Download MSE 310-ECE 340 Micron School of Materials Science and

MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
Problem Set 11 Solutions
ABET Question: j. A knowledge of contemporary issues.
Note: Reading the lab handout and guidelines, paying particular attention to the figures, and the
Ch. 6 Lecture Notes on Semiconductor Heterostructures as well as the video notes on
Semiconductor Heterostructures will considerably help you with this assignment. Your figures
should be fully labeled with figure captions with legends or data labeled.
1. The data provided with this problem set was obtained by analyzing data obtained from a
contemporary optoelectronic device, a light emitting diode (LED), that is made from a
contemporary semiconductor materials system: the III-V semiconductor nitride materials
system. By analyzing the data, your job is to determine certain device parameters of the LED,
propose a device configuration and draw a schematic of it, and relate your knowledge of SP3
and band gap engineering to aspects that are important to device performance and design. You
will do so by answering the questions below. The data are in .csv files. Use a mathematical
program (not Excel) to analyze and plot the data.
a. Using a mathematical program, plot the I-V data and fit the data to extract a turn-on voltage.
Plot should include the data, fit, and turn-on voltage with text denoting the latter and a
legend denoting the data and the fit. Convert the turn-on voltage to the turn-on energy in
eV. Your fit should include goodness of fit parameters including R2, adjusted R2 and
residuals with comments on the goodness of fit provided. Feel free to use any online
reference material to describe the goodness of fit parameters you use. For example, the
online blog called The Minitab Blog (http://blog.minitab.com/) provides solid descriptions
of goodness of fit parameters. NOTE: to extract the Turn-on Voltage, one needs only to fit
part of the data set, NOT the entire data set.
Figure 1: Current-Voltage data from a Blue LED showing a turn-on voltage of
about 2.56 V. The R2 of the fit is 0.9904.
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
Both R2 and the adjusted R2 are better than 0.99 and thus show that the
goodness of fit is very good. The standard error for both fit parameters is
better than 10% so also quite good.
Goodness of Fit - Definitions:
A data set can be described as observed behavior of a dependent variable that
is affected by an independent variable. When performing a regression fit on the
data, the resulting fit function is an unobservable function (i.e., one just observes
the data, not the function that describes the data) that relates the dependent
variable to the independent variable. Deviations of the observable dependent
variable from the fitted function dependent variable are known as the residuals.
Residuals should show a randomness in the differences. Nonrandom, or
deterministic, behavior suggests that the model does not completely describe
the data (e.g., another fit variable may be required). A residual is typically given
by:
residual= y − y
i
i
where
y mx + b
=
i
i
where yi is the y-value in a particular data point, (xi,yi), from the data set and b
and m are the y-intercept and slope, respectively, are fit parameters, and
obtained from the linear fit.
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
Figure Caption: The residuals of the fit function show some nonrandom pattern
which may be due to some nonlinearity in the data that were fit. This may be due
to the low number of data points that were fitted.
The definition of R2 is fairly straight-forward; it is the fractional percentage of
the dependent variable variation that is explained by a linear model. Or
mathematically:
2
n
( residuals )
1−
R2 =
Total Sum of Square
2
 yi − y i 
∑


1 − i =1
=
2
∑( y − y)
n
i =1
i
where n is the number of data points and:
1 n
y = ∑ yi and again y i= b + mxi and.
n i =1
where m is the slope of the fit and b is the y-intercept. If yi –(b+mxi) is small, R2
approaches 1 which indicates a good fit. Here, we are assuming that the variance
for each data point is, on average, equal. The adjusted R2, considers degrees of
freedom (df) in the fit and is given by:
n
( residuals ) / df error
=
1−
2
R
2
Total Sum of Square/dftotal
 yi − y i 
∑


