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Chapter 2

CARRIER AND DOPING DENSITY

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2.1 Introduction

Carrier density is related to the resistivity but is usually measured independently.

Carrier density and doping density are identical for uniformly doped material, but are different for nonuniformly doped materials.

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2.2 CAPACITANCE-VOLTAGE MEASUREMENTS

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Differential Capacitance

(a) A reverse-biased Schottky diode, (b) the doping density and majority carrier density profiles in the depletion approximation

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A small ac voltage with a frequency of 10 KHz to 1 MHz and an amplitude of 10 to 20 mV is applied to obtain a charge increment.

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In order to calculate the doping profile, the area must be known precisely. The C-V curve is measured, but the slope dC/dV or d(1/C2)/dV is used to calculate the doping profile.

The calculated doping density is located at

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The above mentioned method can be used in one-sided

pn junction, provided that the heavily doped concentration

is at least two orders of magnitude higher than the lightly

doped one. MOS-C can also be used, but it must be operated in deep

depletion during the measurement. Interface traps and

minority carrier generation may affect the results. The space charge region in MOS-C is modified to

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What really removed by the ac voltage is effective carrier density, that is not the doping density.

The effective carrier density is approximately the majority carrier density, therefore, the equations become:

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(a) C-V curve of a Si diode,(b) - V curve, (c) p(x) – W profile.

(a) C-V curve of a Si diode,(b) - V curve, (c) p(x) – W profile.

(a) C-V curve of a Si n+ p diode. (b) 1/C2 -V curve

A=2.5 X 10-3 cm2, T=300 K .

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(a) C-V curve of a Si diode,(b) - V curve, (c) p(x) – W profile.

(a) C-V curve of a Si diode,(b) - V curve, (c) p(x) – W profile.

No clear information is obtained from the C-V curve. It is obvious that d(1/C2)/dV is constant in figure (b). The carrier density profile is given in figure (c).

(c) p(x) – W profile. A=2.5 X 10-3 cm2, T=300 K .

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A schematic representation of the doping and majority carrier density of a nonuniformly doped layer .( a ) Zero-biased junction, ( b ) reverse-biased junction showing the doping profile, the majority carrier profiles in the depletion approximation and the actual majority carrier profiles for two reverse-bias voltages ·

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• The steeper the doping gradient, the more carrier density differ from the doping density.

• The Debye length is a measure of the distance over which a charge imbalance is neutralized by majority carriers under steady state or thermal equilibrium.• The Debye length sets a limit to the spatial resolution of the measured profile.

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The actually measured is an effective profile which is not the doping density profile but is closer to the majority carrier profile for nonuniformly doped substrate. All three profiles are identical for uniformly doped substrate.

A simplified relation between the doping density and the measured majority carrier concentration is

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A step profile can not be resolved within 2~3 Debye lengths.

Most of the above mentioned equations are derived based on depletion approximation, which assumes zero mobile carrier density in the space charge region, it is not valid for zero- or forward-biased junction.

Neglect majority carrier will introduce errors in MOS-C measurements. Taking majority carrier into consideration can correct uniformly doped substrate only.

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Doping density profiles for three samples. The filled circles are experimental data. The dashed lines indicate the profiles in the absence of interface states. The dot -dash lines show the profiles when the depletion approximation is used.

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Band Offset

(a) Cross-section and band diagram of two semiconductors with different band gaps, (b)schematic C-V and 1/C2-V plots. Real plots are smeared out and do not exhibit the sharp features shown here.

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C-V measurements can be used to measure n-N or p-P heterojunctions. The plateau capacitance Cp1 is related to the thickness of n-type material, and the plateau voltage ΔVp1 is related to the band offset.

For n* being the effective electron density, ND(x) the doping density, the interfacial charge at the heterojunction is

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The conduction band discontinuity is

where n1,n2 are the free electron densities of n- and N-type semiconductors; Nc1, Nc2, the effective density of states of the conduction bands; and xi the location of the heterojunction interface.

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Density plot of n-GaAs/N-Al0.3Ga0.7As heterojunction. The points are experimental data, the straight line is the assumed donor density.

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From the previous data,

Qi/q=2.74×1010cm-2 and

ΔEc=0.248eV were obtained.

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Maximum-Minimum MOS-C Capacitance

C-VG curve for an SiO2/Si MOS capacitor. NA = 1017 cm-3,tox = 10 nm, A = 5 X 10-

4 cm2.

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The average scr doping density of a MOS-C can be obtained by measuring the max. C (the accumulation region Cox) and the min. C (the hf strong inv. Region Cinv).

