Chapter 5. Carrier Transport Phenomena

Transcription

1 Chapter 5 Carrier Transport Phenomena 1

2 We now study the effect of external fields (electric field, magnetic field) on semiconducting material 2

3 Objective Discuss drift and diffusion current densities Explain why carriers reach an average drift velocity Discuss mechanism of lattice and impurity scattering Define mobility, conductivity and resistivity Discuss temperature and impurity dependence on mobility and velocity saturation State the Einstein relation Describe the Hall effect 3

4 Drift current Electric field force on electrons and holes Free states in conduction and valence band net movement of electrons and holes Net movement of charge due to electric field is called drift. 4

5 Drift current density Jp E Average drift velocity for holes J p,drf = e p vd A /cm 2 p vd (Volume) density of holes Jn E for electrons J n,drf = -e n vd A /cm 2 n vd 5

6 Velocity of the particles E +e v drift velocity of the hole in the electric field So does velocity monotonically increase with time? 6

7 Thermal and drift velocities Without E field With E field vth + v Even in the absence of E-field the holes have random thermal velocity (vth) They collide with ionized impurity atoms and thermally vibrating lattice atoms. Let τcp mean time between collisions. with E-field net drift of holes in the direction of the E-field net drift velocity is small perturbation on random thermal velocity. 7 so τcp remains almost unchanged even in the presence of E-field.

8 Thermal and drift velocities Without E field Now, With E field v peak = eτ cp mp * E 8

9 Drift velocity vd v peak = v eτ cp mp * E vd 0 τcp 2τcp 3τcp t Average drift velocity = μp (mobility) Using more accurate model including the effect of statistical distribution, 9

10 Mobility: vd = με For holes: For electrons: 10

11 Mobility: μ = vd /E Unit: cm2/vs μ n (cm2/v-s ) μ p (cm2/v-s ) Silicon Gallium Arsenide Germanium mn*/m0 mp*/m0 Silicon Gallium Arsenide Germanium

12 Scattering: Two main s cattering mechanis ms Lattice s cattering or phonon s cattering Impurity s cattering 12

13 Phonon Scattering Phonons are lattice vibrations (Atoms randomly vibrate about their T>0K) Lattice vibration causes a local volume change and hence lattice constant change. Bandgap generally widens with a smaller lattice constant. Mobility due to lattice scattering vibration of atoms also increases) µl T 3 2 Disruption of valence and conduction band edges scatters the carriers. (as temp increases 13

14 (a) Electron and (b) Hole mobilities in Si vs. T at different doping concentrations. (Inserts show dependence for almost intrinsic Si) 14

15 Ionized impurity scattering Scattering due to coulomb interaction between electrons/holes and ionized impurities. T increases thermal velocity vth increases, so less time spent for scattering µ I T n NI increases the scattering chance 1 increases µ I N I 15

16 High doping is required to overcome short channel effects even though it reduces mobility. Doping level in modern processors Electron and Hole mobility vs. impurity concentration. 16

17 How to combine mobility effects? τi : average time between two collisions with dopant atoms τl : average time between two collisions with vibrating" lattice points dt : Number of collisions in time dt due to impurity scattering τi d t : Number of collisions in time dt due to lattice scattering τl Total number of collisions in dt: 17

18 Electron mobility of Si vs. T for various Na At present doping level, cooling does not improve speed much. 90nm CMOS process Low doping concentration or high T The lattice scattering dominates High doping concentration or low T The impurity scattering dominates 18

19 Drift current density J n d rf = eµne eτ c µ = * m J increases then cut-off frequency increases circuit density increases How can we increase J? 19

20 How can we increase J? Increase mobility Strained silicon effective mass, mobility GaAs or Ge as semiconducting material Increase electric field Shorter gate length 20

21 Strained Si Intel 90nm process Strain decreases the effective mass, increasing the mobility already been used for production 21

22 Conductivity σ =eμnn +eμpp Units (Ω-cm)-1 Units Ω-cm 22

23 If we assume complete ionization, σ = 1 = eµ n N d or eµ p N a ρ Impurity scattering But curve not linear because - Impurity scattering affects mobility. Resistivity vs. impurity concentration in Si at T=300K 23

