Understanding Semiconductors and Recombination in Solid State Physics

1. Lecture: Semiconductors and recombinationProf Ken Durose, University of Liverpool

2. Outline – semiconductors and recombination 1. Band gap representations 2. Types of semiconductors -Adamantine semiconductors (Hume -Rothery 8-N co-ordination rule -Others -Solid solutions 3. Doping and point defects 4. Generation and recombination

3. 1. Band gap and its representation Shockley – Queissler limit and band gap

4. 1.1 Band gap origins EC Eg Energy of an electron (eV) EF Et EV Diagram from M J Cooke Semiconductor devices (Recall the Pauli exclusion principle) x (m) Energy vs space representation of a band diagram. Et is a trap energy level http://photonicswiki.org/images/thumb/2/22/Homocontrol.png/800px-Homocontrol.png

5. 1.1 Band gap origins Electrons in a periodic potential (e.g. Kronig-Penney model) Diagram from AK Dekker Solid State Physics

6. E E k 1.2 E-k reduced zone representation (textbook) k Direct gap e.g. III-V’s and II-VI’s Indirect gap e.g. Si

7. 1.2 E – k band diagram (GaAs) http://th.physik.uni-frankfurt.de

8. 1.2 N(E) vsE – density of states • NB – there is a very low DOS at the band edge and so photons of energy Eg are not the most likely to be absorbed E Eg N(E)

9. 2 Types of semiconductor + solid solutions Hume-Rothery 8-N Co-ordination rule: The co-ordination number in a compound is 8-N, where N is the average valency number. We will use this rule to go looking for semiconductors like silicon, valency 4 i.e. isoelectronic variants of Si. Si and Geare gpIV semiconductors and are tetrahedrally co-ordinated, they have the structure of diamond. Adamantine = diamond like

10. 2 Types of semiconductor + solid solutions III-V semiconductors GaP, GaAs, GaSb, InP, InAs, InSb etc

11. 2 Types of semiconductor + solid solutions II-VI semiconductors ZnO, ZnS, ZnSe, ZnTe, CdO, CdS, CdSe, CdTe etc

12. 2 Types of semiconductor + solid solutions I-III-VI semiconductors – the chalcopyrite family CuInSe2, CuGaSe2, CuInSe2etc

13. 2 Types of semiconductor + solid solutions I-II-IV-VI semiconductors – the kesterite family Cu2ZnGeSe4, Cu2ZnSnS4, Cu2ZnSnSe4 etc

14. Solid solutions • GaPEg ~ 2.3eV • GaAsEg ~ 1.4eV • Ternary semiconductor Ga(AsxP1-x) – Eg in the range 1.4 – 2.3eV Lattice parameter (a0) varies also NB To vary Eg and ao independently, you need a quaternary system, such as GaxIn1-xAsyP1-y

15. a a[BC] a[AC] 0 x 1 Vegards law - linear variation of lattice parameter with x a[AxB1-xC] = a[BC] - x * {a[BC] – a[AC]} Psst! It might not be linear in Practice ...... but it often is.

16. Eg Eg[AB] Eg[AC] Non - ideal bowed 0 x 1 ‘Vegard’s law for band gap’ Ideal – obeys Vegard’s law i.e. is linear Bowed curve represented by a bowing parameter ‘b’ Eg[AxB1-xC] = x * Eg[AC] + (1-x) * Eg[BC] - b * x * (1-x).

17. Solid solutions in two III-V semiconductor series

18. 3 Semiconductor doping • Substitutional doping • Intrinsic doping • Vacancies • Interstitials • Complexes You are going to need the periodic table again…

19. Substitutionaldopants in Si Everything is on a gpIV site PSigpV on a gpIV site – electron excess – this is a donor BSigpIII on a gpIV site – electron deficient - this is an acceptor Substitutional doping in III-V compounds– such as InP e.g. CdIngpII on a gpIII site – electron deficient = acceptor e.g. SP – gpVI on a gpV site = donor C could occupy the gpIII or the gpV site – amphotericdopant 3 Substitutional doping

20. Substitutional doping in II-VI compounds– such as CdTe On the gpII site… e.g. CuCdgpIA on a gpII site – electron deficient = acceptor e.g. InCd – gpIII on a gpII site = donor 3 Substitutional doping ….cont • On the gpVI site… e.g. AsTegpV on a gpVI site – electron deficient = acceptor e.g. ClTe – gpVII on a gpVI site = donor

