Introduction to Dislocation Mechanics

Introduction to Dislocation Mechanics

What is Dislocation Mechanics? (as meant here) The study of the stress state and deformation of continua whose elastic response is mediated by the nucleation, presence, motion, and interaction of distributions of crystal defects called dislocations

For the moment, Dislocation? An imaginary curve in an elastic continuum that induces a stress field in the elastic body is capable of moving and altering its shape kinetics driven by stress Multiple dislocations interact through their stress fields The stress field of one dislocation modifies the stress acting on another thus affecting the latter s placement Cell walls in OFHC Cu, fatigue loading Zhang, Jiang, 07 Dislocation loops in Si http://www.tf.uni-kiel.de/matwis/amat/def_en/kap_5/illustr/i5_4_1.html

What is a Dislocation? Edge dislocation (after G. I. Taylor, 1934) Burgers vector Also Polanyi 34; Orowan, 34 Line direction www.roosterteeth.com 2012books.lardbucket.org coefs.uncc.edu/hzhang3/w-o-m/

What is a Dislocation? Screw dislocation (after J. M. Burgers, 1939) Burgers vector coefs.uncc.edu/hzhang3/w-o-m/ Line direction

What is a Dislocation? Edge-Screw dislocation (after J. M. Burgers) Burgers vector for entire dislocation Dislocation line coefs.uncc.edu/hzhang3/w-o-m/

7 Why should Dislocation Mechanics be studied? Dislocations in crystalline materials are an inevitable consequence of the storage of elastic energy Once formed, they critically affect mechanical electrical electronic optical performance of devices and structures built from crystalline materials

8 Electronic Materials Mechanical stress is a central factor in Fabrication Performance Reliability Source of generation and motion of undesirable threading dislocations Easy diffusion paths for dopants short circuits across layers Electron-hole recombination centers Sites for defect nucleation, growth and multiplication Strain engineering Altering electronic band-gaps of devices by suitably positioning defects

9 Semiconductor Technology: Thin film-substrate Heterostructure Slip plane Interface misfit segment Semiconductor (GaInAs, SiGe, GaN) thin film ~ 10 nm thick film ~ 600 nm Threading segment Interface misfit dislocation Bulk substrate (Si, SiC)

10 Semiconductor Technology: Thin Film-Compliant Substrate System X-grid of screw dislocations InSb, InGaP film ~300-600 nm Twist-bonded Compliant substrate GaAs ~3-10 nm Bulk substrate

11 IC, MEMS Technology Interconnects: Thermal and residual stress Voids and cracks in metallization layers leading to failure - stress migration Dislocation motion induces stress relaxation Important to understand magnitude of relaxation for quantitative failure estimates (reliability) Fatigue failure of MEMS components Experimental results indicate evolving dissipative mechanisms Residual dislocation stress + relaxation by dislocation motion need to be modeled to understand plasticity at micron length-scales.

12 Interconnects grain Passivation layer SiO 2 Al metallization ~500nm X 1000 nm Si substrate

13 MEMS frequency cycles Macroscopic plasticity does not work for structural dimensions of ~10 m - 0.1 m Gradient plasticity inadequate for detailed analysis of local stress concentrations that drive failure processes Dislocation mechanics required to understand stress concentration and relaxation in MEMS structures

14 Structural Components Inhomogeneous deformation - precursor to failure Strength Formability - Ductility Residual Stress Fatigue

Target Predictions of Dislocation Mechanics Capability: Fine Features Dislocation nucleation due to elastic instability s 15 c cohesive reaction b/4 0 b slip Instability leading to nucleation max b Simple cubic lattice shear traction c cohesive reaction

16 Nucleation vs. Motion a) AA, Beaudoin, Miller, 08 b) c) Figure 1. Schematic illustration of dislocation motion and nucleation; a) motion of an existing edge dislocation resulting in an advance of the slipped region; b) nucleation of an edge dislocation; c) nucleation of an edge dislocation dipole. Red lines indicate slipped regions of the crystal; green lines represent unslipped (but possibly deformed) regions; and black lines represent dislocations as the boundary between slipped and unslipped regions.

17 Nucleation vs. Motion AA, Beaudoin, Miller, 08 Figure 6: Nucleation and motion of a dislocation dipole during nano-indentation, with contours showing relative magnitudes of atomic motion (Å). (a) the undefected cystal. (b) nucleation (c) growth to a full Burgers vector and (d)-(f) motion.

