Vol. 36, No. 8
Journal of Semiconductors
August 2015
Thermal time-constant spectrum extraction method in AlGaN/GaN HEMTs
Yang Junwei(杨军伟), Feng Shiwei(冯士维)Ž , Shi Dong(史冬), and Yang Chunhui(阳春辉)
College of Electronic Information & Control Engineering, Beijing University of Technology, Beijing 100124, China
Abstract: The transient temperature rise in the active region in AlGaN/GaN high electron mobility transistors
(HEMTs) is measured using an electrical method. The original data are smoothed and denoised by a nonparametric
fitting algorithm, called locally weighted scatterplot smoothing (LOWESS). The thermal time-constant spectrum is
extracted to analyze the physical structure of the heat-conduction path in AlGaN/GaN HEMTs. The thermal timeconstant spectra extracted using the LOWESS algorithm are richer and the RC network obtained is greater compared
with those with the traditional denoising method (multi-exponential fitting). Thus, the analysis of the heat-flow
path is more precise. The results show that the LOWESS nonparametric fitting algorithm can remove noise from
measured data better than other methods and can retain the subtle variation tendency of the original discrete data.
The thermal time-constant spectra extracted using this method can describe the subtle temperature variations in
the AlGaN/GaN HEMT active region. This will help researchers to precisely analyze the layer composition of the
heat-flow path.
Key words: AlGaN/GaN HEMT; LOWESS algorithm; exponential fitting; time-constant spectrum; heatconduction path
DOI: 10.1088/1674-4926/36/8/084003
PACC: 4410; 7330
1. Introduction
GaN-based high electron mobility transistors (HEMTs)
have broad application prospects in the high-frequency and
high-power fields because of their excellent material propertiesŒ1 . The rise in temperature caused by high-power density
working conditions becomes an important factor that impacts
their reliability. Therefore, it is important to develop a nondestructive method to obtain the temperature as well as a thermal resistance structure for AlGaN/GaN HEMTs. Analyzing
the distributed RC networks using a continuous time-constant
spectrum is an effective methodŒ2 . In this work, we measure
the transient temperature response of AlGaN/GaN HEMTs in
the active region, utilizing the linear relationship between the
forward voltages of the Schottky junction with temperature under a step-pulse power,
a.t / D
X
Ri Œ1
exp. t=i /;
(1)
i
where a.t / is the heating response curve, t is the step-pulse
power applying time, i are time-constants, and Ri are the
corresponding resistance magnitudesŒ2 . Owing to the interferences of the measuring instruments, the devices themselves,
natural or man-made factors and restrictions in the measurement accuracy, a series of measured discrete data that fluctuate around the true value cause the transient response curve to
be rough, impacting the accuracy of the thermal time-constant
spectra. Therefore, it is necessary to smooth and denoise the
original measured data. Because the transient temperature response curve is the sum of many individual exponential terms,
the traditional denoising method uses multi-exponentials to fit
the measured dataŒ3 .
This paper introduces a nonparametric fitting algorithm
called locally weighted scatterplot smoothing (LOWESS),
which is a type of robust nonparametric regression method that
uses weighted least square fitting locally. At present, this algorithm is widely used in many fields, such as in the research
and calculations for the global standard curve of strontium isotopesŒ4 and in the standardization of gene chipsŒ5 .
The transient measured data for the AlGaN/GaN HEMTs
were smoothed and denoised using the two methods outlined
above, respectively. Then, two different sets of thermal timeconstant spectra were obtained using the Bayes iteration deconvolutionŒ6 , which were compared and analyzed for the AlGaN/GaN HEMT heat-flow path structure for each layer. The
results show that the LOWESS algorithm is superior at extracting the thermal time-constant of the structure.
2. Sample structure and experimental methods
2.1. Sample structure and experimental methods for the
transient temperature measurements
A cross-sectional schematic of an AlGaN/GaN HEMT is
shown in Figure 1(a). The device consists of a 25 nm unintentionally doped AlGaN layer, a 1.5 m GaN buffer layer,
a 400 m SiC substrate layer and a 100 nm SiN passivation
layer. Figure 1(b) shows a schematic of a packaged sample and
the testing environment. The heat-flow path is composed of a
series of different materials and geometries, such as the chip,
the solder and the package body. To ensure more accurate results, the packaged AlGaN/GaN HEMT was tested on a thermostat platform.
