ExamplesExamples of of Defects Defects in in Solids fromSolids fromESPRESSOESPRESSO

Alessandra SattaAlessandra Satta

SLACS-INFM/CNRSLACS-INFM/CNRSardinian LAboratory for Computational Materials ScienceSardinian LAboratory for Computational Materials Science

Cagliari, September 2005

Hands-on Tutorial on the Quantum-ESPRESSO Package

Defects in crystal lattices are classified according to their dimension:

• 0-dimensional: point defects (vacancies, interstitials…)• 1-dimensional: dislocations, multiple junctions• 2-dimensional: grain boundaries, stacking faults• 3-dimensional: precipitates, voids

0D0D

2 nm2 nm

1D1D

2D2D

3D3D

Native: Vacancy

Self - Interstitial

Impurity: Substitutional

Interstitial

Native: Vacancy

Self - Interstitial

Impurity: Substitutional

Interstitial

System:

with gap

without gap

System:

with gap

without gap

InsideInside

Point DefectsPoint Defects

vacancy Self-interstitial impurity

Typical description of a defect structureTypical description of a defect structure

•Energy:

Total E Stability, formation, migration, activation

•Structure:

Geometry Atomic Configuration

Electronic structure Bands, DOS, Charge Density

Point Defects as a paradigmPoint Defects as a paradigm

1. Energy: Stability, formation, migration, activation

A single vacancy first: ‘just’ a missing atom

Remove one atom and relax

Initial Guess Final Structure

1. Input: si_b.inp

2. pw.x < si_b.inp > si_b.out

&control calculation='scf', restart_mode='from_scratch', prefix='si_b' pseudo_dir = '/opt/espresso/pseudo/', outdir=’./pwfiles', disk_io='high” / &system ibrav = 1, nat=8, ntyp= 1, celldm(1) =10.21, ecutwfc = 18.0 / &electrons mixing_beta = 0.7 conv_thr = 1.0d-8 /ATOMIC_SPECIES Si 28.086 Si.vbc.UPFATOMIC_POSITIONSSi 0.000 0.000 0.000Si 0.500 0.500 0.000Si 0.500 0.000 0.500Si 0.000 0.500 0.500Si 0.250 0.250 0.250Si 0.750 0.750 0.250Si 0.750 0.250 0.750Si 0.250 0.750 0.750K_POINTS {automatic}4 4 4 0 0 0

1. Input: si_b.inp

2. pw.x < si_b.inp > si_b.out

3. grep ! si_b.out

! total energy = -63.37728748 ryd

4. Input: vacancy

ATOMIC_POSITIONSSi 0.000 0.000 0.000Si 0.500 0.500 0.000Si 0.500 0.000 0.500Si 0.000 0.500 0.500Si 0.250 0.250 0.250Si 0.750 0.750 0.250Si 0.750 0.250 0.750Si 0.250 0.750 0.750

ATOMIC_POSITIONSSi 0.000 0.000 0.000Si 0.500 0.500 0.000Si 0.500 0.000 0.500Si 0.000 0.500 0.500

Si 0.750 0.750 0.250Si 0.750 0.250 0.750Si 0.250 0.750 0.750

N N - 1

&control calculation=’relax', restart_mode='from_scratch', prefix='si_v' pseudo_dir = '/opt/espresso/pseudo/', outdir='./pwfiles', disk_io='high” / &system ibrav = 1, nat=7, ntyp= 1, celldm(1) =10.21, ecutwfc = 18.0 / &electrons mixing_beta = 0.7 conv_thr = 1.0d-8 /ATOMIC_SPECIES Si 28.086 Si.vbc.UPFATOMIC_POSITIONSSi 0.000 0.000 0.000Si 0.500 0.500 0.000Si 0.500 0.000 0.500Si 0.000 0.500 0.500Si 0.750 0.750 0.250Si 0.750 0.250 0.750Si 0.250 0.750 0.750K_POINTS {automatic}4 4 4 0 0 0

