Questions? — Hudson Hall 235 or Hudson Hall 1111 —

Predicting the Structure of Solids byDFT

Hands-On Instructions

Contents

1. Cohesive Energy for Bulk Phases of Si 11.1. Setting up the Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Structure-Dependent Total Energies for a Small k-Point Grid . . . . . . . . . . . 41.3. Plot the Resulting Total Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4. Converging the k-Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5. Converging the Basis Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2. Phase Stability and Cohesive Properties 72.1. Total Energies as a Function of Lattice Parameter . . . . . . . . . . . . . . . . . 72.2. Cohesive Energies as a Function of Volume per Atom . . . . . . . . . . . . . . . . 82.3. Birch-Murnaghan Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . 9

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 correctdescription of the phase stability of Si by Yin and Cohen is one of the early success stories ofcomputational materials science[1].To do this, we require a mechanism to compute the total energy of an infinite, periodic solidwith 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-aimsinput file control.in):

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# Phys ica l s e t t i n g sxc pw−ldasp in noner e l a t i v i s t i c atomic zora s c a l a r

# SCF s e t t i n g ss c a c cu racy rho 1E −4s c a c cu ra cy e ev 1E −2s c a c cu r a cy e t o t 1E −5

# k− g r i d s e t t i n g s ( to be ad jus t ed )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 onething is new: the k grid specification.As outlined in class, the different Kohn-Sham eigenfunctions of a periodic solid, ϕi(r) can beclassified a little further by refining the index i ≡ {k, n}. According to the Bloch theorem, thedifferent (inequivalent) eigenfunctions on the (periodic) lattice can be written as

ϕk,n(r) = exp(ikr) · uk,n(r) , (1)

Here, uk,n(r) is a so-called Bloch function, which has the same periodicity as the crystal itself. Incontrast, the phase factor exp(ikr) need not have any periodicity at all. All that the translationalsymmetry of the crystal dictates is that the Bloch functions are only inequivalent for the setof three-dimensional, real-valued vectors {k} that are located in a unit volume of the so-calledreciprocal space. This space is spanned by vectors {Gi, i = 1, . . . , 3} that are defined by

ai ·Gj = 2πδij (2)

(where ai 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) =∑i

|ϕi(r)|2 ≡∑k

∑n

|ϕk,n(r)|2 , (3)

we must sample a sufficiently large number of points k in a practical calculation to get a wellconverged result.You will note that the number of possible points k is in principle infinite (the possible values ofk 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 onlycompute such an integral as a sum over specific points on a computer (and we have, somewhatprecariously, 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 gridsetting does, by explicitly setting up a number of integer grid divisions nkx, nky, nkz along eachof the reciprocal lattice vectors Gi, as illustrated in Figure 1 for a two-dimensional example. Inthree 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. Siliconwould turn out to be nonmagnetic, so no explicit spin treatment is needed. The relativistictreatment triggered by the relativistic atomic zora scalar setting is not strictly necessary forsilicon. The nuclear charge of silicon (Z = 14) is still small enough to allow for a non-relativistic

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Figure 1: Illustration of the k-grid for the 2D rectangular lattice. The reciprocal vectors G1 andG2 define a rectangular reciprocal cell. A k-space grid of 3× 3 (example) divides thereciprocal cell into 9 sub-rectangles (green lines) and evaluates the total energy basedon 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.

1.1. Setting up the Structures

The first step towards studying periodic systems with FHI-aims is to construct periodic ge-ometries in the FHI-aims geometry input format geometry.in and visualize them. As a nextstep, we set basic parameters in control.in for periodic calculations. Finally, we compare totalenergies of different Si bulk geometries. Thus:

Task:

• Set up geometry.in files for the Si fcc, bcc, and diamond structures (see Ap-pendix B) Use the approximate lattice constants a of 3.8 A for fcc, 3.1 A forbcc, and 5.4 A for the diamond structure.

• Visualize the resulting structures (e.g., using jmol).

To set up a periodic structure in FHI-aims, all three lattice vectors as well as the atomic positionsin 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 co-ordinates si are dimensionless and the coefficients for the linear combination of thelattice vectors ai. Written out as a formula, this linear combination reads as follows

R = s1 · a1 + s2 · a2 + s3 · a3 (4)

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where R is the Cartesian position of the specified atom.

