1. 1 CHAPTER 8 MOBILITY

2. 2 8.1 INTRODUCTION

3. 3 High mobility material has higher frequency response and higher current. Electron-electron or hole-hole scattering has no first- order effect on the mobility. Electron-hole scattering reduces the mobility. Minority carriers has ionized impurity scattering and electron-hole scattering, majority carriers has ionized impurity scattering.

4. 4 8.2 CONDUCTIVITY MOBILITY

5. 5 CONDUCTIVITY MOBILITY Measure the majority carrier concentration and the conductivity/resistivity is sufficient to calculate the conductivity mobility. For p-type material.

6. 6 8.3 HALL EFFECT AND MOBILITY

7. 7 Basic Equations for Uniform Layers or Wafers Schematic illustrating the Hall effect in a p-type sample.

8. 8 Hall angle is defined as the angle between E x and E y.

9. 9 For p-type sample. For n-type sample. When both carriers present.

10. 10 Temperature and magnetic field dependent Hall coefficient for HgCdTe showing typical mixed conduction behavior. T=220~300K, n=n i 2 /p, R H is independent of B. T=100~200K, mixed conduction causes R H to Decrease and depends on B. T<100K, p dominates, RH is independent of B.

11. 11 Hall coefficient and electron density for GaAs. No mixed conduction, and it is independent of B.

12. 12 The above results are based on the assumption of energy-independent scattering mechanisms.If it is relaxed, the Hall scattering factor r must be Included:

13. 13 (a) Bridge-type Hall sample, (b) lamella-type van der Pauw Hall sample. R 12,34 =V 34 /I 12, V 34 =V 4 -V 3 F is a function of R r =R 12,34 /R 23,41 For symmetric samples F=1. ΔR 24,13 is the difference with and without the magnetic field.

14. 14 The van der Pauw F factor plotted against R r. For uniformly doped samples with thickness d, the sheet Hall coefficient is R HS =R H /d and μ H = ︳ R HS ︳ /ρ s, ρ s =ρ/d.

15. 15 Nonuniform Layers Hall measurements give average values, assuming r=1:

16. 16 The depth profile can be measured by etch and measure method or by a pn junction controlled depletion width. Schottky-gated thin film van der Pauw sample, (a) top view, (b) cross section along the A-A showing the gate two contacts and the space-charge region of width W. The Hall measurement is performed in the region d-W.

17. 17 The spatial varying Hall parameters is determined by: This is the so called differential Hall Effect, DHE.

18. 18 Dopant density profiles determined by DHE, spreading resistance profiling, and secondary ion mass spectrometry.

19. 19 DHE encounters difficulty with large parameter variation in multi layer system. Assume a upper layer has carrier density p 1 and mobility μ 1, and a lower layer has carrier density p 2 and mobility μ 2. The Hall effect measured weighted values are:

20. 20 Multi layers A two layer structure has a upper layer thickness of d 1 a conductivity of σ 1, and a lower layer thickness of d 2 a conductivity of σ 2, the Hall constant is given as: At low magnetic field it becomes:

21. 21 Multi layers At high magnetic field the Hall constant becomes: where R H1 is the layer 1 Hall constant, R H2 is the layer 2 Hall constant, d=d 1 +d 2 and σ is:

22. 22 Hall coefficient of a p-type substrate with an n-type layer as a function of n 1 t 1 for two magnetic fields. For low and high n 1 d 1 the Hall coefficient is independent of the magnetic field. If the upper layer is more heavily doped or type inverted by surface charges, then the surface Hall parameters are measured   (d 1 / d)  1 ; R  R H1 (d / d 1 )

23. 23 Sample Shapes and Measurement Circuits (a) Bridge-type Hall configuration, (b)-(d) lamella-type Hall configuration.

24. 24 Van der Pauw Hall sample shapes.

25. 25 Effect of non-ideal contact length or contact placement on the resistivity and mobility for van der Pauw samples.

26. 26 Hall sample with electrically shorted regions at the end; (a) top view with the gate not shown, (b) cross section along cut A-A.

