Intrinsic Point Defects: Interstitials

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Intrinsic Point Defects: Interstitials

Atomic configurations of interstitials in metalsEquilibrium concentration of self-interstitialsEntropy and enthalpy of interstitial formation in metalsExperimental determination of enthalpy of self-interstitial formationSelf-interstitials in SiGeneration of nonequilibrium self-interstitials and vacancies

References:Allen & Thomas, Ch. 5, pp. 249-257Kelly, Groves, Kidd, Ch. 9, 261-289

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

There are two types of interstitial sites in fcc, bcc and hcp crystal structures: octahedral and tetrahedral

Atomic configurations of interstitials in metals

Octahedral interstice is are bound by 6 atoms that form an octahedron

Tetrahedral interstice is in the center of a tetrahedron formed by four atoms

The number of octahedral interstitial sites per unit cell is 4 for fcc, 2 for hcp, 6 for bcc

The number of tetrahedral interstitial sites per unit cell is 8 for fcc, 4 for hcp, 12 for bcc

Example:

octahedral sites in fcc unit cell

tetrahedral sites in fcc unit cell

The size of the octahedral interstice (ratio of the diameter of the maximum hard sphere which can be fitted to the diameter of the host atom) is larger in fcc than sizes of the tetrahedral and octahedral sites in bcc → solid solubility in bcc is much lower than in fcc

For self-interstitials in metals, the size of both interstices is too small - strong repulsion between atoms (remember steep increase of the interatomic energy at small distances) result in very high Δhf. It can be reduced by forming extended dumbbell or crowdion configurations.

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Equilibrium concentration of self-interstitials

Similarly to the derivation of the equilibrium vacancy concentration, the equilibrium fraction of self-interstitials is defined by the competition between the enthalpy of the vacancy formation and configurational entropy:

In metals (densely packed fcc, hcp, bcc structures) self-interstitials introduce large distortions in the surrounding lattice ⇒ the energy of self-interstitial formation is ~ 2-5 times larger as compared to vacancies ⇒ the equilibrium concentration of self-interstitials is very low (less than one self-interstitial per cm3 at room T) and can be neglected.

Self-interstitials in metals cannot be introduced in a thermodynamically reversible manner, by changing T.

Experiments: no direct evidence of equilibrium interstitials

Atomistic simulations: Δhfi ~ 3-4 eV in close-packed metals

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ksA

Nn

B

if

B

iv

ieq expexp

ifhΔ

A is a pre-factor accounting for the fact that the number of interstitial sites and configurations may not be equal to the number of lattice sites.

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Entropy and enthalpy of interstitial formation in metalsNordlund and Averback, PRL 80, 4201, 1998:

MD simulations of Cu with EAM potential

100/vi cc <

vf

if hh Δ≈Δ 5.2

mTT >

vv

iv ss Δ≈Δ 5.6 - reflects the “size” of the region affected by the point defect. If more atoms

changed their vibrations frequencies, Δsv is larger.

( )zBv vvks 'ln=Δrecall: for interstitials, ν’ > ν 0<Δ i

vs? 0 >Δ i

vswhy

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Entropy and enthalpy of interstitial formation in metals

( )zBv vvks 'ln=Δ

ν’ < ν 0>Δ ivs

Formation of the interstitial is commonly associated with reorganization of the atomic and electronic structure, which extends over several unit cells -interstitials adopt dumbbell or crowdion configurations that have lower formation energy compared to octahedral site

The dumbbell may adopt one of several orientations → positive configurational contribution to Δsv (it may be more appropriate to call it the formation or excess entropy, rather than vibrational entropy).

The formation of extended (dumbell or crowdion) configurations results in the appearance of local vibrational modes, some of which have low frequencies → positive vibrational contribution to Δsv.

interstitial<110>-dumbbell

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=

Tkh

ksA

Nn

B

if

B

iv

ieq expexp

The increase in Δsv and decrease in Δhf → noticeable concentration of interstitials at high T (close to Tm)

Matsukawa and Zinkle, Science 318, 959, 2007

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Entropy and enthalpy of interstitial formation in metalsDerlet, Nguyen-Manh, Dudarev

PRB 76, 054107, 2007DFT calculations for bcc metals

structure of point defects in bcc metals <111>-type configurations have lower Δhf in nonmagnetic bcc metals from groups 5B and 6B; <110> dumbbell configuration has lower Δhf for ferromagnetic bcc Fe