To visualize the resulting files geometry.in files in jmol, please type

/opt/jmol-14.0.7/jmol.sh geometry.in

To get periodically repeated units of the lattice, open the console in Jmol and type

load ”geometry.in” {3 3 3}

The numbers give the repetition of the structure along the corresponding lattice vector.

1.2. 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 oflattice constant.

Task:

• Prepare a control.in file using 3×3×3 k-points and the settings given in theintroduction.

• Use a shell script (see below) to calculate total energies of the fcc, bcc, anddiamond phases of Si as a function of lattice constant a. Consider 7 differentvalues of a in steps of 0.1 A, centered around the lattice parameters given above,for each structure.

The basic settings in control.in were given near the beginning of this documents. To startout, please use:

k g r i d 3 3 3

We will later find that, for a small unit cell, this k-point density is by no means enough – butlet 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-aimsby

mpirun -n 4 aims.031214.scalapack.mpi.x | tee aims.out

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 preservethe exact input files along with the output files. Here, however, most of the calculations can bestarted using the shell script described below, which takes care of these things and which onlyneeds slight adjustments.The example script below calculates the total energy of fcc Si with different lattice constants.

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#!/ bin /bash − ls e t −e # Stop on errorf o r A in 3 .5 3 .6 3 .7 3 .8 3 .9 4 .0 4 . 1 ; do

echo ” Proce s s ing l a t t i c e constant $A Angstrom .”mkdir $A

# Use t h i s cons t ruc t f o r s imple c a l c u l a t i o n s . As va lue s# are rep laced verbatim , always put them in to ”(” , ”)” .A2=$ ( python −c ” p r i n t ($A)/2 . 0 ” )

# Write geometry . incat >$A/geometry . in <<EOF# fcc s t r u c t u r e with l a t t i c e constant $A Angstrom .l a t t i c e v e c t o r 0 . 0 $A2 $A2l a t t i c e v e c t o r $A2 0 .0 $A2l a t t i c e v e c t o r $A2 $A2 0 .0atom frac 0 .0 0 .0 0 .0 S i

EOF

# Write con t ro l . incp con t r o l . in $A/ con t r o l . in

# Now run FHI−aims with 4 proces sor s in d i r e c t o r y $Acd $Ampirun −n 4 aims . 031214 . s ca lapack . mpi . x > aims . outcd . .

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 script.sh

if your script is named script.sh.To retrieve the total energies, you could use the following script for post-processing:

#!/ bin /bash − ls e t −e # Stop on errorecho ”# l a t t i c e cons tant s in Angstrom , energ i e s in eV” > energ i e s . datf o r A in 3 .5 3 .6 3 .7 3 .8 3 .9 4 .0 4 . 1 ; do# Check fo r convergence o f c a l c u l a t i o n .i f ( ! grep −−qu i e t ” Se l f−con s i s t ency cy c l e converged . ” \<$A/aims . out ) | | \( ! grep −−qu 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 the l i n e with ” Total energy o f the DFT ”.E=$ (gawk ’ /\ | Total energy o f the DFT/ { pr in t $12 } ’ $A/aims . out )

# Write r e s u l t s to data f i l e .echo ”$A $E” >> en e r g i e s . dat

done

This script extracts the total energies and writes them to a file called energies.dat, along with thelattice constants. You will need to adapt this script to the other phases of silicon. In particular,adjust the lattice constants.

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Note that, in order to compare total energies for different phases of Si, it is advantageous towrite out the total energy per atom, not per unit cell. This makes a difference for the diamondstructure. For example, use the expression Eatom=$(python -c ”print ($E)/2.0”) for the diamondstructure and puts it into the bash variable $Eatom. You can then write this variable insteadof $E to the data file.

1.3. Plot the Resulting Total Energies

Task:

• Plot the resulting total energies per atom(!) for fcc, bcc, and diamond siliconas a function of the lattice constant (e.g., using xmgrace).

• What is the most stable bulk phase of Si according to your results?

Thus, plot your data (given in fcc/energies.dat, bcc/energies.dat, and diamond/energies.dat) bytyping:

xmgrace -legend load fcc/energies.dat bcc/energies.dat diamond/energies.dat

You might find that, with the current computational settings, the diamond Si phase is unfavor-able compared to the other two phases. However, the experimentally most stable phase is thediamond structure. We will next show that the too coarse 3× 3× 3 k-grid is the reason for thisdisagreement.