The schematic of the testing system is as shown in Figure 2(a). The forward voltage of the gate–source Schottky junc-
† Corresponding author. Email: shwfeng@bjut.edu.cn
Received 28 January 2015, revised manuscript received 16 March 2015
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© 2015 Chinese Institute of Electronics
J. Semicond. 2015, 36(8)
Yang Junwei et al.
Figure 1. (a) Cross-sectional schematic of an AlGaN/GaN HEMT. (b) Schematic of a packaged sample and the testing environment.
Figure 2. (a) The schematic of the testing system. (b) Calibration curve of the forward VGS for an AlGaN/GaN HEMT with respect to temperature
at a current of 1.0 mA. (c) The sequence diagram of measuring the temperature coefficient. (d) The sequence diagram of the testing system.
tion was selected as an electrical temperature sensitive parameter (TSP) in this workŒ7 . The drain electrode in the device
under test (DUT) was open and the forward testing current was
chosen to be 1.0 mA to avoid self-heating effects. The gate–
source forward voltage drop was recorded at different platform
temperatures, varying from 30 to 90 ıC. The temperature calibration curve obtained is as shown in Figure 2(b), which is
linear, and the sequence diagram of measuring the temperature
coefficient is as shown in Figure 2(c). The temperature coeffi-
cient (K) was calculated to be 4.42 mV/K according to the linear fit of the slope. After measuring the temperature coefficient
K of DUT, the channel temperature response can be obtained
by pulsed switching technique and by utilizing the relation,
VF .t / D K T .t /;
(2)
where VF .t / is the forward voltage of the TSP. A test current of
1.0 mA was used when measuring the temperature. The Schottky gate–source voltage (VGS / was chosen to 2 V and the op-
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Yang Junwei et al.
erating voltages were set on 5, 6, 7 and 8 V respectively in the
operation state. A Schottky junction must be forward biased
when the temperature is being measured. Thus, it was necessary to turn off the drain–source voltage (VDS / firstly to prevent
burning the devices, then to forward bias VGS and to sample the
variation of TSPŒ8 . Figure 2(d) shows the sequence diagram of
the testing system. The DUT was set to a constant temperature
(30 ıC). The channel temperature of the DUT reached a steady
state after heating for 30 s. Then VDS was turned off to collect the variations in VGS during the 60 s cooling process, so a
transient temperature response curve could be obtained.
2.2. Smoothing method for the sampling data using the
LOWESS algorithm
With the LOWESS method, the value of each point is a
weighted regression through a nearby data point within a specified span, which is similar to the moving average technique.
The steps of the robust estimation are as follows:
(1) An estimated point (x, y/ is centered to establish the
span width, which depends on
r D f n;
(3)
where r is the number of points that participate in the local regression and n is the total number of smoothing data points.
First, it is necessary to select an appropriate value for f according to the degree that the data fluctuates before the thermal transient response curves are smoothed. Then the value of
r is determined using Equation (2). If the value of f is large,
the fitting curve is smooth. However, potential subtle variations in the data may be masked. Therefore, a proper f value
that matches the experimental data should be determined with
a large number of experiments.
(2) The initial weight of any point (xi , yi / is defined within
the span width, namely the height of the weight function curve
in xi . A cubic weighting function was selected in this work:
!i D
8
ˆ
ˆ
<
1
ˆ
:̂
0;
ˇ
ˇ xi
ˇ
ˇx
r
ˇ !3
x ˇˇ3
;
xˇ
Figure 3. Channel temperature cooling response curve and partial
magnified images at VDS D 5, 6, 7 and 8 V.
ˇ
ˇ xi
ˇ
ˇx
r
ˇ
x ˇˇ
< 1;
xˇ
(4)
otherwise:
(3) The weighted linear least square fit and a first-order
polynomial with the formula y D a C bx are used to estimate
the value of .x; y/
O at x.