i =1
2
n* − 1
=
1−
2
∑( y − y)
n
i =1
i
n*
If intercept is included in the linear model, n*=n-1; otherwise, n*=n and the total
sum of squares is uncorrected.
The standard errors for the slope (m) and the y-intercept (b) are calculated
somewhat differently.
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
n
ε b = sε
n
∑ xi2
i =1
n
(
n∑ xi − x
i =1
)
2
& ε m = sε
1
∑ ( x − x)
n
i =1
(
)
n
(
∑ yi − y
∑ yi − y
=i 1 =i 1
where sε
=
=
2
df error
n* − 1
2
)
2
.
i
References:
1. Wiki - Errors and residuals: https://en.wikipedia.org/wiki/Errors_and_residuals
2.
Origin
Labs:
http://www.originlab.com/doc/Origin-Help/LRAlgorithm#The_Linear_Regression_Model
3.
http://blog.minitab.com/blog/adventures-in-statistics/regression-analysishow-do-i-interpret-r-squared-and-assess-the-goodness-of-fit
4.
http://blog.minitab.com/blog/adventures-in-statistics/why-you-need-tocheck-your-residual-plots-for-regression-analysis
b. Plot the photoluminescence data (i.e., Optical Power versus wavelength spectrum) and then
extract the peak wavelength using a mathematical program for both plotting and extracting.
The plot should include the data and, using text and arrows or a single data point, the peak
wavelength and its location in the data.
Figure 2: The optical power versus wavelength of a blue LED showing the peak
wavelength and peak energy.
From the data in figure 2, the maximum peak wavelength was extracted from the
data and determined to be 463.1 nm. Using the relationship between energy and
wavelength, λ , which is given by:
hc
E=
λ
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
where h is Planck’s constant and c is the speed of light, the peak energy was
calculated to be 2.677 eV. The peak energy suggests an energy band gap of
between 2.6 to 2.7 eV.
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
c. Given the extracted turn-on voltage (i.e., turn-on energy) and peak wavelength (convert
both to eV), approximate the energy band gap of the material. Using this data, determine
the composition and lattice constant of the active region and of the cladding regions
assuming both are comprised of the III-V nitride system. Support your choice of
compositions. Additionally, suggest a substrate on which to grow the LED. Support your
choice of the substrate material. Provide references.
Solution:
We will use the following plot to aid in determining the composition:
Figure 3a: energy versus lattice constant of the In-Ga-N III-V semiconductor
materials system giving the composition percentages for Ga1-xInxN and AlxIn1-xN.
(Reference: E.F. Schubert, Physical Foundations of Solid State Devices (2009))
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
Figure 3b: Similar to figure 3a but shows SiC polymorphs, some of which may be
used as substrates.
Table 1: Range of wavelengths and range of energies for the visible colors.
Using figure 3a and b and table 1, and the peak energy from the LED indicating
an energy band gap of between 2.6-2.7 eV, the band gap energies are in the range
of blue light emission. Given the energy band gap range, the composition of the
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
active region for the Ga1-xInxN would be an In composition of between 14%-19%
or X ranging from 0.14-0.19 with a lattice constant of about 3.25 A. In the figure
below, I use: Ga0.84In0.16N.
The cladding layers should have larger energy band gaps, perhaps about 0.4 – 0.5
eV greater, so a band gap energy of around 3.1 eV. This band gap energy equates
to an In composition of about 5% giving a nitride system Ga0.95In0.05N with a
lattice constant of about 3.22 A. Suitable substrates that are both transparent
to visible light, particularly of this LED (Eg ~2.67eV), and are fairly well lattice
matched are: (1) sapphire with a lattice constant of 4.77 A and an energy band
gap of 9eV or (2) single crystal ZnO with a lattice constant of 3.33A and an
energy band gap of 3.2eV, (2) single crystal SiC (4H and 6H) with a lattice
constant of 3.18A with an energy band gap of between 2.9-3.2eV. All of these
substrates are available in from suppliers such as Semi Wafer
(www.semiwafer.com).
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
d. Consider that the materials system are the nitrides. Given the attached paper, create:
i. A schematic of the cross-section of the device including the substrate, cladding
regions, active region and contacts.
ohmic contact
Cladding layer
hν
Cladding layer
n-Ga0.95In0.05N
i-Ga0.84In0.16N Active region
p-Ga0.95In0.05N
ohmic contact
Sapphire Substrate
Figure 4: Cross section schematic of the blue LED showing composition.
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
ii. A fully labeled energy band diagram of the cladding regions, active region and
contacts for the conditions below. Fully label your band diagrams and include either
the flatband voltage or the applied voltage depending on the condition.
1. Flatband Condition
2. Equilibrium Condition
3. Forward Bias Condition (schematically show electron and hole injection with
recombination and emission)
Applied bias conditions.
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
e. The following question concerns SP3 relative to band gap engineering and your knowledge
of the fundamentals of quantum mechanics, energy band structure, statistical
thermodynamics and classical thermodynamics. Create a bulleted list of the following
categories (i.e., as titles) for each list: Properties, Processing, Structure, Performance. In
the list, provide aspects of importance to achieve specific device performance parameters
including brightness, color, and reliability. Include a one to two sentence explanation of
the aspect of importance and its connection to the category and to device performance. If
there is overlap between an aspect of importance to the categories, then list it under multiple
categories and provide supporting arguments in doing so.
Properties
Structure
• Band gap of active region to tune the output color
• Composition of the active region to tune the energy
band gap of the active region
• Composition of the cladding layers to tune the
energy band gap of the cladding layers
• Band gap of the cladding region
• Band offset between active region and cladding
layer for efficient recombination between e-’s & h+’s
but not too great that a greater turn-on voltage
would be required because of the resulting larger
band gap of the cladding layer
• Active region needs to be a direct band gap
semiconductor to improve recombination efficiency
and limit nonradiative recombination
• Ohmic contacts so current injection into the LED is
not impeded.
• Substrate transparent to emitted light to improve
LED brightness.
• Lattice matching between substrate, cladding layer
and active region to reduce misfit and threading
dislocations
Processing
Performance
• Growth of cladding layer and active region using the
most appropriate (most defect free at a reasonable
cost) growth technique which is chemical vapor
deposition.
• Proper doping of cladding layers to ensure a p-i-n
diode structure.
• Temperature low enough not to cause significant
dopant diffusion
• Composition of the active region to tune the energy
band gap of the active region for the correct color
emission.
• Composition of the cladding layer to tune the
energy band gap of the cladding layer and the band
offsets relative to the active region for efficient
electron-hole recombination while achieving a turnon voltage that is not excessive to reduce power
consumption.
•
•
•
•
LED must be Bright
Tuned to the correct color
Reliable thus limit defects
Relatively low power to keep battery recharge low.
MSE 310-ECE 340
Electrical Properties of Materials
Micron School of Materials Science and Engineering
Fall 2016/Bill Knowlton
Non-ABET Question:
2. Draw, and fully label, the energy band diagram of an pMOS capacitor for the following
conditions:
a. Flatband Condition
Equilibrium Condition