1.4. Converging the k-Grid

Next, we will explicitly check total energy convergence with respect to the k-grid and to thebasis 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.

Task:

• Calculate the total energies for only one of the Si phases as a function of thelattice constant for k-grids of size 8 × 8 × 8, 12 × 12 × 12, and 16 × 16 × 16.Otherwise, use the same computational settings and the same lattice constantsas before.

• Prepare a plot with all total energies drawn against lattice constant. Add thepreviously calculated 3× 3× 3 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 calculationsshould be done exactly as in the last problem but with the appropriate changes to control.in.Discuss the resulting curves and decide which k-space grid would have been “good enough” foryour results.

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1.5. Converging the Basis Set

In the following, just use a 12× 12× 12 k-grid for all three phases.

Task:

• Calculate the total energies for your chosen phase of Si as a function of thelattice constant for the minimal basis and for the full tier1 basis sets. Use thesame lattice constants and computational settings as before together with the12× 12× 12 k-grid.

• Again, prepare a plot with the total energies. Add the results for the mini-mal+spd basis set (the default for the light species settings) from the k-pointconvergence test above.

In order to change the basis size settings, look into the species dependent settings within con-trol.in. There, you will find a line starting with

# ”Fi r s t t i e r ”

Each line after this defines a group of basis functions (radial function type and angular momen-tum) which is added to the minimal basis. In the “light” defaults for Si, there is one additionalradial 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 prepend-ing a ‘#’ character.To run FHI-aims with a full tier1 basis set, uncomment all four lines following # “First tier” byremoving the initial ‘#’ character.Can you make a statement about the accuracy of the total energy (how strongly does it changeand 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, extentof 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 bulksilicon.

2.1. Total Energies as a Function of Lattice Parameter

Task:

• Calculate the total energy of fcc, bcc, and diamond Si as a function of latticeconstant a. Use a k-grid of 12×12×12 points, the minimal+spd basis set (lightdefaults) 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 structureof silicon is most stable in LDA among the crystal structures studied here.

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2.2. Cohesive Energies as a Function of Volume per Atom

The cohesive energy (Ecoh) of a crystal is the energy per atom needed to separate it into itsconstituent neutral atoms. Ecoh is defined as

Ecoh = Ebulk − Eatom, (5)

where Ebulk is the crystal’s total energy per atom(!). Eatom is the energy of an isolated atom.We thus need to recompute the appropriate energy of the isolated Si atom.

Task:

• Perform a total energy calculation of the free silicon atom

• Calculate the cohesive energies and the volume of the crystal per atom for allthe structures treated in this section so far.

• Plot the cohesive energies of all three phases into one plot, using the atomicvolume as the x axis.

For the free atom calculation in the LDA, simply use the following settings: (this will lead to aspherically symmetric atomic state, but we do not need to worry about this here)

# Phys ica l s e t t i n g sxc pw−ldasp in c o l l i n e a rd e f au l t i n i t i a l momen t hundr e l a t i v i s t i c atomic zora s c a l a r

In the species defaults, adjust the following keywords

. . .cut pot 8 1 .5 1 .0b 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 latticeconstant behavior of all phases on equal footing. One possibility to do so is to express the latticeconstant in terms of the volume per atom. This atomic volume can be calculated quite easilyfrom the lattice constant a. The simple cubic (super-)cell has the volume Vsc = a3. This number

has to be divided by the number of atoms Nsc in this cell Vatom = a3

Nsc. Please verify that there

are two, four, and eight atoms in the simple cubic supercell in the case of the bcc, fcc, and thediamond structure, respectively.After plotting E(V ) (where V is the volume per atom) using xmgrace, the diamond structure isindeed the lowest-energy phase. Yet, it is considerably more space consuming than the two close-packed phases. At high pressure, the lower-volume phases might become favorable according tothe Gibbs free energies of the different phases,

G = E − TS + pV . (6)

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2.3. Birch-Murnaghan Equation of State

The lowest-energy lattice parameter a0 is an important quantity which we can calculate fromour data. In principle, this could be done with a quadratic fit for E(a) or E(V ). Here, wewill discuss and use a thermodynamically motivated and more accurate fitting function, theBirch-Murnaghan Equation of State[3, 4]. The energy per atom (E = −Ecoh) is expressed as afunction of the atomic volume (V = Vatom)

E(V ) = E0 +B0V

B′·

((V0/V )B

′

B′ − 1+ 1

)− B0V0B′ − 1

(7)

V0 and E0 are the lowest-energy atomic volume and energy per atom, respectively. B0 is theso-called bulk modulus and B′0 its derivative with respect to pressure. Equation (7) can bederived by assuming a constant pressure derivative B′0.