1. Input: si_b.inp

2. pw.x < si_b.inp > si_b.out

3. grep ! si_b.out

4. Input: vacancy

5. pw.x < si_v.inp > si_v.out

6. grep ! si_v.out

! total energy = -55.21468569 ryd unrelaxed! total energy = -55.21488549 ryd! total energy = -55.20851601 ryd! total energy = -55.21524357 ryd! total energy = -55.21536466 ryd! total energy = -55.21536471 ryd relaxed

Total force = 0.012467 Total SCF correction = 0.000014Total force = 0.010535 Total SCF correction = 0.000045Total force = 0.038545 Total SCF correction = 0.000001Total force = 0.005252 Total SCF correction = 0.000016Total force = 0.000012 Total SCF correction = 0.000007Total force = 0.000130 Total SCF correction = 0.000006

6. grep ! si_v.out

7. grep “Total force” si_v.out

We now have all the ingredients to calculate theEnergy of Formation of a single vacancy in silicon

€

E f = Etot N − 1;1[ ] − N − 1N

Etot N;0[ ]

[ -55.21468569 - 7/8 *( -63.37728748 )] Ry =

= 0.24044086 Ry = 3.27 eV

[ -55.21536471 - 7/8 *( -63.37728748 )] Ry =

= 0.23976184 Ry = 3.26 eV

unrelaxed

relaxed

3.17-3.63.17-3.6aa3.26 (3.27)3.26 (3.27)Si[V]Si[V]

othersothersourourEEf f ((eV eV ))

a- M. Probert et al. PRB 67, 075204 (2003),

O. Pankratov, et al. PRB 56, 13172 (1997), and refs. therein

Simulation cell: too small!

•Energy: Total E formation, migration, activation

The vacancy formation functions at P = const: the Gibbs energy Gf,the enthalpy Hf , and the entropy Sf, are defined as differencesbetween

(defective crystal) - (perfect crystal)

Gf  =  Hf  –  T · Sf

at P = 0 formation energy and enthalpy areindistinguishable :

€

Hf = Etot N − 1;1;Ω[ ] − N − 1N

Etot N;0;NΩ0[ ]

The calculation of the migration barrier Emrequires the knowledge of the saddle point

EmNudged Elastic Band

Migration and Diffusion parameters

Next step to obtain the diffusioncoefficient: migration barrierNext step to obtain the diffusioncoefficient: migration barrier

(next days!)

… back to the structure

•Structure:

Geometry : Atom displacements from bulk positions

xcrysden : calculate the distances between the atomsneighboring the defect: nearest neighbors (NN),

second NN …

and compare with : EXAFS, RBS-C, XSW …

•Structure:

Geometry : Atom displacements from bulk position

Electronic structure : Bands, DOS, Charge Density

We calculate the Density of States and theelectronic charge density in crystalline siliconwith and without defects

&control calculation='scf', restart_mode='from_scratch', prefix='si_b' pseudo_dir = '/opt/espresso/pseudo/', outdir='/scratch/si_vacancy/pwfiles', disk_io='high” / &system ibrav = 1, nat=8, ntyp= 1, celldm(1) =10.21, ecutwfc = 18.0, occupations=‘tetrahedra’ / &electrons mixing_beta = 0.7 conv_thr = 1.0d-8 /ATOMIC_SPECIES Si 28.086 Si.vbc.UPFATOMIC_POSITIONSSi 0.000 0.000 0.000Si 0.500 0.500 0.000Si 0.500 0.000 0.500Si 0.000 0.500 0.500Si 0.250 0.250 0.250Si 0.750 0.750 0.250Si 0.750 0.250 0.750Si 0.250 0.750 0.750K_POINTS {automatic}4 4 4 0 0 0

&control calculation=’nscf', restart_mode='from_scratch', prefix='si_b' pseudo_dir = '/opt/espresso/pseudo/', outdir='/scratch/si_vacancy/pwfiles', disk_io='high” / &system ibrav = 1, nat=8, ntyp= 1, celldm(1) =10.21, ecutwfc = 18.0, occupations=‘tetrahedra’ / &electrons mixing_beta = 0.7 conv_thr = 1.0d-8 /ATOMIC_SPECIES Si 28.086 Si.vbc.UPFATOMIC_POSITIONSSi 0.000 0.000 0.000Si 0.500 0.500 0.000Si 0.500 0.000 0.500Si 0.000 0.500 0.500Si 0.250 0.250 0.250Si 0.750 0.750 0.250Si 0.750 0.250 0.750Si 0.250 0.750 0.750K_POINTS {automatic}6 6 6 0 0 0

