Defects in Crystalline Solids: the Basics Arkady Krasheninnikov - - PowerPoint PPT Presentation
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 1 Jul 09, 2017 Defects in Crystalline Solids: the Basics Arkady Krasheninnikov http://users.aalto.fi/~ark/ Ion Beam Centre, Helmholtz-Zentrum Dresden-Rossendorf
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 1 Jul 09, 2017
Arkady Krasheninnikov Ion Beam Centre, Helmholtz-Zentrum Dresden-Rossendorf Germany, and Department of Applied Physics, Aalto University, Finland, and National University of Science and Technology "MISIS", Russian Federation
http://users.aalto.fi/~ark/
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 2 Jul 09, 2017
defects in crystalline solids?
systems, their classification and origin.
thermodynamic equilibrium;
defect identification
aches to get insights into defect behavior
interesting?
Courtesy of J. M eyer Courtesy of K. Suenaga
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 3 Jul 09, 2017
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 4 Jul 09, 2017
What are defects?
In general, structural imperfection in a crystal, deviation from the perfect order (periodicity) vacancy interstitial
Why do we care about defect?
They frequently govern materials properties:
Ashcroft-Mermin: “Like human defects, those of crystals come in a seemingly end-
less variety, many dreary and depressing, and a few fascinating.”
Note that defects can also be defined in amorphous systems and quasicrystals — beyond the scope of the lecture
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 5 Jul 09, 2017
A structural defect is a configuration in which an atom (or group of atoms) does not satisfy the structure rules pertaining to the ideal reference system or material.
Are defects in solids really the “bad” guys?
Ø
Deterioration of mechanical properties
Ø
Break down of electronic devices
Ø
Non-radiative transitions
Ø
Undesirable magnetism
Improvement of mechanical properties in some systems
Ø
Doping
Ø
Providing bright colors
Ø
Pinning of magnetic vortices in type II superconductors
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 6 Jul 09, 2017
Mechanical properties of carbon nanotube paper Experiment: Simulations: ~ 1 cm ~ 4 µm ~ 15 nm ~ 1 nm
~ 50 µm
The continuum theory model with parameters derived from atomistic simulations
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 7 Jul 09, 2017
How can we classify them?
According to their dimensionality:
0D defects
1D defects
3D defects dislocation Disclination (in 2D) Note that in 2D systems 2D and 3D defects do not exist According to their origin:
vacancy interstitial
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 8 Jul 09, 2017
How can we classify them?
According to their thermodynamics (concentration):
defects only)
According to sample chemical content (elemental solids, compounds):
According to the local number of atoms:
interstitials)
defects in graphene, rotational defects in TMDs, metastable I-V complexes in Si
5 7 5 7
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 9 Jul 09, 2017
Schottky defect
formation energy: Ef = E(N-1) + E(1) – E(N) Ef = E(N-1) + E(N)/N – E(N) Ef > 0 !!! N is the number of atoms in the system to infinity
to an empty site at the surface
Frenkel defect
formation energy: Ef = E(N with iv pair) – E(N) Ef = Ef(vac) + Ef(inter) Ef > 0 !!!
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 10 Jul 09, 2017
In general, let’s define formation energy of an electrically neutral point defect (vacancy/interstitial) as :
Ef = Etot(with defect) – Etot(without defect) ± µ
where µ is chemical potential of the missing/extra atom For substitutional impurities (e.g., N atom in graphene):
For defects in compounds (e.g. C impurity in BN) another condition should be met:
Ef = Etot(with defect) + µC – Etot(without defect) –µN Ef = Etot(with defect) + µN – Etot(without defect) –µC Ef = Etot(with defect) + µBN – µB – Etot(without defect) –µC
In general, Ef is a function of chemical potentials (choose ones matching the experimental situation) So called, N-rich and B-rich conditions;
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 11 Jul 09, 2017
Defect formation energy is positive: it costs energy to create a defect;
Why do we have then defects in solids at finite temperatures? vacancy concentration is low (defects do not interact) T = const Assume: P = const The Gibbs free energy of the crystal: G = G0 + nEf – TS, where G0 = Gibbs energy of the ideal crystal; n = defect concentration; Ef > 0 = energy required to create a vacancy S = configurational entropy
n = Nv/ N
S = kB ln N! (N − Nv)!Nv!