Task:

• Fit the cohesive energy data for the three phases to the Birch- MurnaghanEquation of State using the program murn.py.

• Determine the lattice constant a0, the bulk modulus B0, and the cohesive energyper atom Ecoh at minimum energy for each phase.

• Compare the above quantities for the diamond phase with the experimentalvalues of a = 5.430 A, B0 = 98.8 GPa, and Ecoh = 4.63 eV[5].

• Plot the cohesive energies E(V) with respect to the atomic volume for all threephases.

The fitting program murn.py is part of the FHI-aims distribution. You can get some documen-tation by typing

murn . py − −help

The script takes an input file with two columns, the first containing the volume and the secondtotal energies. Using the file name murn.in as an example, one can simply use the script with

murn . py murn . in −o f i t . dat

The program then writes the parameters V0, E0, B0, and B′0 for the given data set to the outputfile, here called ’fit.dat’.As a quick plausibility check of the fit, you can use the option -p to get a visual impression. Thescript performs no unit conversions, so the bulk modulus B0 is given in units of eVA−3 becausethe cohesive energies and atomic volumes were provided in eV and A3, respectively. You canuse GNU units to convert to SI units. For example, use

un i t s −v ”0 .5 eV/angstromˆ3” ”GPa”

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to convert 0.5 eVA3 to about 80 GPa. The optimal lattice constant can be calculated from theequilibrium atomic volume V0 by

a0 = 3√NscV0 (8)

with Nsc 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 fittedcurves saved in fit.dat 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 smallerthan the volume at the lowest energy. It is, in fact, the Gibbs free energy that is minimized atconstant 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 thusgiven 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 slopep) corresponds to a constant Gibbs energy. The phase with the lowest tangent “wins” (is themost stable phase at given pressure p).What is particularly interesting are pressure values for which two phases have a “commontangent”. 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 volumebecomes stable.Thus, the slope of a common tangent between the E(V ) curves of two different phases marks atransition pressure, i.e., 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 thecommon tangent between two phases, one with lower energy and higher volume, the other withhigher 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 diamondand bcc Si could coexist according to our calculations. Hint: The value should be somewherebetween 10 GPa and 20 GPa. This is somewhere around 100 times the ambient pressure ofabout 100 kPa.Note, however, that there are additional possible crystal structures for silicon which we havenot calculated here. In reality, the Si β-tin phase is a more stable high-pressure phase than thebcc phase. See, for instance, Reference [1].

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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/2l a t t i c e v e c t o r a/2 0 .0 a/2l a t t i c e v e c t o r a/2 a/2 0 .0atom frac 0 .00 0 .00 0 .00 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/2l a t t i c e v e c t o r a/2 −a/2 a/2l a t t i c e v e c t o r −a/2 a/2 a/2atom frac 0 .00 0 .00 0 .00 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/2l a t t i c e v e c t o r a/2 0 .0 a/2l a t t i c e v e c t o r a/2 a/2 0 .0atom frac 0 .00 0 .00 0 .00 S iatom frac 0 .25 0 .25 0 .25 S i

References

[1] M. T. Yin and M. L. Cohen, “Microscopic theory of the phase transformation and latticedynamics of si,” Physical Review Letters, vol. 45, pp. 1004–1007, Sept. 1980.

[2] J. P. Perdew and Y. Wang, “Accurate and simple analytic representation of the electron-gascorrelation-energy,” Physical Review B, vol. 45, pp. 13244–13249, Jan. 1992.

[3] F. Birch, “Finite elastic strain of cubic crystals,” Physical Review, vol. 71, no. 11, pp. 809–824, 1947.

[4] F. D. Murnaghan, “The compressibility of media under extreme pressures,” Proceedings ofthe National Academy of Sciences of the United States of America, vol. 30, pp. 244–247,July 1944.

[5] C. Kittel, Introduction to Solid State Physics. Hoboken, NJ: John Wiley & Sons, Inc, 8 ed.,2005.

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