DOSscf nscf

&inputpp outdir='./pwfiles/' prefix='si_b' fildos=’./si_bulk.dos' /

si_b.dos.inp : the input for DOS calculation in silicon bulk

dos.x < si_b.dos.inp > si_b.dos.out

# E (eV) dos(E) Int dos(E) … … … -5.709 0.2477E+00 0.1036E-01 -5.699 0.2948E+00 0.1331E-01 -5.689 0.3426E+00 0.1673E-01 … … …

6.071 0.2235E+00 0.3199E+02

6.081 0.1834E+00 0.3200E+02 6.091 0.1417E+00 0.3200E+02 … … …

The code will printout the calculatedDOS in si_b.dosthat can beplotted withxmgace or gnuplot EF =

The Fermi energy is thereference value we set

as the top of thevalence band.

along the x-axis:

E = E - EF =

E - 6.081 eV

Now the procedure is similar for the crystalcontaining the defect

s sp p

&control calculation='scf', restart_mode='from_scratch', prefix='si_v' pseudo_dir = '/opt/espresso/pseudo/', outdir=’./pwfiles', disk_io='high” / &system ibrav = 1, nat=7, ntyp= 1, celldm(1) =10.21, ecutwfc = 18.0, occupations=‘tetrahedra’ / &electrons mixing_beta = 0.7 conv_thr = 1.0d-8 /ATOMIC_SPECIES Si 28.086 Si.vbc.UPFATOMIC_POSITIONSSi 0.003081476 0.003081476 0.003081476Si 0.496918524 0.496918524 0.003081476Si 0.496918524 0.003081476 0.496918524Si 0.003081476 0.496918524 0.496918524Si 0.750000000 0.750000000 0.250000000Si 0.750000000 0.250000000 0.750000000Si 0.250000000 0.750000000 0.750000000K_POINTS {automatic}4 4 4 0 0 0

&control calculation=’nscf', restart_mode='from_scratch', prefix='si_v' pseudo_dir = '/opt/espresso/pseudo/', outdir=’./pwfiles', disk_io='high” / &system ibrav = 1, nat=7, ntyp= 1, celldm(1) =10.21, ecutwfc = 18.0, occupations=‘tetrahedra’ / &electrons mixing_beta = 0.7 conv_thr = 1.0d-8 /ATOMIC_SPECIES Si 28.086 Si.vbc.UPFATOMIC_POSITIONSSi 0.003081476 0.003081476 0.003081476Si 0.496918524 0.496918524 0.003081476Si 0.496918524 0.003081476 0.496918524Si 0.003081476 0.496918524 0.496918524Si 0.750000000 0.750000000 0.250000000Si 0.750000000 0.250000000 0.750000000Si 0.250000000 0.750000000 0.750000000K_POINTS {automatic}6 6 6 0 0 0

scf nscf

&inputpp outdir=’./pwfiles/' prefix='si_v' fildos=’./si_v.dos' /

si_v.dos.inp : the input for DOS calculation in defectivesilicon

dos.x < si_v.dos.inp > si_v.dos.out

si_v.dos:A new peak in the gap due to the presence of

the vacancy

Why is the peak so broad?

Increasing the size of simulation cellfrom 8 to 64 lattice sites to 216 …

bulk

vacancySplitting of thedegeneracy dueto Jahn-Tellereffect

unrelaxed relaxedunrelaxed relaxed

The electronic charge density

&inputpp prefix = 'si_b' outdir=’./pwfiles/' filplot = ’./si_b.charge' plot_num= 0 /

&inputpp prefix = 'si_b' outdir=’./pwfiles/' filplot = ’./si_b.charge' plot_num= 0 /

bulk : si_b.pp_rho.inpbulk : si_b.pp_rho.inp

pp.x < si_b.pp_rho.inp > si_b.pp_rho.outpp.x < si_b.pp_rho.inp > si_b.pp_rho.out

The bulk first…

… and the vacancy

&inputpp prefix = 'si_v' outdir=’./pwfiles/' filplot = ’./si_v.charge' plot_num= 0 /