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 12 Jul 09, 2017
The Gibbs free energy of the crystal: G = G0 + nEf – TS Entropy gives rise to appearance of point defects at finite temperatures
n,defect concentration energy
~ hn ~ -TS G = hn -TS
equilibrium concentration
¶G/¶n = 0 Ef - T¶S/¶n = 0 At thermodynamic equilibrium: ln x! » x lnx -x, x à ¥
S = kB ln N! (N − Nv)!Nv!
n = Nv / NL = exp −E f / kBT
n = Nv/ N; Nv<< N
~Efn
G = Efn - TS
Synthetic materials (or e.g., subjected to irradiation) concentration
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 13 Jul 09, 2017
Al: melting Tm = 933K T = 300K, n = 3 10-12 T = 800K, n = 2 10-4 exp(2.4) » 10 (prefactor)
B f B f L v
k s T k h N N n / / exp / D + D
=
Small exercise SW defect in graphene, Ef = 4-5 eV How many defects at 300K?
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 14 Jul 09, 2017
Ac Accelerator Ac Accele- ra rator sputtered sample atoms
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 15 Jul 09, 2017
Optical spectroscopy Indirect methods:
Main idea: detection of optical transitions associated with defects
Electron paramagnetic resonance
Main idea: detection of localized electron magnetic moments
Raman spectroscopy
Main idea: detection of new phonon modes associated with defects
X-ray absorption spectroscopy and related methods (XAFS,XANES)
Main idea: probing the electronic states associated with defects by exciting core electrons
Scanning probe microscopy Direct methods:
Main idea: getting an image of a particular defect on the surface Main idea: getting an image of a particular defect in a thin slice
Transmission electron microscopy
finite defect concentration required Detection of individual defects is possible
Positron annihilation spectroscopy
Main idea: a positron gets stuck on the vacancy;we detect the photons which appear upon its annihilation with an electron
(does not work for 2D materials)
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 16 Jul 09, 2017
The main idea: We can detect the magnetic moment associated with the spin of the unpaired electron
Assume we have unpaired localized electrons Example: dangling bond at atoms near vacancies in semiconductors
) 2 / ) 2 ; 2 / 1 magneton Bohr (the Factor Lande (the
e B e s B e s
m e g m B g m E E ! = » ± = + = µ µ
Let’s apply a steady magnetic field B0
E0
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 17 Jul 09, 2017
E E0 B g h
B eµ
n = 2 / 1 + =
s
m 2 / 1
s
m we will have absorption! B0 B g h
B eµ
n =
Let’s apply alternating EM field B1 of frequency n perpendicular to B0 If B0 ~ 0.3 T
n ~ 9 GHz (microwave region)
adsorption peak its derivative (with opposite sign)
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 18 Jul 09, 2017
Raman spectroscopy
spectra; the associated processes are not allowed in pristine graphene Concado et al., Nano Lett. 11 (2011) 3190
area;
peaks appear
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 19 Jul 09, 2017
Raman spectroscopy
In principle, even identification
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 20 Jul 09, 2017
MoSe2 graphene Seeing is believing … (many more examples later on)
Courtesy of J. M eyer Courtesy of U. Kaiser Courtesy of K. Suenaga
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 21 Jul 09, 2017
Courtesy of J. M eyer Courtesy of U. Kaiser
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 22 Jul 09, 2017
Vacancy in graphite:
The defect size on the image is much larger than the actual size. The vacancy appears as protrusion!