Cs=KsεoA/W

ψs,inv=2ψF+(4~6)kT/q

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From the measured C we can calculate W and then find out NA.

or use Cinv instead of C2ψF

Where R=Cinv/Cox

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An empirical equation is given as

where C1=RCox/(A(1-R))

The results are shown below.

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Doping density versus Cinv/Cox as a function of oxide thickness for the SiO2/Si system at T = 300 K.

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(a) MOSFET connection to determine the doping density of the gate, (b) resulting C-V curve calculated, ND = 5 X 1019 cm-3,tox = 10 nm.

MOSFET can also be used.

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Integral Capacitance

Differentiation often results in noisy profile,

especially for low dose ion implanted material.

• Integrate part of the C-V curve to obtain a partial dose of the implantation. • The chosen region can not extend into the uniformly doped substrate, nor into the region within 2~3 Debye lengths from the surface. • The partial dose is given by

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The area dependence is linear not square dependent.Besides, the ion implantation projected region R can be decided by

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Mercury Probe Contact

• C-V measurement needs to form a junction, sometimes, high temperature process is not desirable, then a temporary contact is in need.

• The mercury contact is a well defined orifice with a known area. It can be as small as in the um range to perform the wafer mapping.

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Electrochemical C-V Profiler

• Depth profiling can be obtained by etching the semiconductor surface between C-V measurement. • Early measurements divided the measurement and etch processes, in this method they are combined into one operation.• A dc current is passed through the I terminal, the dc voltage is monitored through the V terminal, and in order to reduce the series resistance, the ac signal is applied through the ac terminal.

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Schematic diagram of the electrochemical cell showing the Pt, saturated calomel and carbon electrodes and the pump to agitate the electrolyte and disperse bubbles on the semiconductor surface.

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The measured carrier density is

Where Δv is the ac voltage, typically 100~300mV, 30~40Hz; and ΔW is the resulting scr width change.W is determined by measuring the imaginary part of the current.

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Dissolution of the semiconductor depends on the presence of holes.For p-type semiconductor holes are obtained by forward bias the electrolyte-semiconductor junction. For n-type semiconductor holes are obtained by illuminating and reverse bias the junction.

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The etching depth depends on the dissolution current Idis.

where M is the semiconductor molecular weight,z is the dissolution valency (number of charge carriers required to dissolve one semiconductor atom), F the Faraday constant (9.64×104C), ρ the semiconductor density, and A the contact area.

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This method is mostly suitable for III-V materials because z=6 is well defined. For Si z varies from 2~5.

Hydrogen bubbles may cause nonuniformity and degrade the etching rate. This problem is solved by using a pulsed jet of the electrolyte.

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Profiles obtained with the ECV profiler and with SIMS. (a) p+(B)/p(B) Si, (b) n+(As)/p(B) Si. One drop of Triton X-100 is added to 100ml electrolyte gives z=3.7±0.1 for Si.

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2.3 CURRENT-VOLTAGE MEASUREMENT

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MOSFET Substrate Voltage-Gate Voltage

C-V measurement are performed at frequencies of 0.1~1MHz in order to reduce stray capacitance and increase the signal to noise ratio.

Typical diode area for C-V measurement is 0.1~1 mm diameter.

I-V measurement performed on MOSFET are used to obtain similar information in small area.

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The MOSFET is operated in linear region with small VDS 100mV. ≦

Apply a substrate voltage to change the scr width under the gate.

Adjust VGS whenever VSB is changed to keep IDS constant to ensure a fixed inversion charge density.

The relevant equations are

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Operational amplifier circuit for (a) the MOSFET substrate / gate voltage method, (b) the MOSFET threshold voltage method.

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Some considerations that affect the accuracy of this

method. Constant IDC corresponding to constant inversion

charge density is only a first order approximation. When VGS changes the mobility changes. The profile is affected by short channel effects. The profile can not be obtained within 3 Debye length

to the surface, the Debye limit.

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MOSFET Threshold Voltage

The threshold voltage is measured as a function of VSB.

where γ=(2qKsεoNA)1/2/Cox

The measured depth is

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First, measure VT vs. VSB, plot VT against (2ψF+VSB)1/2.

The slope gives γ.

NA can be calculated from a known value of γ.

ψF is a function of NA, therefore, iteration method is

necessary to obtain accurate results.

Constant drain current (typically 1 μA) method is used to

determine VT (so, it can be read directly from the

measuring circuit).

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Dopant profiles determined by MOSFET threshold voltage, SRP, pulsed C-V , and SUPREM3. Reprinted after Ref. 62 by permission of IEEE.

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Operational amplifier circuit for (a) the MOSFET substrate / gate voltage method, (b) the MOSFET threshold voltage method.

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Spreading Resistance

Spreading resistance method discussed in CH 1 is considered as an I-V method of the profiling measurement.

