1 CHAPTER 7 CARRIER LIFETIMES. 2 7.1 INTRODUCTION Lifetime: Generation lifetime, recombination...

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CHAPTER 7

CARRIER LIFETIMES

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7.1 INTRODUCTION

• Lifetime: Generation lifetime,

recombination lifetime.

• Occur in bulk: lifetime τg, τr..

• Occur in surface: surface velocity sg, sr.

• The measured lifetime are always effective values consisting both components.

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Recombination and Generation

(a) Forward-biased and (b) reverse-biased junction.

4

• Consider p-type semiconductor in this chapter.

• Consider minority electrons.• Consider linear, quadratic, and third-order

terms.

7.2 RECOMBINATION LIFETIME-SURFACE RECOMBINATION VELOCITY

5

RECOMBINATION LIFETIME-SURFACE RECOMBINATION VELOCITY

n=no+Δn, p=po+Δp, no<<po, Δn=Δp.

(7.1)

(7.2)

(7.3)

(7.4)

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Recombination mechanisms: (a) SRH, (b) radiative, (c) Auger.

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(7.5)

(7.6)

(7.7)

(7.8)

(7.9)

8

low-level injection (ll):Δn<<po,

high-level injection (hl): Δn>>po.

(7.10)

(7.11)

(7.12)

(7.13)

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Table 7.1 Recombination Coefficient

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Recombination lifetime versus majority carrier density for p-Si [Cp=1.1×10-30

cm6/s (dashed lines), Cp =10-31cm6/s (solid lines)] and for n-Si (Cn=2.8×10-31

cm6/s). Data from Ref. 10.

Recombination Lifetime Versus Majority Carrier Density

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SRH (multi-phonon) recombination

(7.14)

(7.15)

(7.17)(7.16)

(7.18)

(7.19)

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sr Versus Injection Level η As a function of σps for Nit=1010cm-2, pos=1016cm-3, ETs=0.4eV, σns =5×10-14cm2. Data from ref. 15

sr Versus Injection Level η

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• For heavily doped Si, Auger recombination dominates the carrier lifetime.

• For lightly doped Si, SRH (multi-phonon) recombination dominates the carrier lifetime.

• Radiative recombination plays almost no role in Si, but is important in direct bandgap material like GaAs.

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7.3 GENERATION LIFETIME-SURFACE GENERATION VELOCITY

• Each of the recombination process has a generation counterpart.

SRH recombination --- Thermal generation

Radiative recombination --- Optical generation

Auger recombination --- Impact ionization

• When pn>ni2 recombination dominates,

when pn<ni2 generation dominates.

• The smaller the pn product, the higher is the generation rate.

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Usually, τg>τr. For Si τg (50~100)τ≒ r.

when psns<ni2

For ET≠Ei, sr > sg.

(7.20)

(7.21)

(7.22)

(7.23)

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7.4 RECOMBINATION LIFETIME-OPTICAL MEASUREMENTS

n(t)/t = G – R = G – n(t)/ τeff(7.24)

τeff(n) = n(t)/ [G(t) – dn(t)/d(t)] (7.25)

The continuity equation for uniform ehp generation and zero surface recombination

In the Transient Photoconductance Decay (PCD) method, with G(t) << dn(t)/d(t)

τeff(n) = – n(t)/ dn(t)/d(t) (7.26)

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τB is the bulk recombination lifetime, D the minority carrier diffusion constant under ll injection or the ambipolar diffusion constant under hl injection, d the sample thickness.

The excess carrier density for low level injection is given by Δn(t)= Δn(0)exp(-t/τeff)

(7.28)

(7.29)

In the steady-state method, with G(t) >> dn(t)/d(t)

τeff(n) = n(t)/ G (7.27)

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Effective lifetime versus wafer thickness as a function of surface recombination velocity. D=30cm2/s.

For thick samples τeff ~ τB.For thin samples τeff depends on d and sr.

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(7.30)

(7.31)

• For Sr~0, plot 1/τeff vs. 1/d, the slope is 2sr, the intercept is 1/τB

• For Sr~∞, plot 1/τeff vs. 1/d2, the slope is πD2, the intercept is 1/τB.