21. Metal i.e. cation vacancies e.g. Cd vacancies in CdTe Cd oxidation state 2+ Te oxidation state 2- If you heat CdTe it loses Cd when neutral Cd leaves it takes two electrons with it leaving a doubly +ve charged VCd VCdis a double acceptor 3 Native defect or ‘intrinsic defect’ doping - vacancies • Non-metal i.e. anion vacancies • e.g. S vacancies in CdS • Cd oxidation state 2+ • S oxidation state 2- If you heat CdS it loses S • when neutral S leaves it takes two electrons with it leaving a doubly -ve charged VS VS is a double donor heat heat + Cd(g) + S(g) CdTe with VCd CdS with VS CdTe CdS

22. Metal i.e. cation interstitials e.g. Cd interstitials in CdTe Cd oxidation state 2+ Te oxidation state 2- Add neutral Cd to CdTe as an interstitial – to achieve its usual oxidation state it must lose two electrons. Cdi is assumed to be a donor 3 Native defect or ‘intrinsic defect’ doping - interstitials • Non-metal i.e. anion vacancies • e.g. Te interstitials in CdTe Add neutral Te to CdTe as an interstitial – to achieve its usual oxidation state it must gain two electrons. Teiis assumed to be a donor

23. e.g. the ‘A-centre’ Add neutral Cd to CdTe as an interstitial – to achieve its usual oxidation state it must lose two electrons. CdTe’ single donor VCd•• double acceptor [VCd – ClTe]• single acceptor This is the ‘A-centre’ 3 Complex centres

24. 3 Energy levels in the gap of silicon Diagram from Solid State Electronic Devices, Streetman and Banerjee

25. 3 Kroger – Vink nomenclature for point defects • If you need to get specific about point defects and their reactions and equilibria, then check out Kroger-Vinknomenclature… http://en.wikipedia.org/wiki/Kr%C3%B6ger%E2%80%93Vink_notation

26. 4 Generation and recombination • Trapping • Recombination • Direct and indirect • Recombination via trap states (‘Shockley Hall Reed’ mechanism) • Kinetics for recombination in direct gap materials

27. 4.1 Recombination types Direct recombination (a) It is radiative Indirect recombination (b-d) is not usually radiative. (Auger recombination not shown is also ‘indirect’) via a trap (“Shockley Hall Reed”) direct Diagram from Intro to Electronic Devices Michael Shur

28. 4.2 Trapping centres • Centres below the Fermi level at Erare full of electrons. • For them to act as ‘traps’, either a) holes are temporarily trapped there then re-emitted or b) electrons are temporarily trapped there then re-emitted This is the Shockley Hall Reed mechanism Diagram from Solid State Electronic Devices, Streetman and Banerjee NB strictly this is what ‘trapping’ is. However the term ‘trap’ is used more widely than this – as follows now…

29. 4.2 Recombination via traps a) holes are trapped b) electrons annihilate with the trapped holes overall there is one electron hole pair less plus some heat This is most often called Shockley Hall Read recombination Diagram from Solid State Electronic Devices, Streetman and Banerjee

30. 4.2 Recombination via traps Rate for Shockley Hall Reed recombination • The recombination rate is maximised when the trap energy Et is mid-gap. • These are “killer traps” or “lifetime killers” e.g. AuSi • - Where Et is mid-gap, the diode factor has a value of n = 2 Treatment from Intro to Electronic Devices M Schur

31. 4.3 Kinetics of direct recombination At equilibrium Symbols G = generation rate R = recombination rate n = negative carriers p = positive carriers ni = intrinsic carrier concentration  Under steady state conditions (e.g. under illumination), there is additional generation: r = rate constant for recombination m3s-1

32. 4.3 Kinetics of direct recombination For the case where there is additional generation of the recombination rate is written This can be simplified by substituting (subtract this from both sides)

33. 4.3 Recombination in direct gap semiconductors Examples

34. 4.3 Recombination in direct gap semiconductors There is a numerical example in M J Cooke, page 69.

35. Example – generation/recombination Example from M J Cooke – Semiconductor Devices, p 69-70

36. Example cont… Example from M J Cooke – Semiconductor Devices, p 69-70

37. Books used to compile this lecture(including picture credits) • Semiconductor Devices, M J Cooke • Intro to Electronic Devices, M Shur • Solid State Electronic Devices, B G Streetman and S K Banerjee • Solid State Physics, AK Dekker