(4) The robust weight function is defined to calculate the
new weight using the residual of the estimated formula. The
robust weighting is:
8
2
ˆ
< 1 jei j2 ; ei 6 1;
ıi D
(5)
:̂0; otherwise;
i
and ri D yi yOi is the error
where ei D 6Median.jr1rj;jr
2 j; ;jrn j/
of each y value.
(5) Steps (3) and (4) are repeated with the new weighting
such that the smoothed value of any point .xi ; yi / could be obtained after several cycles. The theoretical cycles were repeated
three timesŒ9 .
Figure 4. Comparison of the original data, exponential fitting data and
LOWESS fitting data at VDS D 7 V.
3. Results and discussion
The channel temperature cooling response curve of a DUT
and partial magnified images are shown in Figure 3. In the
working state, VGS was held constant at 2 V, while the VDS
values were 5, 6, 7 and 8 V, with corresponding operating currents of 303.46, 356.14, 418.56 and 467.34 mA respectively.
The sampling frequency of the data was 1 s in this experiment. From Figure 3, the sampling delay was 4 s and the trend
of the sampled data completely followed the actual physical
significanceŒ10 . However, the data fluctuated around the true
value in a small range. Therefore, it was necessary to smooth
and denoise the discrete sampled data.
f D 0.07 was selected according to Equation (3), and VDS
D 7 V in the experiments. The difference between the fourthorder exponential fitting and the LOWESS nonparametric fitting algorithms is shown in Figure 4. This shows that the overall trends for the two fits using two fitting methods are consistent with the original data and meet the strict unchanging requirement, which has a realistic physical meaning. However,
Figure 4 shows that the exponential fitting curve (line 2) is
slightly offset from the actual data (line 1). This is because
of the strict exponential form of Equation (1), which ignores
the subtle changes in the original data. On the contrary, the
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Yang Junwei et al.
Figure 5. Comparison between two thermal time-constant spectra after exponential fitting and LOWESS fitting. (a) VDS D 5 V. (b) VDS D 6 V.
(c) VDS D 7 V. (d) VDS D 8 V.
LOWESS smoothing curve (line 3) is closer to the actual data.
This is because there is no fixed function limiting the trend
of the data, which describes the cooling process of the channel
temperature of a DUT more accurately along the entire chip and
package body and to the thermostatic platform, and reduces the
error between the fitted and original data. Thus, the LOWESS
nonparametric fitting algorithm is more accurate than the traditional exponential fitting method when the thermal transient
responses of AlGaN/GaN HEMTs are analyzed.
The characteristics of the electrical method caused the
heating state and measurement state to separate, making it impossible to acquire the total heating process of a DUT at one
time. Therefore, the heating response curve can be obtained
by acquiring the cooling response curve, whose characteristics
and Equation (1) for the heating response curve of the heatflow path have a complementary relationship. This curve is
made up of the thermal resistances and thermal capacitances
of various components. The process is similar to that for a
multi-stage series RC circuit, whose step response shows the
characteristics of multiple discrete time constants, determined
by the thermal resistance and thermal capacitance valuesŒ2 .
Therefore, the physical structure of the heat removing path in
AlGaN/GaN HEMTs can be analyzed by extracting the thermal time-constant spectra. As shown in Figure 5, the timeconstant spectra were extracted using the transient temperature
response curves, which were denoised by exponentially fitting
and LOWESS fitting, respectively. The VDS values were 5, 6,
7 and 8 V.
Figures 5(a) and 5(b) show that the thermal time-constant
spectra extracted by exponential fitting (line 1) only have two
discrete time constants at 60 s and 2.4 105 s for VDS D 5 V
and 6 V. This shows that the transient response curve after exponential fitting only reflects the two-layer physical structure
of the heat conduction path. However, there were four discrete
time constants in the time-constant spectrum (line 2), which
were extracted using the LOWESS algorithm. The first and
second time constants for line 2 were on both sides of the first
time constant of line 1, but their magnitude was reduced by
about half. This was mainly because the solder layer was too
thin and its temperature variation was relatively small, flowing through the solder layer when heat was conducted from the
channel to the temperature platform. As a result, the temperature variations in the channel-to-chip and chip-to-solder layers
were combined after exponential fitting, resulting in one time
constant and an increase in the magnitude. On the contrary, the
LOWESS algorithm can capture the subtle variations in the solder layer temperature, allowing the thermal time constant of the
solder layer to be extracted.
