2. Point Defects. R. Krause-Rehberg

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1 R. Krause-Rehberg 2. Point Defects (F-center in acl) 2.1 Introduction 2.2 Classification 2.3 otation 2.4 Examples 2.5 Peculiarities in Semiconductors 2.6 Determination of Structure and Concentration 2.7 acancies in thermodynamic Equilibrium 2.8 Irradiation-induced Point Defects 2.9 Aspects of Defect Chemistry

2 2.7 acancies in thermodynamic equilibrium statistical considerations change of free enthalpy during formation of Schotty-type vacancies (.. number of atoms; W vacancy formation energy) F W T S (positive energy term can be compensated by gain of entropy) S is complete entropy gain; is calculated in the following probability to form vacancies in atoms is equal to probability to choose atoms out of atoms (numerator): G ( 1) (! + 1) (! )! The factor! in the denominator excludes those cases which differ only by the different order of pic-out of atoms. using the Boltzmann-Equation and the Stirling approximation:! S B lng B ln (! )!! ln x! xln x x RKR 2001 Structure of imperfect solids - Defect chemistry 2

3 RKR 2001 Structure of imperfect solids - Defect chemistry acancies in thermodynamic equilibrium T W B ln ] ln ) )ln( ( ln [ B S ln 0 B T T W F it follows: in thermal equilibrium: F is extreme value in the lattice: << (in real crystal / > 10 3 ) T W B exp thus: vacancies must exist in an ideal crystal at T>0! [1]

4 2.7 acancies in thermodynamic equilibrium example: T1000K and W 1 e / 10-5 T1000K and W 3 e / not detectable real example: vacancies in Au W 0.98 e, but in Si W > 3.6 e vacancy concentration is slightly larger compared to Eq. [1] further factors to be taen into account: - interaction of vacancies - influence of point defects to S - volume wor for dilatation of lattice defect density often much larger: crystal far from thermal equilibrium excess vacancies due to e.g. irradiation by fast particles also: vacancies can be quenched-in by very fast quenching quenching rate must be about 10 4 s -1, then a large fraction of thermal vacancies remain during slow warming-up: vacancies become mobile (migration energy required) RKR 2001 Structure of imperfect solids - Defect chemistry 4

5 experimental results positron annihilation detects vacancies in thermal equilibrium increase of pea height H/H 0 of angular correlation is due to increasing number of vacancies slope of Arrhenius plot gives vacancy formation enthalpy H (HU+p) only possible in metals in semiconductors: H is too large, so number of thermal vacancies is too small to be detected RKR 2001 Structure of imperfect solids - Defect chemistry 5

6 2.8 Irradiation-induced point defects Generation of defects generation often done by electron irradiation displacement energy in metals and semiconductors is about e electron threshold energy still about 500 e reason: only a small fraction of electron energy can be transferred to atom (collision law) maximum transfer during central collision: T m 4m1m ( m + m ) 2 E incident [2] computer-simulated impact of an electron (1 Me) in Cu (T m 40 e); generated vacancy is only stable when outside of spontaneous recombination volume example: m 1 m e g; m 2 m Cu g T m 4m e /m Cu E incident E incident (some relativistic corrections are required) using electrons up to several Me, mostly Frenel pairs are generated RKR 2001 Structure of imperfect solids - Defect chemistry 6

7 2.8.1 Generation of defects Ion implantation is most important doping method in semiconductor technology mass m 1 and m 2 in Eq. 2 are comparable (situation lie billiard balls) however: penetration depth for ions of 100 e is about 1 µm (1 Me electrons > 0.5mm) ions produce extended defect cascades; energy large enough for 10 4 displacement events however: only a few defects survive (stationary state reached after 1 ps) also larger point defect clusters are generated computer-simulated defect cascade in {100} plane of Cu. T250 e, t a 0.06 ps, t b 0.35 ps, t c 1.5 ps stationary result after 1.5 ps: 5 vacancies and 5 interstitials RKR 2001 Structure of imperfect solids - Defect chemistry 7

8 ion implantation important dopant species in Si are As and B in order to obtain homogenous doping depth profiles: multiple implantation steps with different energy depth distribution of implanted B (As) atoms in silicon depth distribution of displaced atoms in B-implanted Si when implantation dose large enough: lattice becomes amorphous amorphisation dose is function of ion mass, target species, and temperature RKR 2001 Structure of imperfect solids - Defect chemistry 8

9 2.8 Irradiation-induced point defects temperature dependence of amorphisation dose is strong at elevated temperature: defects anneal during irradiation at room temperature: boron implantation will not lead to amorphisation in technology: defects must be annealed often: rapid thermal annealing (RTA) in Si: 30s at 950 C done by light illumination by strong halogen lamps (few W) temperature dependence of amorphisation dose for different ions in Si RKR 2001 Structure of imperfect solids - Defect chemistry 9

