Regression Case Study: Abstract
Statistical Engineering Division
Case Studies Series
A Regression Case Study: The Non-Linear Modeling of a
2-Dimensional Family of Curves Involving p-type
Semiconductor Electron Mobility
Herb Bennett
Semiconductor Electron Devices, EEEL
James J. Filliben
Statistical Engineering Division, ITL
This is the first in a series of talks which presents--in a
tutorial style-- the details of a collaborative case study
between members of ITL's Statistical Engineering Division
and members of the NIST scientific/engineering staff. The
purpose of this series is to present methodologies,
techniques, principles, and strategies for data-analytic
problem-solving which may be potentially applicable to other
NIST problems beyond the specific case study at hand.
The Problem
This first case study focuses on the regression modeling of the
electron mobility for p-type gallium aluminum arsenide (GaAlAs)
in the "minority electron" case--that is, for the case where
there are fewer electrons than holes. Quantum mechanical
non-linear integral-differential equations give a self-consistent
description of carrier transport and mobility in
Ga_{1-y}Al_{y}As/GaAs heterostructures, where y is the mole
fraction of AlAs. Many hours of NIST Cray CPU time were used to
solve these complex equations. The results are usually given via
numerical tables for describing quantitatively how the electron
mobility varies with dopant density and aluminum arsenide mole
fraction, but interpolatory use of such look-up tables in
semiconductor device simulators on engineering workstations is
computationally inefficient--particularly for industry. We are
thus led to the desired output from the data analysis, namely, a
closed-form 2-dimensional analytic function f such that
mobility = f(dopant density, mole fraction)
Importance: Device Simulators
If such a function can be derived, then it will represent a
significant increase in computational efficiency via the inclusion
of more physically correct mobility models in commercial
semiconductor device simulators. The combination of the existing
NIST Cray-generated mobility data and the derived 2-dimensional
analytic function will lead to computer simulators that are at
once both more parsimonious (fewer unknown or variational
parameters) and more accurate (improved predictability).
Example: Cell Phones
The talk itself will consist of the step-by-step sequencing
through the analysis, with emphasis on the approach being
used and pitfalls to be avoided. Sub-topics include
transformations, admissible non-linear models, separable
functions, and melding functions. The data set to be
analyzed in this talk is found in BENNETT.DAT and BENNETT2.DAT
of version 98.11 of Dataplot (use LIST and COPY to extract
this file).
For your convenience, copies of these files are included here:
BENNETT.DAT
This is Dataplot data file BENNETT.DAT
Electron mobility for p-type AlGaAs
Herb Bennett
October 1998
Reference--Figure 3 of Journal of Applied Physics 80
page 3851, 1996
Response variable = electron mobility for p-type AlGaAs
Number of observations = 21
Number of variables per line image = 8
Order of variables on a line image--
1. Response variable 1 = mobility for AlAs mole fraction = .00
2. Response variable 2 = mobility for AlAs mole fraction = .05
3. Response variable 3 = mobility for AlAs mole fraction = .10
4. Response variable 4 = mobility for AlAs mole fraction = .15
5. Response variable 5 = mobility for AlAs mole fraction = .20
6. Response variable 6 = mobility for AlAs mole fraction = .25
7. Response variable 7 = mobility for AlAs mole fraction = .30
8. Factor 1 = coded acceptor density (in cm**-3)
(true density = coded density x 10**13)
To read this file into Dataplot--
SKIP 25
READ BENNETT.DAT Y1 Y2 Y3 Y4 Y5 Y6 Y7 X
Y1 Y2 Y3 Y4 Y5 Y6 Y7 X
------------------------------------------------------------------------