Two limiting cases are of particular interest:

sr 0 gives tan(d/2)d/2 and

sr gives tan(d/2) or d/2/2

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Determination of bulk lifetime, surface recombination velocity, and diffusion coefficient from lifetime measurements.

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The above discussions holds for samples with one dimension much smaller than the other two dimensions. For samples with none of the three dimensions (a, b, and c) very large, then for sr~ ∞

(7.32)

(7.33)

(7.34)

(7.35)

(7.36)

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Calculated normalized excess carrier density versus time as a function of surface recombination velocity. d=400μm, α=292 cm-1 (optical absorption coefficient).

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7.4.1 Photoconductance Decay (PCD)

Schematic diagram for photoconductance decay measurements.

= q (nn+ pp) (7.37)

n= / q (nn+ pp) (7.38)

In PCD, the conductivity

n=n0+n, p=p0+p under low level injection

n = p, the excess carrier density is related to the conductivity by

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Schematic diagram for photoconductance decay measurements.

Where Δg =ΔσA/L.

(7.39)

(7.40)

(7.41)

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For constant voltage method, choose R smaller than the sample resistance, in ll case RΔg<<1 or ΔV<<Vo, then

For constant current method, R is very large, if rdkΔg<<1 or ΔV/ Vo/<<rdk /R, then

(7.42)

(7.43)

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• For ΔV to be proportional to Δn, it is necessary that ll injection must prevail, otherwise corrections must be applied.

• The contacts should not inject minority carriers, nor should they raise sample temperature.

• The electric field in the sample should be held at E 0.≦3/(μτr)1/2, μ is minority carrier mobility.

• The excitation light should be penetrating through the sample.

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(a) RF bridge measurement (b) Microwave Reflectance Measurement

PCD Measurement Schematic for Contactless

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Low Sr can be achieved by: Oxidized Si surface, Sr=20cm/s. Immersing Si in HF (Sr=0.25cm/s), in iodine/methanol

(Sr=4cm/s). Low temperature SiN remote plasma CVD deposition

(Sr=4-5cm/s).Inorganic sulfides passivation on GaAs (Sr=1000cm/s),

P / P = C

In the microwave reflection method

(7.44)

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7.4.2 Quasi-Steady-State Photoconductance (QSSPC)

The sample is illuminated with a flash lamp with a ms decay time, the sample is under quasi-steady-state during the measurement as the light intensity varies from the maximum to zero. The generation rate is

For eff < flash , the solution of Eq. (7.25) is

(7.45)

(7.46)

(7.47)

G = f / d

G(t) = 0 for t 0; G0exp(-t/flash) for t > 0

n(t)=eff/(1-eff /flash )G0(exp(-t/flash )-exp(-t/eff )

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Effective Recombination Lifetime Versus Injection Carrier Density

Obtained with the QSSPC technique. Adapted from ref. 37.

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7.4.3 Short-Circuit Current / Open-Circuit Voltage Decay (SCCD/OCVD)

For the short-circuit current,

For the open-circuit voltage

(7.48a)

(7.48b)

(7.49)

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• The current decays exponentially with time, the time constant depends on excess carrier density.

• The voltage decay is influenced by the junction RC time constant. This effect is reduced by measuring the small-signal voltage decay with a steady-state bias light to reduce R.

• For samples with thickness much larger than the minority carrier diffusion length, sr is no longer important

Isc, Voc ~ (exp(-t/τB))/(t1/2) (7.50)

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7.4.4 Photoluminescence Decay

• Direct band gap material has larger PL density than indirect band gap material.• It becomes complicate when self-absorption occurs .

(7.51)

• The first term is the non-radiative lifetime,γ is the photon recycling factor.• Self-absorption is not important for indirect semiconductor.

(7.52)

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7.4.5 Surface Photovoltage (SPV)

Sample cross section for SPV measurements. The optically transparent, electrically conducting contact to the left of the sample allow light to reach the sample and the voltage to be measured.

