L5: Surface Recombination, Continuity Equation & Extended Topics tanford University

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1 L5: Surface Recombination, Continuity Equation & Extended Topics EE 216 : Aneesh Nainani 1

2 Announcements Project Select topic by Jan 29 (Tuesday) 9 topics, maximum 4 students per topic Quiz Thursday (Jan 31) 6-7pm, CIS-X 101 Syllabus: Material covered till Monday (Jan 28) 2

3 Recap: Shokley-Read Hall U n p 2 thnt pn ni 2n cosh[( E E i T i ) / kt] 3

4 Quiz Why not photon alone not sufficient for indirect bandgap? 4

5 Photon Energy and Wave Vector Photon has large energy for excitation through bandgap, but its wave vector is negligible compared to size of BZ 5

6 Phonon Energy and Wave Vector Phonon has large wavevector comparable to BZ, but negligible energy compared to bandgap 6

7 Localized Traps and Wave Vector Trap provides the wavevector necessary for indirect transition 7

8 Current Continuity Equations We have discussed 1. Carrier transport (drift and diffusion) The Current Transport Equations are J J N P qnn q p p e e D N D P dn dx dp dx For electrons For holes (19) (20) 2. Generation of carriers and recombination of carriers The mechanism which ties all of these together is the condition of continuity of current (or conservation of charge). 8

9 Current Continuity Equations rate of change of number of holes in X net hole generation in X net holes flowing into X dp dt X ( G U ) X 1 q J P 1 q X J X X P 9

10 Current Continuity Equations dp dt X G p p p o X 1 q dj dx P X dp dt G p p p o 1 q dj dx P (21) Similarly for electrons, dn dt G n n n o 1 dj dx These are the Continuity Equations. q N (22) 10

11 Current Continuity Equations Though fully equivalent, the continuity equations can be more generally expressed as n t 1 q J N n t thermal RG n t other processes p t 1 q J P p t thermal RG p t other processes 11

12 Continuity Equation: a good analogy 12

13 One final equation needed to describe device operation is Poisson's equation. (Electrostatic field due to charge) e d dx Where K s r e o E = electric field r = space charge density (coul/cm 3 ) K s = dielectric constant Poisson's Equation e o = permittivity of free-space = 8.85 x (F/cm) 13

14 In general, the net charge density in a semiconductor is given by r e 2 d V 2 dx q p n N dv dx q K e s o Poisson's Equation D N A n p N N A D (23) Equations (19) to (23) constitute a complete set of equations to describe carrier, current and field distributions. Given appropriate boundary conditions, we can solve them for an arbitrary device structure. Generally, we will be able to simplify them based upon physical approximations and judicious elimination of terms. 14

15 J J N P qnn q p p e e D D n t 1 q J n N t thermal RG p t 1 q J p P t thermal RG N P Key Results dn dx dp dx n t other processes p t other processes Current Transport Equations Continuity Equations d 2 V dx 2 q K s e o n p N A N D Poisson s Equation 15

16 Surface States 16

17 Surface States Multiple levels of surface states 17

18 Surface States: Quiz? Recombination using 2 states not possible as they states are separated far apart physically in the lattice 18

19 Surface Recombination To this point we have considered bulk recombination processes. Recombination at the surface is often enhanced over bulk values, particularly in bare, unpassivated surfaces. Incompletely bonded Si atoms at the surface can act as efficient traps for recombination. SiO 2 passivates Si, but there is NO EQUIVALENT to SiO 2 for compound semiconductors like GaAs or even Ge. E c E i Allowed states E v 19

20 Let us consider the case where light is shining on a N-type semiconductor. At the surface, recombination of electrons and holes will take place. By analogy, the net recombination rate at the surface is: U th Surface Recombination TS N p(0) (25) where N TS = density of surface recombination centers (cm -2 ) S = V th N TS = surface recombination velocity (cm/sec) (26) p(0) - p o = excess carrier density at surface. p o If the rate of recombination at the surface is higher than that in the bulk (as it often is), then the excess carrier concentration will be lower at the surface and there will be a diffusion of electrons and holes (in equal number for charge neutrality i.e., J = 0) towards the surface. 20