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2.4 MEASUREMENT ERRORS AND PRECAUTIONS

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Debye Length and Voltage BreakdownMobile majority carriers do not follow the doping profile if the doing density varies within the Debye length. What being measured is the effective carrier density.

How close we can measure to the surface:MOS devices: 3LD

Schottky diode: the zero bias scr width W0V

pn junction: junction depth+W0V

The measured profile depth upper limit is the breakdown scr width, WBD. There is no depth limit in Electrochemical C-V Profiler

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Spatial profiling limits. The “3 LD” line is the lower limit for conventional MOS-C profiling, the zero bias “W0V” line is the lower limit for pn and Schottky diode profiling, and the “WBD” line is the upper profile limit governed by bulk breakdown.

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For degenerate semiconductor, the resolution is limited by Thomas-Fermi screening length LTF :

where h is Planck`s constant and m* is the effective mass .

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For δ-doped semiconductor or quantum well, the resolution limit is

where N2D is the two dimensional doping density in cm-2.

Materials with higher carrier effective mass have better resolution.

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Series Resistance

(a) Actual circuit, (b) parallel equivalent circuit, (c) series equivalent

circuit for pn or Schottky diode.

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True capacitance is measured for Q 5, where ≧Q=ωC/G.

Wafer placed on a probe station without back

metallization has series contact resistance.

This can be checked by reducing the measurement

frequency, if CP is increased then there is rs.

C can be determined by:

Where Cs1 is measured at ω1, Cs2 is measured at ω2.

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1S212S

22 CC

C

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Equivalent circuits with ( a ) contact resistance , ( b ) contact capacitance . The elements within the dashed rectangles represent the intrinsic device .

In MOS devices if the back contact resistance rc causes problem, it may be advantageous to leave the back oxide on, the contact capacitance Cc may be performed as a short circuit at the measurement frequency.

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Series resistance changes the phase angle between the measured V and I, that interferes the profile measurements, if rsG ＜＜ 1 then from the relation between Cp and C, we can find

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Measured dopant profiles for a GaAs epitaxial layer on a semi-insulating substrate. The series resistance was obtained by placing resistors in series with the device.

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Minority Carriers and Interface Traps

(a) Equilibrium C-VG curve of an MOS-C, deep-depletion curves for (i) the sweep rate is 5 V/s and for (ii), (iii) is 0.1V/s, (b) the carrier density profiles determined from (a). Cox ＝ 98 pF, tox = 120 nm.

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For curve (i) the minority carrier generation is negligible.

For slower sweeping rate the minority carrier generation causes errors.

Cooling the device in LN2 can reduce the generation rate.

The dC/dV for curve (ii) is lower than the dC/dV for curve (i).

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Interface states cause C-V curves to stretch out. It can be corrected by high- and low-frequency

measurements

Pulsed C-V measurement at 30M Hz or cooling the device can reduce the interface effect.

Interfacial layer causes errors in Schottky diode C-V measurement, n 1.1 is satisfactory for profiling. ≦

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Diode Edge and Stray Capacitance

Accurate C-V profiling needs a precise contact area.

The effective area should take lateral space charge region into consideration.

If lateral extend equals to the vertical extend, then

where C=KsεoA/W, A=πr2, r is the contact radius,

For Si and GaAs b=1.5, for Ge b= 1.46.

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The edge effect is negligible if r 100bW.≧ The effective doping density is related to the

actual doping density by

The minimum recommended radius depends on the doping concentration is

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Ceff=AC+PCper+NCcor

A is the area, p the perimeter, N the number of corners.

Stray capacitance comes from:

1. cable (calibration before measurement)

2. probe (calibration before measurement)

3. bonding pads (calculation)

4. gate protection diode in MOSFET

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Excess Leakage Current

The voltage across entire diode is

V=Vjunction + Ileakage × quasi-neutral region resistance.

If excess leakage current occurs, such that considerable voltage drops in the quasi-neutral region, then errors are introduced in calculating the depletion width.

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Deep-Level Dopants / Traps

For C-V profiling measurements, the ac frequency should be high enough such that the traps are unable to follow it; and the dc voltage should be changed slow enough such that the traps can respond to it.

The deep-level dopants may not be totally ionized, the emission time constant is

where σp is the capture cross section area, vth the thermal velocity, Nv the effective density of states in the valence band.

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Band diagram of a reverse-biased Schottky diode showing complete ionization in the space-charge region (scr) but only partial ionization in the quasi-neutral region (qnr). (a) V=V1 (b)﹐ V=V1 +ΔV

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If τe﹤1/2πf, then carriers can be emitted during the positive ac half cycle, normal C-V curve can be obtained.