27. 27 (a)Hall sample with electrically shorted end regions, (b) ratio of measured voltage V Hm to Hall voltages V H. G=V Hm /V H. V Hm : V H for W/L<3. V H : V H for W/L<3.

28. 28 8.4 MAGNETORESISTANCE MOBILITY

29. 29 MAGNETORESISTANCE MOBILITY (a)Hall sample, (b) short, wide sample, Hall voltage is nearly shorted; (c) Corbino disk, Hall voltage is shorted. They can be used to measure the magnetoresistance effect.

30. 30 Physical magnetoresistance effect (PMR): The sample resistance increases when it is placed in a magnetic field. Because the conduction is anisotropic, involves more than one type of carrier, and carrier scattering is energy dependent. Geometrical magnetoresistance effect (GMR): The charge carrier path deviates from a straight line. ξ=( 〈 τ 3 〉〈 τ 〉 / 〈 τ 2 〉 2 ) 2 is the magnetoresistance scattering factor.  GMR =   H

31. 31 Geometric magnetoresistance ratio of rectangular samples versus μ GMR B as a function of the length-width ratio.

32. 32 For Corbino disc. For rectangular samples with low L/W ratio and μ GMR B<1 For determining the error in μ GMR to be <10%, then L/W must be <0.4.

33. 33 8.5 TIME-OF-FIGHT DRIFT MOBILITY

34. 34 (a)Drift mobility measurement arrange- ment and normalized output voltage pulse (μ p =180cm 2 /V ． s,τ n =0.67 μs, T=423K, =60V/cm)

35. 35 (b) output voltage pulses (μn=1000 cm 2 /V ． s,τ n =1μs, T=300K, =100V/cm, N=10 11 cm -2 ), (c) output voltage pulses (μn=1000cm 2 / V ． s, d=0.075cm, T=300K, =100V/cm, N=10 11 cm -2 ). t d =0.75us.

36. 36 TIME-OF-FLIGHT DRIFT MOBILITY This method was first demonstrated in Haynes- Shockley experiment. The pulse shape is : Where N is the electron density in the packet at t=0. The first term in the exponent is drift and diffusion part and the second term is the recombination part.

37. 37 TIME-OF-FLIGHT DRIFT MOBILITY The minority carrier mobility is determined as The diffusion constant is where the pulse width Δt is measured at half the maximum amplitude.

38. 38 The lifetime is determined by measuring the electron pulse at t d1 and t d2, corresponding to two drift voltages V dr1 and V dr2. If there is no minority carrier trapping, the output pulse is V 01 and V 02, then If there is minority carrier trapping, the pulse area is A p, then plot log(A p ) vs. delay time t d, the slope should be -1/τ n.  = I/ q  tn

39. 39 (a)Time-of-flight measurement schematic, (b) output voltage for t t «RC, The dashed lines indicate the effect of carrier trapping. (c) output voltage for t t »RC, (d) implementation with a p+πn+ diode, both carriers can be measured.

40. 40 From the above system, Q N =qN=Q A +Q C At t=0 Q A =0, at t=t t Q A =Q N, where t t is the transit time When Q A changes from 0 to Q N the external current is:

41. 41 The output voltage I×R is For t t <<RC For t t >>RC t t can be determined from either case.

42. 42 Two drift mobility measurement implemen- tations as discussed in the text.

43. 43 In Fig. a, V 1 is applied to the gate and the diode, so there won’t have inversion layer under the gate. V 2 ’s period is 100us, its pulse width is 200ns. The poly-Si resistivity is 10KΩ/ □. Holes drift into the substrate, electrons drift along the surface. The time difference between two peaks gives the drift velocity. The field dependence of the mobility is obtained by varying V 2. The gate voltage dependence of the mobility is obtained by varying V 1.

44. 44 For short channel devices with R s Solve for I D,sat and drop high order terms Plot 1/ I D,sat vs. L m, gives (1/ I D,sat ) int and L m,int

45. 45 Substituting (1/ I D,sat ) int into L m,int gives The above equation contains no μ eff.