21 recombination p(0) p Fn p GL p o 0 x Surface Bulk The continuity equation governing the diffusive flow of holes, assuming a negligible drift component is dp p po 1 dj P GL dt q dx G L p p p Surface Recombination p o D P 2 d p 2 dx F p diffusion 21

22 In steady state, dp 0 dt In the bulk, the boundary condition is p p o p G L At the surface, the boundary condition is D P diffusion flux = recombination rate dp dx x0s p 0 Surface Recombination p o The solution to the differential equation is S p / LP x / LP px po pgl 1 e 1 S p / LP (27) 22

23 Surface Recombination where L P D P p Minority carrier difussion length (28) p e X / Lp p o Surface p o + p G L S p / Lp Bulk X S p/ Lp 23

24 Surface Recombination Note: 1. The distance of the "disturbance" of the excess carrier concentration into the bulk, L P, depends upon D P and p. 2. The value of the excess concentration at the surface depends upon p, D P and S. In the limit as S (infinite recombination rate at the surface), p(0) p o which is often the case for III-V materials like GaAs. This is quite important as devices are scaled to small dimensions as everything comes closer and closer to the surface. 24

25 Surface Recombination sometimes the interface is the device 25

26 Extended Topics E-k and E-x diagram Band diagram of real semiconductors Degeneracy factor for donors and acceptors 26

27 E-x (band-diagram) & E-k (band-structure) 27

28 E-x (band-diagram) & E-k (band-structure) 28

29 E-x (band-diagram) & E-k (band-structure) 29

30 Relationship b.w real and k-space Good way to think about k-space is to think about momentum 30

31 Bandstructures (E-k) of real semiconductors 31

32 Brillouin Zone in Cubic Lattice 32

33 Brillouin Zone in Real FCC Cubic Lattice Note unlike cubic lattice, zone edge is not at π/a 33

34 Analogy to E-k diagram: 4D info through 2D plots EE 216 : Aneesh Nainani 34

35 E k along Γ X Direction EE 216 : Aneesh Nainani 35

36 E k along Γ L Direction EE 216 : Aneesh Nainani 36

37 E k diagram EE 216 : Aneesh Nainani 37

38 E k diagram for GaAs EE 216 : Aneesh Nainani 38

39 Analogy for E-k diagram EE 216 : Aneesh Nainani 39

40 Constant E surface for Conduction Band EE 216 : Aneesh Nainani 40

41 Quiz: what is the degeneracy factor for DOS of electrons in Ge / Si? EE 216 : Aneesh Nainani 41

42 Constant E surface. Off-diagonal terms of effective mass = 0 EE 216 : Aneesh Nainani 42

43 Constant E surface for Valence Band Off-diagonal terms may not be zero EE 216 : Aneesh Nainani 43

44 Measurement of Energy Gap Red direct bandgap, Blue - indirect bandgap EE 216 : Aneesh Nainani 44

45 Temperature dependence of bandgap EE 216 : Aneesh Nainani 45

46 Measurement of effective mass EE 216 : Aneesh Nainani 46

47 Measurement of Effective Mass: Ge 47

48 Measurement of Effective Mass: Ge 48

49 Degeneracy factor for donors and acceptors Why N D + and N A - have different statistics than FD? I mentioned this in L2 (slide 34) also.. Now lets understand why 49

50 Localized vs. Delocalized States 50

51 Statistics of Donors 2 combinations out of total 4 A more intuitive way to think about this is that certain configurations which would have been allowed if this state was a part of band are not allowed in this localized state due to Coulomb interaction which leads to decrease in probability of occupancy.. which comes from the degeneracy factor 51

52 Statistics of Acceptors 52

53 Statistics of Donors / Acceptors 53

54 Statistics of Donors / Acceptors Degeneracy factor 2 for donors Degeneracy factor 4 for acceptors Video on Fermi Dirac statistics: devices.class2go.stanford.edu 54

55 Heavy doping: Bandtail states 55

56 Heavy doping: Hopping conduction 56