Vacancy in GaAs Vacancy in GaP We can detect point defects! (and surely line defects)
simulations experiments
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 23 Jul 09, 2017
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 24 Jul 09, 2017
Compare the relative energies of several atomic configurations (conformations) Find the equilibrium positions of the atoms corresponding to the absolute minimum of energy (the structure) Calculate the properties (mechanical/ electronic/magnetic) of the system
Assume we know how the atoms interact, i.e., we know the energy of the system as a function of atomic coordinates: E = E(R1, R2, R3, R4, …)
force
Model the dynamical behavior of the system at finite temperatures (molecular dynamics) Calculate thermodynamic properties making use of statistical mechanics
mi ∂2 ∂t2 Ri = − ∂ ∂Ri E(R
1, R2,..., Ri,....RN )
SV Adatom
E = E(R)
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 25 Jul 09, 2017
Always a compromise between the computational efficiency and the accuracy
Overview of computational methods in materials science.
Number of atoms in the system Level of sophistication 10 10 10
1 2 3
10
6
HF post HF DFT TB Empirical potentials
Ø
Time-dependent density-functional- theory methods (beyond Born- Oppenheimer approximation)
Ø
Density-functional- theory methods
Ø
Tight-binding methods
Ø
Molecular dynamics with analytical potentials
Ø
Kinetic Monte-Carlo methods 101 102 103 104 108
TD- DFT analytical
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 26 Jul 09, 2017
Empirical (analytical) potentials: E is an analytical function of atom coordinates Semi-empirical (tight-binding) methods:
Parameters are normally chosen to match the experiments or the results of more accurate calculations Parameters of the Hamiltonian are chosen in such a way that the results of calculations match the experiments or ab inito calculations
First-principles (ab initio) methods:
No information from experiments is required: The Schrödinger equation is solved using some approximations (the same for all the materials).
Atomistic simulations: we need to know the energy of the system as a function of atomic coordinates: E = E(RA, RB, RC, RD, …)
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 27 Jul 09, 2017
Log Log Ener nergy gy Log Log ssl
slowing Co Collisions wi with electrons Co Collisions wi with electrons Co Collisions wi with nuclei Co Collisions wi with nuclei h Collisions with nuclei: MD on the BO surface h Collisions with electrons: Time dependent DFT h Both: MD beyond BO approximation (TD-DFT
combined with MD for the atoms)
Simulations:
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 28 Jul 09, 2017
Ø An atom-level
computer simulation can make it much clearer what the process really looks like
Io Ion Ma Material
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 29 Jul 09, 2017
Animation by K. Nordlund, U. Helsinki
Bulk systems 2D materials (free standing) 2D materials (free standing)
the ion and target atoms
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 30 Jul 09, 2017
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 31 Jul 09, 2017
(Such as graphene, BN, transition metal dichalcogenides, etc.)
Ø Defects (point and line defects) are ubiquitous; The thermodynamics
is responsible for defect appearance at finite temperatures;
Ø Man-made materials (e.g., CVD-grown) may be not in equilibrium; Ø As 2D materials have ‘surface’ only, effects of the environment are strong! Ø Defects can be created by ion and electron irradiation; Ø Defects affect material properties and may be beneficial; For an overview see AVK and F. Banhart, Nature Mater. 6 (2007) 723-733. AVK and K. Nordlund, Appl. Phys. Rev. 107 (2010) 071301.
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 32 Jul 09, 2017
Ø Exciting new physics
h Large surface area h All of the above is relevant to graphene: production of defects in a 2D material h Energy conversion mecha-
nism from the projectile to atomic stochastic motion is different from that in bulk solids due to nano-scale system size
h Only a small amount of energy of the projectile is
deposited into the system, but this may give rise high local temperature
Example: 30 eV transferred to a C atom in C60 would rise “temperature” up to ~ 2000K
Arkady Krasheninnikov HZDR, Germany and Aalto University, Finland slide 33 Jul 09, 2017