• The surface scr is induced by surface charges, not due to a bias voltage.

• The incident light wavelength is varied, the sample back is kept in dark.

• Some minority carriers diffuse toward the illuminated surface, building a SPV which is proportional to Δn(W).

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• SPV method is used to determine the minority carrier diffusion length.

• To simplify data extraction, we need Undepleted wafer much thicker than Ln

The scr width smaller than Ln

α(d-W)>>1>> αW, α the absorption coefficient Sample diameter > sample thickness

(7.53)

The diffusion equation solution is

(7.54)

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The Δn(W) is related to the voltage by

which yields

Vspv is proportional to Δn for Vspv<0.5kT/q

(7.55)

(7.56)

In constant surface photovoltage method, Vspv=const, Δn(W) is constant, for different wave lengths with different α, adjust the photon flux density to obtain constant Vspv, this yields

37

Plot Φ vs. 1/α, the intercept is Ln and the slope is related to s1.

(7.57)

In constant photon flux density method,

Plot 1/ Vspv vs. 1/α, the intercept is Ln.

(7.58)

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(a) Constant voltage, (b) constant photon flux SPV plots for Si samples.

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The relation between λ and α is important for the accuracy of SPV method

For Si, λ=0.7~1.1μm.

For InP, λ=0.8~0.9μm.

For GaAs, λ=0.75~0.87μm.

For Si.

The condition W<<Ln is usually true for single-crystal Si, yet, for GaAs and amorphous Si this condition is not satisfied, then the intercept is given as

(7.59)

(7.60)

(7.61)

(7.62)

(7.63)

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7. 4. 6 Steady-State Short-Circuit Current

Measurement schematic for (a) the short-circuit current diffusion length measurement method, (b) for the ELYMAT double surface method.

The sample should be a pn junction or Schottky diode.The Jsc is measured with varying wavelength.

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For an n+p diode the second term is negligible.

Adjust Φ such that Jsc is kept constant.

Where C1=Jsc/q(1-R)Ln, plot Φ vs. 1/α.

(7.64)

(7.65)

(7.66)

Or write

Where X=q(1-R) Φ/Jsc, plot 1/α vs. (X-1).

(7.67)

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Electrolytical metal tracer method:IF: short penetration depth, moderate diffusion length,negligible back s

urface recombination.IR: penetrating light, low front surface recombination.

A is the area, sf is the front surface recombination velocity.

(7.68a)

(7.68b)

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7.4.7 Free Carrier Absorption

(a) Schematic free carrier absorption arrangement, (b) schematic for lifetime mapping.

(a) (b)

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Pump beam energy hυ>Eg, probe beam energy hυ>Eg. The transmitted photon flux is

d is the sample thickness, αfc is free carrier absorption coefficient, R the reflectance. For n-type semiconductor Kn~10-18cm2/μm2, Kp~(2-2.7)10-18cm2/μm2.

(7.69)

fc=Kn2n (7.70)

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The change in a transmitted pulsed beam is

The change in the absorption coefficient is

Δn is related to the minority carrier lifetime.The fractional change in transmitted flux is

(7.71)

(7.72)

(7.73)

46

Knowing the laser generation rate G’=(1-R)(cm-2s-1) and the sample thickness d, the effective lifetime is

(7.74)eff = dn/G’

Free carrier absorption lifetime map

The black body emits over a wide wavelength range with a peak at peak = 3000/T m (7.75)

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7.4.8 Electron Beam Induced Current (EBIC)

E/Eehp=Nehp+γ(Ebs/Eehp)

Ebs is the mean backscattering electron energy, Eehp~3.2Eg, =3.64eV for Si. γis the backscattering coefficient. The electron penetration depth is

ρis the semiconductor density. The generation volume is pear shaped for Z<15. for 15<Z<40 a sphere, for Z>40 a semispherical.

(7.76)

(7.77)

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Electron Beam Induced Current

Assume a sphere volume of (4/3) π(Re/2)3, the generation rate is

For Si with a beam current of 10-10A. Eehp=3.64eV, E=104eV, G=3X1024 ehp/cm3.s IEBIC can be collected by a junction. The beam position and energy can be changed. G’=IbNehp/q.

