement between DFT and experimental data is not our goal for this exercise DFT-LDA is an approximation, and we here see how well (or not) it works After performing the Birch-Murnaghan fit for all three phases, please plot the resulting fitted curves saved in fitdat into one figure A Transitions Between Different Phases This here is a simple extra exercise (if one has a printer and a ruler) not part of the official exercise By exposing the crystal to different pressure, one can enforce different atomic volumes smaller than the volume at the lowest energy It is, in fact, the Gibbs free energy that is minimized at constant pressure and temperature Thermodynamically, the pressure can be written as p = E V (9) If there were only a single curve E(V ), the volume at equilibrium at a certain pressure is thus given by exactly the above relation However, there is more than one possible phase for Si and each has a different relation E(V ) For a given pressure p, we can thus draw a tangent with p = E V at each of these curves Simply looking at the definition of the Gibbs free energy, each of these tangents (constant slope p) corresponds to a constant Gibbs energy The phase with the lowest tangent wins (is the most stable phase at given pressure p) What is particularly interesting are pressure values for which two phases have a common tangent In these cases, the Gibbs energy of these two phases is the same At lower pressures, the phase with higher volume becomes stable; at higher pressures, the phase with lower volume becomes stable Thus, the slope of a common tangent between the E(V ) curves of two different phases marks a transition pressure, ie, the pressure at which a phase transition between the two would occur One can find such a transition pressure quite simply in our plot: Take a ruler and find the common tangent between two phases, one with lower energy and higher volume, the other with higher energy and lower volume This is called the Maxwell construction From the slope of this line (a common tangent), deduce the transition pressure at which diamond and bcc Si could coexist according to our calculations Hint: The value should be somewhere between 10 GPa and 20 GPa This is somewhere around 100 times the ambient pressure of about 100 kpa Note, however, that there are additional possible crystal structures for silicon which we have not calculated here In reality, the Si β-tin phase is a more stable high-pressure phase than the bcc phase See, for instance, Reference [1] 10

11 B Information on the BCC, FCC, and Diamond Lattices The fcc lattice for Si with a lattice constant a is defined by l a t t i c e v e c t o r 0 0 a/2 a/2 l a t t i c e v e c t o r a/2 0 0 a/2 l a t t i c e v e c t o r a/2 a/2 0 0 a t o m f r a c S i The bcc lattice for Si with a lattice constant a is defined by l a t t i c e v e c t o r a/2 a/2 a/2 l a t t i c e v e c t o r a/2 a/2 a/2 l a t t i c e v e c t o r a/2 a/2 a/2 a t o m f r a c S i The diamond lattice for Si with a lattice constant a is defined by l a t t i c e v e c t o r 0 0 a/2 a/2 l a t t i c e v e c t o r a/2 0 0 a/2 l a t t i c e v e c t o r a/2 a/2 0 0 a t o m f r a c S i a t o m f r a c S i References [1] M T Yin and M L Cohen, Microscopic theory of the phase transformation and lattice dynamics of si, Physical Review Letters, vol 45, pp , Sept 1980 [2] J P Perdew and Y Wang, Accurate and simple analytic representation of the electron-gas correlation-energy, Physical Review B, vol 45, pp , Jan 1992 [3] F Birch, Finite elastic strain of cubic crystals, Physical Review, vol 71, no 11, pp , 1947 [4] F D Murnaghan, The compressibility of media under extreme pressures, Proceedings of the National Academy of Sciences of the United States of America, vol 30, pp , July 1944 [5] C Kittel, Introduction to Solid State Physics Hoboken, NJ: John Wiley & Sons, Inc, 8 ed,