&inputpp prefix = 'si_v' outdir=’./pwfiles/' filplot = ’./si_v.charge' plot_num= 0 /

vacancy :si_v.pp_rho.inpvacancy :si_v.pp_rho.inp

pp.x < si_v.pp_rho.inp > si_v.pp_rho.outpp.x < si_v.pp_rho.inp > si_v.pp_rho.out

&inputpp --------------/&plotnfile = 1 filepp(1) = ’./si_b.charge' weight(1) = 1.0 iflag = 2 output_format = 2 fileout = ' ./si_b.2D.rho.dat' x0(1) =0.0, x0(2)=0.0, x0(3) = 0.0, e1(1) =1.0, e1(2)=1.0, e1(3) = 0.0, e2(1) =0.0, e2(2)=0.0, e2(3) = 1.0, nx=56, ny=56 /

&inputpp --------------/&plotnfile = 1 filepp(1) = ’./si_b.charge' weight(1) = 1.0 iflag = 2 output_format = 2 fileout = ' ./si_b.2D.rho.dat' x0(1) =0.0, x0(2)=0.0, x0(3) = 0.0, e1(1) =1.0, e1(2)=1.0, e1(3) = 0.0, e2(1) =0.0, e2(2)=0.0, e2(3) = 1.0, nx=56, ny=56 /

2-Dimensional bulk : si_b.pp_rho2D.inp

pp.x < si_b.pp_rho2D.inp > si_b.pp_rho2D.outpp.x < si_b.pp_rho2D.inp > si_b.pp_rho2D.out

./si_b.2D.rho.dat

./si_b.2D.rho.psn0 0.09 7

./si_b.2D.rho.dat

./si_b.2D.rho.psn0 0.09 7

plotrho.x < si_b.plotrho.inp > si_b.plotrho.outplotrho.x < si_b.plotrho.inp > si_b.plotrho.out

bulk : si_b.plotrho.inpbulk : si_b.plotrho.inp

[110]

[001]

&inputpp --------------/&plot nfile = 1 filepp(1) = ’./si_v.charge' weight(1) = 1.0 iflag = 2 output_format = 2 fileout = ' ./si_v.2D.rho.dat' x0(1) =0.0, x0(2)=0.0, x0(3) = 0.0, e1(1) =2.0, e1(2)=2.0, e1(3) = 0.0, e2(1) =0.0, e2(2)=0.0, e2(3) = 1.0, nx=56, ny=56 /

&inputpp --------------/&plot nfile = 1 filepp(1) = ’./si_v.charge' weight(1) = 1.0 iflag = 2 output_format = 2 fileout = ' ./si_v.2D.rho.dat' x0(1) =0.0, x0(2)=0.0, x0(3) = 0.0, e1(1) =2.0, e1(2)=2.0, e1(3) = 0.0, e2(1) =0.0, e2(2)=0.0, e2(3) = 1.0, nx=56, ny=56 /

2-Dimensional vacancy : si_v.pp_rho2D.inp

pp.x < si_v.pp_rho2D.inp > si_v.pp_rho2D.outpp.x < si_v.pp_rho2D.inp > si_v.pp_rho2D.out

./si_v.2D.rho.dat

./si_v.2D.rho.psn0 0.09 7

./si_v.2D.rho.dat

./si_v.2D.rho.psn0 0.09 7

plotrho.x < si_v.plotrho.inp > si_v.plotrho.outplotrho.x < si_v.plotrho.inp > si_v.plotrho.out

vacancy : si_v.plotrho.inpvacancy : si_v.plotrho.inp

[110]

[001]

Increasing the size of thesimulation cell up to 216lattice sites

Rebonding effectthrough pairing ofSi-Si in the [110]zigzag chain

3-Dimensional bulk : si_b.pp_rho3D.inp

&inputpp --------------/&plot nfile = 1 filepp(1) = ’./si_b.charge' weight(1) = 1.0 iflag = 3 output_format = 3 fileout = ’./si_b.3D_bis.rho.xsf' x0(1) =0.0, x0(2)=0.0, x0(3) = 0.0, e1(1) =1.0, e1(2)=1.0, e1(3) = 0.0, e2(1) =1.0, e2(2)=-1.0, e2(3) = 0.0, e3(1) =0.0, e3(2)=0.0, e3(3) = 1.0, nx=56, ny=56, nz=56 /