(7.78)

(7.79)

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(a) Conventional EBIC implementation, (b) depth modulation by electron beam energy.

In figure (b),

For high Sr,

(7.80)

(7.81)

50

pn junction I-V curve showing the space-charge and quasi-neutral region currents.

7.5 RECOMBINATION LIFETME-ELECTRICAL MEASUREMENTS

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7.5.1 Diode Current-Voltage

The forward current of a diode has two parts:

for a p+n diode τp can be calculated

(7.82)

(7.83)

J = J0,scr(exp(qV/nkT)-1) + J0,qnr(exp(qV/kT)-1)

J0,scr = qniW/scr;

J0,qnr = qni2F (Dn/NALn + Dp/NDLp)

J0,qnr qni2F Dp/NDLp

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PN Junction Cross Sections

(a) Recombination mechanisms in the n-substrate for d<Lp, (b) Denuded zone on precipitated substrate, (c) Epitaxial layer on substrate, and (d) SOI wafer

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The correction factor:

(7.84)

(7.85)

(SrLp/Dp)cosh(d/Lp) + sinh(d/Lp)F= cosh (d/Lp)+ (SrLp/Dp)sinh(d/Lp)

(1+NA2/NA1)exp(Dp/Lp)+(1-NA2/NA1)exp(-Dp/Lp) F (1+NA2/NA1)exp(Dp/Lp)-(1-NA2/NA1)exp(-Dp/Lp)

For the ex[itaxial device in Fig. 7.17(c)

Jr=-qniW/g-qni2F(Dn/NALn+Dp/NDLp)

-qniW/g-qni2FDp/NDLp

(7.86)

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Reverse Leakage Current Versus SCR Width for Measured and Corrected Capacitance

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7.5.2 Reverse Recovery (RR)

(a) Reverse recovery circuit schematic, (b) current and voltage waveforms for abruptly switched current, (c) current and voltage waveforms for ramped currents.

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Qs is the charge remained at t=ts. Usually, Qs/Irτr is a constant.

(7.87)

(7.88)

(7.89)

For power diodes, the current can not change abruptly, there is a current slope during turn off. The life time is calculated as:

(7.90)

57

Storage time versus (1+ If/Ir). Reprinted after Kuno (Ref. 87) by permission of IEEE.

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7.5.3 Open-Circuit Voltage Decay (OCVD)

Open-Circuit voltage decay (a) circuit schematic, (b) voltage waveform.

The carrier concentration at the junction edge is

(7.91)

59

The corresponding voltage is

The diode voltage is Vd=Vj+Vb. Vb, the Dember Voltage, is caused by μn and μp difference.

Vb has to be taken into consideration at hl, but is negligible at ll. For substrate thickness d>>Ln, and ll the diode voltage is

(7.92)

(7.93)

(7.94)

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Open-circuit voltage decay waveform.

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For t>4τr ,

At hl, the life time is

The slope of the above curve is

(7.95)

(7.96)

(7.97)

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7.5.4 Pulsed MOS Capacitor

Pulsed MOS capacitor recombinationlifetime measurement. The device behavior at various voltages is shown in (a), (b), (c), and the C-VG and VG-t curves in (d).

Method 1

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For tp>τr, the capacitance change is CACBCC .

For tp<<τr, the capacitance change is CACBCA .

Change the pulse width and measure the CC, plot ln((1/CA

2)-(1/CC2)) vs. tp, τr is obtained from the slope.

(7.98)

(7.99)

(7.100)

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The C-VG and C-t behavior of and MOS-C pulsed into deep depletion.

Method 2

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The thermal generation components of a deep-depleted MOS capacitor.

1. Bulk scr generation τg, 2. lateral surface scr generation sg, 3. surface scr generation under the gate sg’, 4. qnr generation Ln, 5. back surface generation sc.