The thermal time-constant spectra with VDS D 7 and 8 V
are shown in Figures 5(c) and 5(d) respectively. In both figures,
the number of peaks in the thermal time-constant spectra extracted through exponential fitting (line1) became three, which
were at 23, 222 and 4 105 s. Among them, the first two
peaks appear to overlap with that of the LOWESS smoothed
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Yang Junwei et al.
Figure 6. Series network model of the Foster thermal resistance and thermal capacitance.
Figure 7. (a) Transient response. (b) Time-constant spectrum.
curve (line 2). This is because the channel temperature rise was
caused by the increased VDS , which led to temperature variations flowing through the solder layer, becoming larger in the
heat conduction process. However, the third time-constant and
its magnitude of line 1 are both larger than these of line 2. The
reason is similar to the first time-constant of line 1 in Figures
5(a) and 5(b), namely, the temperature variations in the solderto-package body and packaging body-to-temperature platform
were combined after exponential fitting. In this experiment, the
exponential fitting reflects the temperature variations between
the chip and solder layer, but other phenomena may exist, as
shown in Figures 5(a) and 5(b) in other experiments.
Figure 5 shows that four thermal time-constant spectra extracted using the LOWESS algorithm all have four peaks and
the fourth time constant is 2.3 106 s. Hereby, the heat conduction path under test consists of four main parts, the channelto-chip, chip-to-solder, solder-to-package body, and packaging body-to-temperature platform, which correspond to the selected known model in Figure 1. The LOWESS algorithm successfully smoothed the transient temperature response curve,
regardless of whether the drain–source voltage was large or
small. The accurate thermal time-constant spectra were then
extracted to analyze the reasonable physical structure of the
heat conduction path.
To verify whether the time-constant spectra can describe
the heat conduction path structure correctly, a lumped circuit
was measured, which is an artificial four-order Foster series
network model based on known thermal resistance and thermal capacitance valuesŒ10 . The four-order distributed RC oneports is equal to four layers structure of DUT, and resistance
and capacitance values of each stage are assumed according to
the equation i D Ri Ci . In order to correspond to the DUT,
i are the four discrete time constants of line 2 shown in Figure 5(c). The proportion of the four assumed Ri is the same
as that of the four discrete time constants magnitude of line 2
shown in Figure 5(c), which is 0.24 : 0.145 : 1.2 : 0.17. Thus,
the capacitance value can be calculated. The model is shown
in Figure 6. The transient temperature response curve measured by simulation software and the extracted thermal timeconstant spectrum are shown in Figure 7. Figure 7(a) simulates the temperature variation of the RC network after step
pulse. Figure 7(b) shows that there were four discrete time constants extracted by the model, which correspond to Figure 5.
Therefore, it is reliable to analyze the physical structure of the
heat conduction path of AlGaN/GaN HEMTs using the thermal
time-constant spectra.
4. Conclusions
The channel temperature cooling process in AlGaN/GaN
HEMTs was collected using a cooling response curve measurement method. The discrete data collected were smoothed
and denoised utilizing the LOWESS nonparametric fitting algorithm. Compared with the traditional multi-exponential fitting method, the LOWESS algorithm is not subject to the limitations of fixed expressions. It is highly robust and can describe the collected temperature variations in detail. According to the complementary symmetry relationships between the
heating response curves and the cooling response curve, the
channel temperature heating response curve of an AlGaN/GaN
HEMT could be deduced, which were smoothed by exponential fitting and with the LOWESS algorithm. Then the thermal time-constant spectra were extracted using a Bayesian
deconvolution. The results show that the number of discrete
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Yang Junwei et al.
time constants was greater for the LOWESS nonparametric
fitting algorithm than for exponential fitting, and the thermal
time constants well describe the physical structure constitution of the heat conduction path in actual AlGaN/GaN HEMTs.
From the thermal time-constant spectra, it can be seen that
the heat conduction path in AlGaN/GaN HEMTs contain four
parts, which are the channel-to-chip, chip-to-solder, solder-topackage body, and the package body-to-constant temperature
platform.
[5]
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[7]
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