5.7420 5.1360 4.5520 4.0030 3.5000 3.0510 2.6580 1
5.1060 4.5840 4.0810 3.6060 3.1710 2.7790 2.4340 2
4.7180 4.2440 3.7880 3.3570 2.9610 2.6040 2.2880 3
4.2250 3.8100 3.4110 3.0340 2.6860 2.3710 2.0920 5
3.9030 3.5240 3.1610 2.8180 2.5010 2.2140 1.9590 7
3.5650 3.2240 2.8970 2.5880 2.3030 2.0450 1.8140 10
2.9260 2.6540 2.3940 2.1480 1.9210 1.7150 1.5300 20
2.5690 2.3340 2.1100 1.8990 1.7030 1.5250 1.3660 30
2.1450 1.9530 1.7710 1.5990 1.4410 1.2960 1.1650 50
1.8890 1.7210 1.5640 1.4160 1.2790 1.1540 1.0410 70
1.6430 1.4990 1.3640 1.2390 1.1230 1.0160 0.9198 100
1.2580 1.1500 1.0500 0.9575 0.8725 0.7948 0.7241 200
1.1010 1.0060 0.9191 0.8397 0.7670 0.7007 0.6403 300
0.9936 0.9046 0.8249 0.7530 0.6880 0.6292 0.5760 500
0.9901 0.8971 0.8152 0.7420 0.6764 0.6175 0.5647 700
1.0680 0.9601 0.8660 0.7832 0.7097 0.6449 0.5873 1000
1.5390 1.3720 1.2270 1.0980 0.9863 0.8881 0.8013 2000
1.9640 1.7460 1.5560 1.3900 1.2430 1.1150 1.0020 3000
2.4890 2.2120 1.9680 1.7530 1.5630 1.3970 1.2500 5000
2.6920 2.3920 2.1250 1.8920 1.6870 1.5060 1.3470 7000
2.7200 2.4180 2.1520 1.9180 1.7110 1.5280 1.3680 10000
BENNETT2.DAT
This is Dataplot data file BENNETT2.DAT
Electron mobility for p-type AlGaAs
Herb Bennett
October 1998
Reference--Figure 3 of Journal of Applied Physics 80
page 3851, 1996
Response variable = electron mobility for p-type AlGaAs
Number of observations = 154 (= 21 points/curve x 7 curves)
Number of variables per line image = 3
Order of variables on a line image--
1. Response variable = mobility (in xm**2/v*s)
2. Factor 1 = coded acceptor density (in cm**-3)
(true density = coded density x 10**13)
3. Factor 2 = mole fraction of AlAs (7 levels--0, .05, .10, ..., .30)
To read this file into Dataplot--
SKIP 25
READ BENNETT.DAT Y X TAG
Y X Tag
Mobility Density Mole Fraction
of AlAs
------------------------------------
5.742 1 .00
5.106 2 .00
4.718 3 .00
4.225 5 .00
3.903 7 .00
3.565 10 .00
2.926 20 .00
2.569 30 .00
2.145 50 .00
1.889 70 .00
1.643 100 .00
1.258 200 .00
1.101 300 .00
0.9936 500 .00
0.9901 700 .00
1.068 1000 .00
1.539 2000 .00
1.964 3000 .00
2.489 5000 .00
2.692 7000 .00
2.720 10000 .00
5.136 1 .05
4.584 2 .05
4.244 3 .05
3.810 5 .05
3.524 7 .05
3.224 10 .05
2.654 20 .05
2.334 30 .05
1.953 50 .05
1.721 70 .05
1.499 100 .05
1.150 200 .05
1.006 300 .05
0.9046 500 .05
0.8971 700 .05
0.960 1000 .05
1.372 2000 .05
1.746 3000 .05
2.212 5000 .05
2.392 7000 .05
2.418 10000 .05
4.552 1 .10
4.081 2 .10
3.788 3 .10
3.411 5 .10
3.161 7 .10
2.897 10 .10
2.394 20 .10
2.110 30 .10
1.771 50 .10
1.564 70 .10
1.364 100 .10
1.050 200 .10
0.9191 300 .10
0.8249 500 .10
0.8152 700 .10
0.8660 1000 .10
1.227 2000 .10
1.556 3000 .10
1.968 5000 .10
2.125 7000 .10
2.152 10000 .10
4.003 1 .15
3.606 2 .15
3.357 3 .15
3.034 5 .15
2.818 7 .15
2.588 10 .15
2.148 20 .15
1.899 30 .15
1.599 50 .15
1.416 70 .15
1.239 100 .15
0.9575 200 .15
0.8397 300 .15
0.7530 500 .15
0.7420 700 .15
0.7832 1000 .15
1.098 2000 .15
1.390 3000 .15
1.753 5000 .15
1.892 7000 .15
1.918 10000 .15
3.500 1 .20
3.171 2 .20
2.961 3 .20
2.686 5 .20
2.501 7 .20
2.303 10 .20
1.921 20 .20
1.703 30 .20
1.441 50 .20
1.279 70 .20
1.123 100 .20
0.8725 200 .20
0.7670 300 .20
0.6880 500 .20
0.6764 700 .20
0.7097 1000 .20
0.9863 2000 .20
1.243 3000 .20
1.563 5000 .20
1.687 7000 .20
1.711 10000 .20
3.051 1 .25
2.779 2 .25
2.604 3 .25
2.371 5 .25
2.214 7 .25
2.045 10 .25
1.715 20 .25
1.525 30 .25
1.296 50 .25
1.154 70 .25
1.016 100 .25
0.7948 200 .25
0.7007 300 .25
0.6292 500 .25
0.6175 700 .25
0.6449 1000 .25
0.8881 2000 .25
1.115 3000 .25
1.397 5000 .25
1.506 7000 .25
1.528 10000 .25
2.658 1 .30
2.434 2 .30
2.288 3 .30
2.092 5 .30
1.959 7 .30
1.814 10 .30
1.530 20 .30
1.366 30 .30
1.165 50 .30
1.041 70 .30
0.9198 100 .30
0.7241 200 .30
0.6403 300 .30
0.5760 500 .30
0.5647 700 .30
0.5873 1000 .30
0.8013 2000 .30
1.002 3000 .30
1.250 5000 .30
1.347 7000 .30
1.368 10000 .30
