SEMICONDUCTORS I.Methods of calculating point defects in semiconductors: SEMICONDUCTORS I. Methods of calculating point defects in semiconductors
OUTLINE
1. Why are semiconductors important and how do they work?
2. The role of identifying point defect complexes.
3. The target quantities for calculation.
4. Models of the defective semiconductor. Size-convergence.
5. Choice of Hamiltonians and basis sets
Illustrative examples
II. Tips for modeling point defects in semiconductors Peter Deák
Slide2: SEMICONDUCTORS: most important contribution of physics to society!
Started with theory:
Nobel prize in physics 195?: to Bardeen, Brattain & Shockley for the transistor
Lead to technology:
Nobel prize in physics 2000: to Alferov and Kroemer for semiconductor heterostructures used in
high-speed and optoelectronics and to Kilby, for the integrated circuit
… and on the side: boosted the development of basic science:
Nobel prize in physics 1985: to von Klitzing for the quantized Hall effect
In the mean time relied heavily on:
Nobel prize in physics 1999: to Kohn for density-functional theory SEMICONDUCTORS: basic hardware of (electronic) information society!
data processing semiconductor microelectronics
data transfer semiconductor optoelectronics
automation micromachined semiconductor sensors and actuators
Slide3: PUZZLE 1: Si, GaAs, GaP, InP, Ge are the most often used semiconductors. What is the percentage of Si in applications? 95 % PUZZLE 2: how much of the world’s industrial production value comes from electronically controlled processes? Why? problems with silicon !
operating at high temperature
controlling high power
switching at high frequency
emitting light
withstanding corrosion and radiation damage 10 %
The most important attribute of a semiconductor: BAND GAP: The most important attribute of a semiconductor: BAND GAP How does a semiconductor work? If it is pure (intrinsic) — it does not! DOPING IS NEEDED!
2. THE ROLE OF IDENTIFYING POINT DEFECT COMPLEXES.: 2. THE ROLE OF IDENTIFYING POINT DEFECT COMPLEXES.
THE PROCESS OF IDENTIFICATION: THE PROCESS OF IDENTIFICATION
3. THE TARGET QUANTITIES FOR CALCULATION: 3. THE TARGET QUANTITIES FOR CALCULATION 3.1. DOES THE MODEL EXIST? and find (meta)stable configuration for all models in all charge states: - calculate:
Slide8: assumption of thermal equilibrium i: there are many different defects related to an impurity! Calculation of the Fermi-level, EF (doping-level):
3.2.WHAT ARE THE SPECTROSCOPIC PROPERTIES?: 3.2.WHAT ARE THE SPECTROSCOPIC PROPERTIES? calculate:
calculate: occupation levels Hall, TSC, DLTS
calculate: one-electron energies optical abs., PL, photoconductivity
calculate: one-electron eigenstates hf parameters EPR, ENDOR, ODMR : IR, Raman, PL sidebands (isotope substitution, effect of uniaxial stress)
4. MODELS OF THE DEFECTIVE SOLID: 4. MODELS OF THE DEFECTIVE SOLID
4.1. The Molecular Cluster Model (MCM): 4.1. The Molecular Cluster Model (MCM) - the states of the valence band can be transformed into a set of well localized two-center bonds
• bonds on the boundary of MCM replaced by host-atom — hydrogen bonds
- the conduction band cannot be described sufficiently well with a set of localized states
• gap “big” and converges slowly— problem with defect states having strong CB admixture
simple direct space formalism
• only one (pseudo)eigenvalue equation has to be solved
• any “molecular” code can be used (many-body theory !)
- loss of symmetry
• sites are no longer equivalent
• artificial charge transfer
• surface dipole layer
SOLUTION: MAKE IT BIG (~ 102) !
4.2. The Supercell Model (SCM): 4.2. The Supercell Model (SCM) - standard band theory for a periodic array of supercells (superlattice) - the density must be integrated over the Brillouin-zone (BZ) the Bloch-condition and the Born - von Kármán conditions are satisfied: making the number of K vectors in the BZ equal to N1 N2 N3 = N SPECIAL K-POINT THEOREM: Monkhorst-Pack sets:
4.3. The K=0 () approximation:: 4.3. The K=0 () approximation: direct space: reciproc space: • the BZ of the “perfect” supercell is a reduction of the primitive BZ: each vector K in the RBZ
represents L3 points in the PBZ. • THIS IS ONLY TRUE IN THE PERFECT SUPERCELL
or if
• INTERACTION AMONG THE REPEATED DEFECTS ARE
NEGLIGIBLE! CYCLIC CLUSTER MODEL:
neglect of interactions beyond the boundary of the Wigner-Seitz cell.
4.4. Size-Convergence: 4.4. Size-Convergence A/ the amplitude of the localized wave functions
least stringent: dispersion correction can be applied
B/ the deviations of host atom positions
depends on defects; size convergence must be tested!
C/ the deviations of charge density
most severe: need for good quality K-set, or large size if K=0
Slide15: Size-Convergence in SCM Size-Convergence in MCM: Si122H100:VSi < 0.15 eV
Si206H158:VSi-VSi < 0.15 eV
5. CHOICE OF HAMILTONIANS AND BASIS SETS: 5. CHOICE OF HAMILTONIANS AND BASIS SETS ab initio = first-principles ≠ exact Multi-atom system Adiabatic principle Dynamic Jahn-Teller effect
Non-radiative recombination Many-electron problem One-electron approximation Approx. I. Approx. II. Amphoteric defects
Biradicals Hartree-Fock theory Density Functional theory
Slide17: An amphoteric defect, SiC:VSi t2 a1 (3x) one-electron theory many-electron theory
(e.g. MC-SCF) E(S) ≤ E(T) - 0.1 eV A biradical defect, C:Ci
Slide18: Hartree-Fock theory Error Approx. III. correlation: Etot, DEij, Ii Correction post-HF, e.g. CI or MPT
exact if basis infinite
DFT gap-error: DFT gap-error TRUE DFT “Scissor-operator” = rigid shift of iC energies How about defects?
• shift unoccupied D as iC and leave occupied ones alone?
• shift all defect levels in proportion to (iC - D)/Eg ?
• !! shift D in proportion to
Slide20: BASIS SETS DFT-SCM: PLANE WAVES CCM, MCM or HFR: GAUSSIAN-TYPE ORBITALS Advantages
Good description of delocalized states, and easy convergence tests.
Disadvantages
Not so good for localized states, inflexible with atom types.
Surface calculations or comparative calculation for atoms, molecules are difficult, and
calculation of hyperfine data are cumbersome. Advantages
Good description of localized states, flexibility with atom types.
Calculation of surfaces or isolated entities easy, hyperfine data straightforward.
Population analysis is a big help in understanding defect phenomena.
Disadvantages
Not so good in the interstitial region (high order polarization functions needed).
Difficult to test convergence, and overcompletness may lead to numerical instabilities.
Slide21: WHERE TO START? Demos: Computer Simulation of Materials at Atomic Level
eds. P. Deák, Th. Frauenheim, M. R. Pederson [Wiley-VCH, Berlin 2000]