66

The mechanism 2 is involved when pulsing from accumulation, pulsing from inversion is usually used to obtain low surface generation. The capacitance dependence on VG and Qn is

Solving for VG and differentiate wrt t gives

(7.101)

For pulsed capacitor, dVG/dt=0, that gives

(7.102)

(7.103)

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The dQn/dt is the thermal generation rate, which can be written as

As=2πrW is the area of the lateral scr region.AG=2πr2 is the gate area. Ln’ is effective diffusion length contains bulk and surface effects.

(7.104)

(7.105)

When dQn/dt = -qni2Dn/NALn’ is substituted into Eq.(7.103)

C= Ci/(1-t/t1)1/2(7.106)

where = d/Ln

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Generation rates as function of temperature for Si with τg=1 ms, sg=1cm/s, s'g=0.1cm/s,L'n=10-2cm, sc=100cm/s, W=3 μm, r=0.5mm, NA=1016 cm-3, Dn =37(300/T)1.2 cm2/s, d=750 μm, and ni =9.15×1019(T/300)2exp(-0.5928/kT).

69

7.6 GENERATIN LIFETIME 7.6.1 Gate-Controlled Diode

The gate-controlled diode. D is the n+p diode, G the circular gate, and GR the circular guard ring.

70

Gate-controlled diode in (a) accumulation, (b) depletion, (c) inversion; (d) shows the current-voltage characteristic with points A, B, and C corresponding to (a), (b), and (c). Reprinted with permission after Schroder, Ref. 107.

The generation currents are:1. Diode scr current IJ

2. Gate induced scr current IGIJ

3. Surface under the gate IS

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At point B 0<φs<VD1+2φF, the scr width is

For φs>VD1+2φF, the scr width pins to

The junction scr width is

(7.107)

(7.108)

(7.109)

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Experimental gate-controlled diode current-voltage characteristic.

(7.110)

(7.111)

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7.6.2 Pulsed MOS Capacitor

Zerbst plot The MOS is pulsed into deep depletion and the C-t is measured. The generation rate is

Winv=(4KsεoφF/qNA)1/2

(7.112)

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Pulsed MOS Capacitor

Zerbst plot

τg,eff is defined by considering only the Scr generation rate

(7.113)

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(a) C-t response and (b) Zerbst plot. Reprinted with permission after Kang and Schroder, Ref. 118.

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The relation between scr width and C is

The relation for Zerbst plot is

The slope contains the information of τg and sg.The intercept is related to sg’, Ln, and Sc.

(7.114)

(7.115)

(7.116)

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Because

The Zerbst plot vertical axis is proportional to the ehp generated charge or current, and the horizontal axis is proportional to the scr width.The C-t transient time is quite long, the relaxation time tf is

(7.117)

(7.118)

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Generation lifetime measurement schematic for SOI devices.

(7.119)

(7.120)

79

Current versus inverse capacitance plot for the device whose Zerbst plot is shown in Fig. 7.28 Reprinted with permission after Kang and Schroder, Ref. 118.

(7.121)

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Jscr = [Cox/(Cox-C)]J = qni(W-Winv)/q,eff+qniseff (7.122)

W = Ks0Ag(1/C-1/Cox) (7.123)

qni(W-Winv)g,eff =

dJscr/dW(7.124)

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Inversion, saturation, and deep-depletion MOS-C curves.

Linear sweep method

R=dVG/dt is the sweep rate

(7.126)

(7.125)

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Linear ramp plot for the device whose Zerbst plot is shown in Fig. 7.28. Reprinted with permission after Kang and Schroder, Ref. 118.

When the device enters saturation, Csat changes neither with VG nor with t, the left side of the above Equation becomes 0.

(7.127)

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Corona-pulsed deep-depletion measurement apparatus.

Corona-oxide-semiconductor (COS) method

+ or – charge from a corona source is deposited on the semiconductor surface

84

∵ dVFB/dt=0, ∴

For QG = -Qs = constant

(7.128)

(7.129)

(7.130)

(7.131)

(7.132)

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COS generation lifetime plot.

From dQn/dt, one finds

(7.133)

(7.134)