If τe ＞ 1/2πf, then carriers can not be emitted during the positive ac half cycle, the measured C-V curve can not present accurate doping profile.

During the negative ac half cycle, the scr narrows and the carriers are captured rather than emitted.

Usually, capture time is faster than the emission time.

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Semi-insulating Substrate

Semi-insulating Substrate, SOI, p on n, or n on p, have the following problem. When reverse bias increases on contact 1, the series resistance rs also increases.

The area of contact 2 should be much larger than that of contact 1, such that C2 is short-circuited and the C1 is measured.

Conducting layer on an insulating substrate showing the increasing series resistance with increasing back bias on contact 1.

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Instrumental Limitations

In C-V measurement, usually ΔV is kept constant

so that

When W increases and C decreases, ΔC decreases, and the instrumental limitations may affect the results.

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2.5 HALL EFFECT

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The Hall coefficient is defined as

where b=μn/ μp

and r is the scattering factorfor lattice scattering r=3π/8=1.18for ionized impurity scattering r=315π/512=1.93for neutral impurity scattering r=1

Usually, r is unknown and assumed to be 1.

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The Hall coefficient is determined experimentally as

For extrinsic p-type material, p is larger than n,

and for extrinsic n-type material,

where t is the sample thickness

The carrier type and carrier density can be determined from the Hall Coefficient.Assuming r=1 results in about 30% error.

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If a p-type semiconductor with doping density NA is

compensated with ND, then the hole density is

determined from:

where g 4 is the degeneracy factor for acceptors, ≒and EA the acceptor energy level above valance band.

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The slope of log(p) vs. 1/T is EA or EA/2, depends on the amount of ND.

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Carrier density vs. reciprocal temperature for Si : In with Al and B contamination. NIn = 4.5 X 1016 cm-3, EIn = 0.164 eV, NAl = 6.4 X 1013 cm-3, EAl=0.07 eV, NB = 1.6 X 1013 cm-3,ND = 2 X 1013 cm-3.

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Hall measurement gives the averaged carrier density.

Density profile can be obtained with differential Hall

effect (DHE) measurements. If the sheet Hall coefficient RHs=VH/BI is measured, t

he carrier density profile is

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For a p on n or n on p two layer material with a top layer

thickness t1, conductivity σ1, Hall constant RH1 and a

bottom layer thickness t2, conductivity σ2, Hall constant

RH2 ,The Hall constant is

where t= t1+ t2and

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If the top layer is more heavily dopes than the substrate

or it is formed by inversion through surface charges,

such that σ2 <<σ1 , then

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2.6 OPTICAL TECHNIQUES

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Plasma Resonance

The semiconductor reflection coefficient is

where n is refractive indexk=αλ/4π is the extinction coefficientα the absorption coefficient, λ the wavelength

At plasma resonance

λ=λp , R 1, ≒m* is the effective mass

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The free carrier density p can be determined from λp.

Usually, λp is not well defined, and it is λmin at the

Rmin being determined; λmin < λp .

An empirical equation is used to find out the carrier

density, this technique is used for 1019cm-3>p, n>1018cm-3.

A, B, C are fitting parameters.

This method measures uniformly doped material with

thickness at least equal to 1/ α.

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Free Carrier Absorption

Photons with hν<Eg can be absorbed by free carriers.

The free hole absorption coefficient is

The empirical results are

This technique is used for p, n>1017cm-3

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Free Carrier Absorption

For n-GsAs

Free carrier absorption is also related to sheet

resistance ρs

where T is the transmittance, k=0.15 for n-Si,

k=0.3375 for p-Si.

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Infrared Spectroscopy

(a) Energy band diagram for a semiconductor containing donors at low temperature, (b) energy band diagram showing the donor energy levels, (c)band diagram when both donors and acceptors are present. The “above-band gap” light fills donors and acceptors.

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(a) Donor impurity spectrum for 265 Ω-cm n-Si at T ～ 12K, (b) spectrum for the sample in (a) with " above-band gap " illumination.

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This method is excellent for identifying impurities. Calibrated data is necessary to relate the peak

height and the impurity density. The transmittance of semiconductor through a

sample with thickness t is

for reasonable measurement sensitivity, t 1/α.≒

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Photoluminescence

Incident light on samples to create EHP, subsequent radiative recombination are measured.

Impurity identification is very precise, but density measurement is difficult.

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Comparison between the PL resistivity and electrical resistivity for B and P in Si.

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Secondary Ion Mass Spectrometry (SIMS)

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Lateral Profiling

Two-axis beveled sample. The cross-hatched regions represent the implanted layer in both the vertical and the beveled planes. The probes are indicated by the solid points and their stepping direction by the arrows.