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Experimental determination of enthalpy of self-interstitial formationDue to the very small equilibrium concentration of self-interstitials, Δhf

i and Δsvi cannot be

obtained in a manner similar to vacancies

Irradiation by high-energy particles generate a high concentration of

vacancy-interstitial (Frenkel) pairs

density of Frenkel pairs is found from changes in electrical conductivity

annealing at an elevated T results in annihilation of Frenkel pairs

colorimetric measurements are used to determine the formation enthalpy

of a Frenkel pair, ΔhfFP

vf

FPf

if hhh Δ−Δ=Δ

no experimental methods exist for evaluation of Δsvi

Kubota et al., J. Computer-Aided Mater. Design 14, 367, 2007: MD simulation of 85 keV collision cascade in δ-Pu at 600 K

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Self-interstitials in SiIn more open (compared to metals) crystal structures, Δhf

i can be smaller and equilibrium concentrations of self-interstitials may be significant.

For self-interstitials in Si, Δhfi may be comparable to Δhf

v

and both defects are likely to contribute to high-T diffusion

atomic configurations of self-interstitial in Si [Leung et al., PRL 83, 2351, 1999]

Measurements of self-diffusivity of Si at high T (usingradioactive isotopes of silicon as tracers) → activation energy of ~4.1–5.1 eV

∑=j

jjSD CDD

sTkCD Bii /cm )/84.4exp(914 2−≈

sTkCD Bvv /cm )/03.4exp(6.0 2−≈

contribution of self-interstitials to self-diffusion is increasing with increasing T and becomes dominant above 1300 K

- vacancies

- interstitials

The above values are still being debated and it is difficult to separate Di and Ci.

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Generation of nonequilibrium self-interstitials and vacanciesGeneration of non-equilibrium point defects in material processing or performance

quenching to low T: the high-T concentration of point defects is frozen-in due to the low diffusion - defects (mostly vacancies, interstitials are too mobile) cannot migrate to their sinksClusters of vacancies and pores may form at the initial stage of quenchingThermal stresses may exceed yield strength → generation of dislocations (sinks for point defects)

irradiation: can create thermal spikes, electronic excitations, collision cascades, high densities of Frenkel pairs, pores or disordered (amorphous regions)

plastic deformation: along with dislocations, low-T plastic deformation can introduce significant numbers of vacancies and self-interstitials, e.g., dislocation climb is defined by flux of point defects to/from dislocations, movement of dislocations with jogs leads to the emission or absorption of point defects.

high-energy particles (e.g., neutrons, protons,…) generate high energy “primary knock on atoms” which, in turn, create collision cascades (many Frenkel pairs). Interstitials and vacancies concentrate in outer and central regions of the collision cascade, respectively. May lead to radiation swelling (formation of pores) in nuclear reactor materials.

high-energy electrons (≥ 1 MeV can generate Frenkel pairs - transfer energy of 20-40 eVto an atom). Such electrons can also be generated by high energy γ-rays.

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Generation of nonequilibrium self-interstitials and vacanciesExample: Isochronal recovery of electron-irradiated copper containing an atomic concentration of Frenkel pairs of ~10-6

from Kelly, Groves, Kidd

• annihilation of vacancy-interstitial pairs results in recovery of electrical conductivity• removal of prismatic dislocation loops produced by clustering of vacancies and interstitials

resistivity ρ is assumed to be proportional to the concentration of point defects

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Example: generation of defects by short pulse laser irradiation

175 ps 200 ps 400 ps 450 ps

0 ps 5 ps 10 ps 50 ps 100 ps

MD simulation of irradiation of Cr target by a 200 fs laser pulse at a fluence of 638 J/m2

Lin, Johnson, Zhigilei Phys. Rev. B 77, 214108, 2008

University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei

Distance from the current surface, nmN

umbe

rofv

acan

cies

pera

tom

icpl

ane

0 2 4 6 80

1

2

3

4

5 maximum depthof the melting front

Most of the vacancies are formed at the solid-liquid interface during the resolidification process (cooling rates ~5×1012 K/s at the time of resolidification!)

Additional point defects - from thermally-activated generation of Frenkel pairs followed by a quick escape of highly mobile self-interstitials to the melting front

Very high vacancy concentration:1.7×10-3 in the top 5 nm layer

interstitial (<110>-dumbbell)

cluster of four interstitials (<111>-crowdion)

vacancy

Example: generation of defects by short pulse laser irradiation