&inputpp --------------/&plot nfile = 1 filepp(1) = ’./si_b.charge' weight(1) = 1.0 iflag = 3 output_format = 3 fileout = ’./si_b.3D_bis.rho.xsf' x0(1) =0.0, x0(2)=0.0, x0(3) = 0.0, e1(1) =1.0, e1(2)=1.0, e1(3) = 0.0, e2(1) =1.0, e2(2)=-1.0, e2(3) = 0.0, e3(1) =0.0, e3(2)=0.0, e3(3) = 1.0, nx=56, ny=56, nz=56 /

pp.x < si_b.pp_rho3D.inp > si_b.pp_rho3D.outpp.x < si_b.pp_rho3D.inp > si_b.pp_rho3D.out

xcrysden:

File: open si_b.3D_bis.rho.xsf

Tools: Data Grid

xcrysden:

File: open sisi_b.3D__b.3D_bisbis..rhorho..xsfxsf

Tools: Data Grid

Donors and Acceptors

Boron and Arsenic are typical doping species in silicon

1. choose a pseudopotential for the impurities

2. compose the simulation cell

3. impose a volume relaxation

&control

calculation='vc-relax', restart_mode='from_scratch', prefix='SiAs' dt=100.0, tstress = .true. tprnfor = .true., pseudo_dir = ’/opt/espresso/pseudo/', outdir='/scratch/si_doping/pwfiles', disk_io='high'/ &system ibrav = 1, celldm(1) =10.21, nat=8, ntyp= 2, ecutwfc = 18.0, occupations='smearing', smearing='gauss', degauss = 0.0037, / &electrons mixing_beta = 0.7 conv_thr = 1.0d-8 /&ions ion_dynamics='damp' / &cell cell_dynamics='damp-w'/ATOMIC_SPECIES Si 28.086 Si.vbc.UPF As 74.92 As.gon.UPF …………………………….

……………………..ATOMIC_POSITIONSSi 0.000 0.000 0.000Si 0.500 0.500 0.000Si 0.500 0.000 0.500Si 0.000 0.500 0.500As 0.250 0.250 0.250Si 0.750 0.750 0.250Si 0.750 0.250 0.750Si 0.250 0.750 0.750K_POINTS {automatic}2 2 2 0 0 0

WARNING:

The total number ofelectron is odd!

WARNING:

The total number ofelectron is odd!

Ry units

./pw.x < sias.inp> sias.out

unit-cell volume = 1064.3323 (a.u.)^3 new unit-cell volume = 1064.3324 (a.u.)^3 new unit-cell volume = 1064.3328 (a.u.)^3 new unit-cell volume = 1064.3336 (a.u.)^3 new unit-cell volume = 1064.3352 (a.u.)^3 --------------------------------- new unit-cell volume = 1064.3773 (a.u.)^3 new unit-cell volume = 1064.3871 (a.u.)^3 new unit-cell volume = 1064.3973 (a.u.)^3 final unit-cell volume = 1064.3973 (a.u.)^3

unit-cell volume = 1064.3323 (a.u.)^3 new unit-cell volume = 1064.3324 (a.u.)^3 new unit-cell volume = 1064.3328 (a.u.)^3 new unit-cell volume = 1064.3336 (a.u.)^3 new unit-cell volume = 1064.3352 (a.u.)^3 --------------------------------- new unit-cell volume = 1064.3773 (a.u.)^3 new unit-cell volume = 1064.3871 (a.u.)^3 new unit-cell volume = 1064.3973 (a.u.)^3 final unit-cell volume = 1064.3973 (a.u.)^3

Strain calculation:Strain calculation:

€

=Δaa0

=astrained − a0

a0= +2.357 •10−5

grep volume sias.outgrep volume sias.out

Exp.: -0.0019 ± 0.0003Exp.: -0.0019 ± 0.0003