24 Impurity scattering affecting mobility. Resistivity vs. impurity concentration for Ge, GaAs and GaP 24

25 Electron concentration and conductivity vs. 1/T Lattice scattering 25

26 Electron concentration and conductivity vs. 1/T Assuming a n-type material with donor doping Nd >> ni, σ = e( µ n n + µ p p ) eµ n n n electron concentration σ = 1 = eµ n N d If we also assume complete ionization, ρ Mid temp complete ionization n constant at Nd but μ reduces with temp due to lattice scattering so conductivity drops. At high temperature, intrinsic carrier concentration increases and dominates both n and σ At low temp, due to freeze-out both n and σ reduce 26

27 Ohms Law: Ohm s law 27

28 Velocity saturation vd = μe Random thermal = energy vth for Si Modern processors (90nm,1.2V) vth 107cm/s 28

29 Intervalley transfer mechanism in GaAs Higher effective mass, low mobility At higher E lower mobility lower current negative resistance Lower effective mass, high mobility 29

30 Negative resistance Negative resistance region -ve resistance used in design of oscillators. Oscillator: GaAs 300GHz, GaN 3THz Oscillation frequency depends on transit time in the device. 30

31 Carrier Diffusion Concentration Low concentration High concentration x Positive slope in x-direction a flux towards negative-x direction 31

32 Diffusion current density for holes D: diffusion coefficient for electrons 32

33 Diffusion coefficient (I) l=average mean free path l = (vth + vdr) τ cp If E=0, vdr=0 l=vth τcp Electron concentration versus distance. 33

34 Diffusion coefficient (II) ½ n(l)(-vth) ½ n(l)(vth) ½ n(-l)(-vth) ½ n(-l)(vth) = sum of electron flow in +x direction at x=-l and electron flow in x direction at x=l 34

35 Diffusion coefficient (III) Taylor expansion of n(-l) en n(+l) at x = 0: n(+l) =

36 Diffusion coefficient (IV) Recall that D: diffusion coefficient = vthl (cm2/s)=vth2 τcp 36

37 For holes: Note 37

38 Diffusion of (a) electrons and (b) holes in a density gradient 38

39 Total Current Density DRIFT DIFFUSION Generalized current Density Equation - 39

40 If a semiconductor is non-uniformly doped? Consider a case where donor concentration increases in x-direction Diffusion of electrons Formation of electric field Prevents further diffusion Due to electric field potential difference across the device In the region at lower potential EF-EFi higher 40

41 Induced Electric Field Ex Potential: Electric field Assuming Electron concentration Donor concentration 41

42 Recall Taking log Take the derivative with respect to x Electric field 42

43 The Einstein Relation Consider the general current density equation: n,p Assume n-graded semiconductor material in thermal equilibrium: Assuming n Nd 43

44 Substituting [ ] dn d because 0 dx 44

45 Similarly 45

46 The Hall effect n- or p-type, carrier concentration and mobility can be experimentally measured. Electric and magnetic fields are applied to a semiconductor. 46

47 The Hall Effect (Lorentz force) F = qvx x Bz Hall voltage Buildup of carriers here 47

48 Due to magnetic field, both electrons and holes experience a force in y direction. In n-type material, there will be a build up of ve charge at y=0 and in p-type material, there will be a build up of +ve charge. The net charge induces a electric force in y-direction opposing the magnetic field force and in steady state they will exactly cancel each other. 48

49 Hall Voltage: Ey = vxbz VH = EyW=vxWBz 49

50 How to determine n-type or p-type & doping concentration? For p-type material: Jx vx = ep VH will be +ve Ix = (ep)(wd ) For n-type material: 54 VH will be -ve For n-type material VH will be -ve n will still be +ve! 50

51 How can you determine μ n, μp? 56 51

52 Summary Drift net movement of charge due to electric field. Drift current = J drf = e( nµ n p p+) Eµ Mobility (μ) lattice and impurity scattering 3 2 Mobility due to lattice scattering µ L T T Mobility due to impurity scattering µ I 3 2 NI Ohms law V = L I σ A ρl =I A RI = Drift velocity saturates at high electric field velocity saturation 52

53 Summary Some semiconductors mobility μ reduces at high E negative resistance used in design of oscillators. Diffusion current J diff = edn dn dx dp edp dx Einstein relation relation between diffusion coefficient and Dn D p kt = mobility µ = µ e n p Hall effect can be used to determine semiconductor type, doping concentration and mobility 53