Target Predictions of Dislocation Mechanics Capability: Fine Features Dislocation multiplication - Frank-Read source 18 Screw dislocation

19 Target Predictions of Dislocation Mechanics Capability: Fine Features Hardening due to interactions Short range stress field Forest hardening Dissociation energetically favorable Partial dislocation Stacking fault in crystal Long range stress field Lomer-Cottrell lock

Orientation dependence of work hardening 20 Clarebrough and Hargreaves, AJP, 1960

Stress Target Predictions of Dislocation Mechanics Capability: Coarse Features Arising from lattice stretching Due to presence of dislocations (residual or internal stress) Due to applied loads Hardening Retardation of dislocation motion due to modification of local stress field acting on dislocation due to stress field of others Deformation Elastic stretching Slip due to dislocation motion (permanent deformation) deformation microstructure Time dependence of mechanical response 21

22 Patchy Slip brass Piercy, Cahn & Cottrell, 1955

23 Slip bands, localization Chang & Asaro, 1980 Al Cu

W O S { O } q : = q -q Related kinematical question + - W \ = : W * C + - Characterize the possible jumps in q field on S such that grad q on W \ where A is a given C vector field * = A on W satisfying curl A = 0. * q =-ò A d x C for any closed curve C surrounding O, and this is constant on S. S 1

a d e b c

Discontinuity of a Discontinuity Terminating curve of Displacement discontinuity = DISLOCATION Polar angle of director discontinuity = DISCLINATION

The classical question (Volterra - dislocations, Frank nematic disclinations) W O S { O } + - W \ = : W * 2 üï Minimize ò grad q dv W or ï ý solve div grad q = 0 on W \ S ï ïþ + - subject to q - q = : q = 2 pk on grad q n = 0 on S and say grad q n = 0 on W S 1 # grad q has to blow up like as r 1 # energy density like 2 r total energy in W is unbounded x O

Classical field of a screw dislocation/nematic wedge disclination x 2 x 3 x 1 u e e 3 13 1 23 2 æx ö 2 = q = arctan ç çèx ø 1 b sinq = q, = - 2 4p r 1 b cos q = q, = 2 4p r 1 Discontinuous Displacement (even apart from origin) Except origin, smooth strain field!!!! Moral So, dislocation strain fields are not really the ones from taking a deriv. of the displacement field BUT derivative fields obtained on the Simply-connected domain Induced by the cut

A slightly different, partial, alternate formulation As an alternate problem for grad q, ask to find A s.t. How to do this? curl A =-do2pk e ü z ï ïýï div A = 0 ïþ A n = 0 on W on W Punctured domain etc. not physical and Impossible for practical computation However, this does not say anything about determining q with the required properties. Need formulation that produces finite energy AND an associated q field without requiring cuts, holes etc.

Dislocation-Eigendeformation Formulation # Infinite TOTAL energy classical solution t core n q + q - l is troublesome. Can the problem be forced so as to give finite energy, keeping most global features intact? # Regularize jump across S (but not only q) S l + - ( )( q q ) # l = g t - l n in Sl; 0 otherwise. t = x t and g constant in layer outside core and decays to zero inside core So, l( n ) is only non-zero component \ l( t) = 0 and l( n), t ¹ l( t), n # b : = curl l ¹ 0 and localized in core, and ò curl l ezda = q = 2pK A for any area patch A containing core

Dislocation-Eigendeformation formulation # Replace grad q in classical defect theory by E d : = grad q -l # Replace div grad q = 0 on W \ S by a) div E = 0 on W # Replace grad q n = 0 on W by b) E n = 0 on W # 2 pk smeared over core =- b = curle Recall, alternate problem for grad q field in classical theory curl A =-do2pk e ü z ï ïýï on W div A = 0 ïþ A n = 0 on W morally E = A (where A is a classical construct). but E has finite energy. # \ since we also have that outside S E = gradq by construction, we have managed to define a potential field q whose gradient field grad q matches A in most of the domain. l Main utility of eigendeformation formulation is it provides a new field for specification of dynamics of dislocation lines. d d d

Connection of eigendeformation formulation with classical picture # classical question: solve div é grad qù êc = 0 on W \ S ë úû + - subject to q - q = q = 2 pk on S Cgrad q n = 0 on S and say Cgrad q n = 0 on W without assumption A =gradq t core n S l c q + q - l # Write l = grad z + g üï ï curl g = b = curl l ï ý ïïï div g = 0 ïþ g n = 0 on W SH on W d+ d- q - q = z = q " values of l ³ 0, c ³ 0 g is a smooth field except at origin for c = 0. z + - z - = ò l æ = q + -q - l ö dx as l 0 p : é0,1ù y x 0 yl ë û ' t + - + p 2 n ç è ø x0 > c d # div é ù = div é grad ( q - z ) ù = div é ù ëê CE ûú 0 ëê C ûú ë Cg û d 1 1 q - z = q s.t. div é grad q ù = div é ù for c ¹ 0 ëê C úû ë Cg û d 1 T : = CE (say with q = 0 on W) with q smooth for c ¹ 0 d T = 0 across any surface

Connection of eigendeformation formulation and classical picture c as l 0 In W \ S T = C gradq ; divt = 0 d ( d grad q l) T = C -, but in W \ S l = 0 l d d d \ In W \ Sl T = C gradq ; divt = 0 d d q = q ; T = 0 c But did not require cut surface punctures etc.