46. 46 Plotting L m,int vs. L m,int /(1/ I D,sat ) int has the slope A. Plotting A vs. 1/ (V GS -V T ) gives the line with slope S. The saturation velocity is  sat =1/ (W eff C ox (S-2R s ))

47. 47 8.6 MOSFET MOBILITY

48. 48 MOSFET MOBILITY There are many kinds of mobility due to : lattice (phonon) scattering, ionized impurity scattering,neutral impurity scattering, piezoelectric scattering, or surface scattering. According to Mathiessen’s rule

49. 49 Effective Mobility A MOSFET drain current is given as below, the first term is drift current, the second term is diffusion current

50. 50 Surface conditions for gate-to-channel capacitance measurements for (a) V GS ＜ V T, 2C ov is measured (b) V GS ＞ V Tb 2C ov +C ch is measured.

51. 51 (a)C GC and Q n versus V GS ; (b) I D versus V DS. W/L=10μm/10μm, t ox =10nm, N A =1.6×10 17 cm -3.

52. 52 μ eff versus V GS for the data of Fig. V T =0.5V. Once Q n and g d are obtained,μ eff can be calculated.

53. 53 Split C-V measurement arrangement. Q n can be obtained from I 1. Q b and substrate doping profile can be found.

54. 54 Capacitance as a function of gate voltage.

55. 55 The electric field is vertical field. η=1/2 for electron mobility, 1/3 for hole mobility.

56. 56 (a)Electron and (b) hole effective mobility as a function of effective field. Data taken from the references in the inserts.

57. 57 (a) Effective mobility, (b) normalized mobility versus V GS -V T.

58. 58 Effect of Gate Depletion and Channel Location Simulated Gate-to-channel Capacitance Versus Gate Voltage as a function of poly-Si gate doping density. Oxide leakage current not considered. t ox =2nm, N A =1.69×10 17 cm -3 μ eff =300 cm 2 /V-s

59. 59 MOSFET Cross Section Showing Drain and Gate Current, gate current adds to source current and subtracts drain current C ox C GC = 1+C ox /C G +C ox /C ch I D,eff = I D + I G /2 I D,eff =  I D = I D (V DS2 ) - I D (V DS1 )

60. 60 Drain and Gate Currents Versus Gate Voltage for n-channel MOSFET Gate insulator: HfO 2 ~2nm thick. With permission of W. Zhu and T.P. Ma, Yale University

61. 61 Effect of Inversion Charge Frequency Response Showing source and drain resistance (R S and R D ),inversion layer resistance R ch, overlap, oxide, channel and bulk capacitance (C ov,C ox,C ch, and C b ) C ox C ch tanh( ) C GC = Re () C ox +C ch +C b = (j 0.25  C’R ch L 2 ) ½

62. 62 Simulated Gate-to-channel Capacitance Versus Gate Voltage (a) Frequency(b) Channel Length Gate depletion and oxide leakage current not considered. Oxide leakage current not considered. t ox =2nm, N A =10 17 cm -3 μ n =300 cm 2 /V-s

63. 63 V G =V FB +  s + Q s /C ox ± Q it /C ox C ox (C ch +C it ) C GC = C ox +C ch +C b +C it C it = q 2 D it / (1+  2  it 2 )  it = exp(  E/kT)/  n  th N c = 4x10 -11 exp(  E/kT) [s]

64. 64 Simulated Gate-to-channel Capacitance vs. V G as a Function Interface Trap Density (a) Gate depletion and oxide leakage current not considered (b) Oxide leakage current not considered t ox =2nm, N A =10 17 cm 17, μ n =300 cm 2 /V-s, D it =10 12 cm -2 eV -1, τ it =5×10 -8 s

65. 65 Field-Effect Mobility The MOSFET g m is: Define the field effect mobility as:

66. 66 Effective and field-effect mobilities.

67. 67 If we take the μ eff dependence on V G into consideration, then

68. 68 When MOSFET operates in saturation region the drain current is: B represents the body effect. If m is the slope of (I D,sat ) 1/2 vs. (V GS -V T ), then