Modeling Minority-Carrier Lifetime Techniques that use Transient 

Excess-Carrier Decay

Steven W. Johnston, Gregory M. Berman1, and Richard K. Ahrenkiel2

National Renewable Energy Laboratory • Golden, CO 80401

1Optical Science & Engineering Program, University of Colorado • Boulder, CO 80304

2Colorado School of Mines • Golden, CO 80401

The information contained in this poster is subject to a government license.

Presented at the 33rd IEEE Photovoltaic Specialists Conference • San Diego, CA • May 11–16, 2008 • NREL/PO-520-43308

Model coil and sample Coupled inductors (like transformer)

and sample as 1-turn coil

Sample inductance

Sample

resistance

Coil antenna

Circuit models

L1 L2 RL

kc

RL

L1-Lm L2-Lm

Lm

Lm

= kc L1L2

ZL

100

80

60

40

20

0

ABS(Z)

460 480 500 520 540

Frequency (MHz)

-100

-50

0

50

100Angle(Z)

30

25

20

15

10

5

ABS(Z)

460 480 500 520 540

Frequency (MHz)

-100

-50

0

50

100Angle(Z)

5000

4000

3000

2000

1000

0

ABS(Z)

200 300 400 500 600 700

Frequency (MHz)

-100

-50

0

50

100Angle(Z)

5000

4000

3000

2000

1000

0

ABS(Z)

200 300 400 500 600 700

Frequency (MHz)

-100

-50

0

50

100

Angle(Z)

450

400

350

300

250

200

150

100

50

0

-50

ABS(Z) [ohms]

504 506 508 510 512 514 516

Frequency (MHz)

80

60

40

20

0

-20

Angle(Z) [degrees]

tuning point

35

30

25

20

15

10

5

Real Transformed Impedance (

0

∆R) [

Ω]

0 10 20 30 40 50

Imaginary Transformed Impedance (∆X) [Ω]

 sample 0.01 mm from coil 

 sample 2 mm from coil

 sample 4 mm from coil

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

∆R (ohms)

0.0 0.5 1.0 1.5

∆L (nH)

 tape sample directly to coil

 sample spaced ~2 mm from coil

 sample spaced ~4 mm from coil 

1.0

0.8

0.6

0.4

0.2

0.0

|Γ|

420 425 430 435 440 445

Frequency (MHz)

 |Γ| unloaded

 |Γ| loaded (sample) 

 |Γ| loaded & light

Q=127 Q=140 

0.05

0.04

0.03

0.02

0.01

0.00

|Γ|

432.80 432.85 432.90 432.95 433.00

Frequency (MHz)

Operating frequency

 |Γ| with light on 

 |Γ| with light off 

Γ = 0.0356 

Γ = 0.0007 

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Γ| |

3.5 3.6 3.7 3.8 3.9

Rsample

 modeled |Γ| 

 linear fit

Directional 

Coupler

Power in 

(out port)

Power out 

(in port)

Variable 

air-gap 

capacitor

coil 50 Ω coaxial cable

(transmission line)

Reflected 

power out 

(coupled 

port)

ZL coaxial cable

Add capacitor

Model

capacitor: S(ω)=ωC

2

ωτ

sin(ωτ )

[cos(ωτ )+ cos(ξωτ)]

 

 

 

 

 

 

 

 

 

 

where:

τ =

d

vo

ξ = d2 − d1

d

C = ε oad

h

d1 d2

d

a

h

Add transmission line (coaxial cable) ZL

3.5

3.0

2.5

2.0

1.5

1.0

0.5

0.0

∆R (ohms)

0.0 0.5 1.0 1.5 2.0

∆L (nH)

 kc = 0.25 

 kc = 0.20

 kc = 0.15

L1 = 70 nH

L2 = 4 nH

f = 500 MHz

 RL Å 3 to 15 1⁄2 

Zcap

= 0.05 − 1

S(ω)

i

“Capacitor Modeling at High Frequencies”

J.A.B. Faria, IEEE Trans. on Education 35, 

1992, pp. 214-216.

10-1

100

101

102

103

104

ABS(Z)

0 200 400 600 800 1000

Frequency (MHz)

-100

-50

0

50

100Angle(Z)

Measured 

 Modeled

0.01

0.1

1

Signal (V) 

0 50 100 150 200

Time (μs) 

14 μs

34 μs

 μJ/pulse

 50

 26

 9

 4

 0.4 

1

10

100 Signal (mV) 

0 50 100 150 200

Time (μs) 

13 μs

 μJ/pulse 

 32

 16

 8

 0.8

0.01

0.1

1

Signal (V) 

0 50 100 150 200

Time (μs) 

 μJ/pulse 

 230

 140

 53

 48

29 μs

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Γ| |

0 20 40 60 80

Laser light intensity (μW)

Measured Reflection Coefficient 

vs. 

Light Intensity

Ansoft HFSS Modeling

Ztrans

= ZO

ZL −iZO tan(βz)

ZO − iZL tan(βz)

where β = 2π/λ

z = length of coax

ZO = 50 Ω characteristic impedance

Measured impedance analyzer data Modeled coil on transmission line

Why measure minority-carrier lifetime?