10 Rutherford Bacscattering classical method to investigate ion implantation defects: Rutherford Bacscattering probe atoms (H, He) penetrate into the sample into lowindex directions (channels) defects which are present scatter the probe atoms and raise the bacscattered intensity defect depth profiles can be determined RKR 2001 Structure of imperfect solids - Defect chemistry 10

11 2.8.2 Annealing of excess point defects irradiation defects far from thermal equilibrium but still stable (frozen-in) increasing temperature: defects start migration exp E E m 0 m BT... migration energy annealing of irradiation defects in Cu electrical resistance electrical residual resistance at low temperature: sensitive for defects (electrons are scattered during movement in electric field) RKR 2001 Structure of imperfect solids - Defect chemistry 11

12 Annealing of excess point defects annealing starts at very low temperatures (in metals: interstitials have smallest migration energy) many annealing stages; during annealing: defect reaction (e.g. formation of vacancy clusters) often: several mechanisms lead to disappearance of same defect: vacancies vanish due to migration of interstitials and vacancies itself several stages (A-E) for interstitials: close Frenel pairs (A-C) and separated interstitials (D+E) curve b: electron irradiation curve : plastic deformation curve a: quenched sample stage I + II: interstitial annealing stage III: vacancy annealing isochronal annealing curve of Cu after 3- Me-electron irradiation (T irr 4.5 K) RKR 2001 Structure of imperfect solids - Defect chemistry 12

13 Annealing of excess point defects example: defects after electron irradiation in Ge (E e- 2 Me, T irr 4K) distinct annealing stage at 200K sample with highest dose: formation of divacancies during annealing (they anneal at about 400K) formation of divacancies prove: it is no movement of interstitial but vacancies this is supported by the fact that the vacancies disappear completely in this stage; interstitial stage always incomplete (Polity et al., 1997) RKR 2001 Structure of imperfect solids - Defect chemistry 13

14 Defect reactions defect reactions during annealing sequence is usual effect they could be rather complex typical example: defect annealing in Cz-Si after lowtemperature electron irradiation many different oxygen-vacancy complexes are formed most simple defect is the so-called A-center during annealing: sequence of different x O y complexes are formed defects stable up to 800 C, although a monovacancy anneals at about 200K oxygen stabilizes the defects -O complex (A-center) RKR 2001 Structure of imperfect solids - Defect chemistry 14

15 2.9 Aspects of defect chemistry chemical or defect reaction ν A A + ν B B ν reaction index : λ C C + ν ν C D + ν ν i D D ν...number of moles prior to reaction: λ 0 ; complete reaction λ 1 in thermodynamic equilibrium: 0 < λ < 1 equilibrium condition: G 0 λ, T p minimum of free entalphy reaction runs spontaneously only when: G < 0 λ, T p RKR 2001 Structure of imperfect solids - Defect chemistry 15

16 The mass action law chemical or defect reaction dc dt A dc dt 1 A + B C + 2 RKR 2001 Structure of imperfect solids - Defect chemistry 16 D i... reaction velocity coefficients c i... concentrations B C D 1 ca cb and for return reaction: 2 cc cd in case of thermodynamic equilibrium: velocity in both directions identical dc dc A C dt dt and thus: cc cd 1 c... concentration in equilibriu m i c c equation is called mass action law A not only for chemical reactions: intrinsic conductivity in semiconductors n p n + p mass action law : 0... number of all electrons and thus: n p ' B 2 dc dt dc dt

17 Defect chemistry in HgCdTe HgCdTe is used as infrared detector; dominating defect is 2- Hg (twofold ionized acceptor) vacancies are in equilibrium with vapour p Hg over crystal e + h + + 2h 0 + Hg( gas) 2 + Hg Hg Hg mass action law: electrical conductivity is sum of intrinsic conduction and ionization of Hg acceptors n h i 2 2 [ Hg ] h phg v (1) neutrality condition: intrinsic part n h (at high temperatures) n h + 2 [ 2 Hg ] 0 (2) 2 extrinsic part 2 [ Hg ] h (3) technical use at T < 100K (no intrinsic conduction); combination of (1) to (3) gives: 2 3 i phg 2 2v h77 K h77k + 2ih77K 2 p v Hg 0 (4) RKR 2001 Structure of imperfect solids - Defect chemistry 17

18 Defect chemistry in HgCdTe 2 3 i phg 2 2v h77 K h77k + 2ih77K 2 p v Hg 0 (4) is equation which defines connection between Hg partial pressure and concentration of Hg vacancies in crystal (because of h 77K 2 [ Hg ]) constants i and v were determined by electrical measurements (Hall effect) lines in figure: result of simulation by eq. (4); measured points: experimental data of Hg vacancy concentration obtained by positron annihilation RKR 2001 Structure of imperfect solids - Defect chemistry 18

19 Defect chemistry treatment of quasi-ternary compounds becomes rather difficult very large number of intrinsic defects but in special regions: only a few defects dominate calculation of the dominating defects requires the nowledge of thermodynamic constants which are usually only roughly nown diagram only valid for a constant temperature CuInSe 2 RKR 2001 Structure of imperfect solids - Defect chemistry 19

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