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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

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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

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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:

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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

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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

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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

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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

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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.

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WHERE TO START? Demos: Computer Simulation of Materials at Atomic Level eds. P. Deák, Th. Frauenheim, M. R. Pederson [Wiley-VCH, Berlin 2000]

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