Transient techniques for measuring minority-carrier lifetime in silicon

Microwave Reflection Photoconductive Decay (μ−PCD)

waveguide

sample

light pulse

Resonant-Coupled Photoconductive Decay (RCPCD)

directional 

coupler E/H plane tuner

• Contactless

• Indirect and small bandgap materials can be measured

• Transient technique gives direct measure of decay rate

Lifetime is reduced when defects are present, so the value 

of lifetime can give an estimate of material quality.

• Excess carriers are created by light pulses and increase the conductivity of the sample.

• Small antenna or open-ended waveguide senses changing photoconductivity in the sample.

• Electronic circuitry measures the decay of photoconductivity as carriers in the sample 

recombine to equilibrium concentration.

How do μ-PCD and RCPCD work? 

• Microwave Reflection Photoconductive Decay (μ-PCD)

• Resonant-Coupled Photoconductive Decay (RCPCD)

• Transient Free-Carrier Absorption (FCA)

These techniques are

Couple sample to coil

Experimentally-measured impedance with 

increasing light on sample

Circuit-modeled impedance with changing 

sample resistance

HFSS-modeled impedance with changing 

sample conductivity

and

Measured and modeled values for a 

variable air-gap capacitor set at ~15 pF

Best fit real resistance

Use directional coupler, or circulator, to send power to antenna and monitor reflected 

power due to sample photoconductivity

Circuit photo showing directional coupler, 

shielded tunable capacitor, semi-rigid coaxial 

cable, and coil antenna

Measured impedance analyzer data 

of capacitor, coaxial cable, and coil

Modeled circuit of capacitor, 

coaxial cable, and coil

Zoom in near 500 MHz The combination of coil impedance, coaxial cable 

length, and capacitance resonance leads to a 

circuit impedance resonance near 500 MHz.

At the tuning point, the impedance magnitude 

is 50 Ω, and the impedance angle is 0°. For 

this condition there is no reflected power from 

the directional coupler, Γ = 0.

While samples vary in size, shape, and 

conductivity, the tuning point can be found by 

adjusting the capacitor, frequency, and the 

coupling distance of the sample to the coil 

antenna.

Measured frequency response of 

reflected power for a tuned circuit

Expanded view of operating point 

(photoconductivity causes power reflection)

Modeled response of power reflection 

due to sample resistance

Measured power reflection coefficient 

due to sample photoconductivity

(pulse amplifier saturates when Γ~0.05, so 

measurements are made for 0<Γ<0.05)

RCPCD μ-PCD Transient FCA

Transient Free-Carrier Absorption 

(FCA)

microwave signal 

generator

Microwavefrequency amplifier

diode detector

high-bandwidth 

dc-coupled 

pulse amplifier

Simulated waveguide structure for μ-PCD

Using Ansoft’s HFSS software

μ-PCD block diagram

oscilloscope

The large box represents air for calculation purposes. The 

gray square is a semiconductor sample. The red spot 

represents the laser spot where conductivity is changed. The 

blue waveguide includes E and H plane tuning stubs and has 

the E-field magnitude superimposed on it.

Tuning of the waveguide is accomplished using the E and H 

plane stubs and by adjusting frequency. Tuning to a zero 

point in phase with a magnitude minimum results in a linear 

phase response to changing conductivity, but magnitude 

response is noisy. Tuning to a magnitude maximum results 

in good magnitude response but phase is noisy. Therefore, 

best linearity results from tuning to a frequency between the 

minimum and maximum values of the waveguide impedance 

magnitude.

Magnitude (Ohms) Phase (degrees)

The graphs below show the magnitude and phase of the 

waveguide’s impedance as the conductivity in the spot of 

the wafer is changed. The magnitude shows a large linear 

range; however, the phase approaches a minimum. This 

leads to a noisy, poorly-resolved solution, and as can be 

seen below, the reflected power also becomes non-linear.

HFSS is used to simulate reflected power, the quantity that is actually 

measured experimentally. As can be seen, there is a region of linearity 

at low injection levels. But, as the spot’s conductivity increases into

Reflected Power vs Conductivity

high injection, the curve levels off, thus limiting the range of 

this method to lower injection levels. It is worth noting, 

though, that the linear region covers over 10% of possible 

reflected power, resulting in a strong, clean signal.

α =

q3λ

2

p

4π2

ε oc 3nm *2 μ

Free carrier absorption in semiconductors is given by

, where λ = wavelength

p = density of free carriers

n = refractive index

m* = effective mass, and

μ = mobility.

D. K. Schroder et al., “Free Carrier 

Absorption in Silicon”, IEEE J. 

Solid-State Circuits SC-13, 1978.

Free carrier absorption is a linear function of the density 

of free carriers. Light with an energy greater than that 

of the bandgap is intrinsically absorbed. This is 

displayed in the left portion of the graph below. For light 

with wavelengths longer those corresponding to the 

bandgap energy, the absorption coefficient is linearly 

proportional to the carrier density. This infrared 

absorption is displayed on the right portion of the graph 

below.

Unpolished surfaces scatter light and reduce the 

infrared beam’s transmission. We have measured 

a double-polished wafer, yet the signal is still small 

compared to background noise and requires higher 

injection levels than μ-PCD and RCPCD.