Table Of ContentA111D3 60M2^1 £
400-89
NIST NIST SPECIAL PUBLICATION
PUBLICATIONS
U.S. DEPARTMENT OF COMMERCE/Technology Administration
National Institute of Standards and Technology
Semiconductor Measurement Technology:
TXYZ
Version 2.0 of the Thermal
Analysis Program: TXYZ20
400-89
1992
C.2
7
he National Institute of Standards and Technology was established in 1988 by Congress to "assist
industry in the development of technology . . . needed to improve product quality, to modernize
manufacturing processes, to ensure product reliability and to facilitate rapid commercialization ... of
. . .
products based on new scientific discoveries."
NIST, originally founded as the National Bureau of Standards in 1901, works to strengthen U.S.
industry's competitiveness; advance science and engineering; and improve public health, safety, and the
environment. One ofthe agency's basic functions is to develop, maintain, and retain custody ofthe national
standards of measurement, and provide the means and methods for comparing standards used in science,
engineering, manufacturing, commerce, industry, and education with the standards adopted or recognized
by the Federal Government.
As an agency of the U.S. Commerce Department's Technology Administration, NIST conducts basic
and applied research in the physical sciences and engineering and performs related services. The Institute
does generic and precompetitive work on new and advanced technologies. NIST's research facilities are
located at Gaithersburg, MD 20899, and at Boulder, CO 80303. Major technical operating units and their
principal activities are listed below. For more information contact the Public Inquiries Desk, 301-975-3058.
Technology Services Manufacturing Engineering Laboratory
• Manufacturing Technology Centers Program • Precision Engineering
• Standards Services • Automated Production Technology
• Technology Commercialization • Robot Systems
• Measurement Services • Factory Automation
• Technology Evaluation and Assessment • Fabrication Technology
• Information Services
Materials Science and Engineering
Electronics and Electrical Engineering Laboratory
Laboratory • Intelligent Processing of Materials
• Microelectronics • Ceramics
• Law Enforcement Standards • Materials Reliability1
• Electricity • Polymers
• Semiconductor Electronics • Metallurgy
• Electromagnetic Fields1 • Reactor Radiation
• Electromagnetic Technology1
Building and Fire Research Laboratory
Chemical Science and Technology • Structures
Laboratory • Building Materials
• Biotechnology • Building Environment
• Chemical Engineering1 • Fire Science and Engineering
• Chemical Kinetics and Thermodynamics • Fire Measurement and Research
• Inorganic Analytical Research
• Organic Analytical Research Computer Systems Laboratory
• Process Measurements • Information Systems Engineering
• Surface and Microanalysis Science • Systems and Software Technology
• Thermophysics2 • Computer Security
• Systems and Network Architecture
Physics Laboratory • Advanced Systems
• Electron and Optical Physics
• Atomic Physics Computing and Applied Mathematics
• Molecular Physics Laboratory
• Radiometric Physics • Applied and Computational Mathematics2
• Quantum Metrology • Statistical Engineering2
• Ionizing Radiation • Scientific Computing Environments2
• Time and Frequency1 • Computer Services2
• Quantum Physics1 • Computer Systems and Communications2
• Information Systems
'At Boulder, CO 80303.
2Some elements at Boulder, CO 80303.
Semiconductor Measurement Technology:
TXYZ
Version 2.0 of the Thermal
Analysis Program: TXYZ20
John Albers
Semiconductor Electronics Division
Electronics and Electrical Engineering Laboratory
National Institute of Standards and Technology
MD
Gaithersburg, 20899
June 1992
U.S. DEPARTMENT OF COMMERCE, Barbara Hackman Franklin, Secretary
TECHNOLOGY ADMINISTRATION, Robert M. White, Under Secretary for Technology
NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director
National Institute of Standards and Technology Special Publication 400-89
Natl. Inst. Stand. Technol. Spec. Publ. 400-89, 33 pages (June 1992)
CODEN: NSPUE2
U.S. GOVERNMENT PRINTING OFFICE
WASHINGTON: 1992
For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC 20402-9325
Semiconductor Measurement Technology:
Version 2.0 of the TXYZ Thermal Analysis Program: TXYZ20
Table of Contents
Page
Abstract 1
Introduction 2
Review of the Model, the TXYZ Code, and the TXYZ20 Code 2
Examples of Calculations with TXYZ20 6
Benchmark Example 7
Other Examples 10
Acknowledgments 18
References 18
TXYZ
Appendix. Version 2.0 19
List of Tables
Page
1. Annotated input data file (inputl.dat) for TXYZ20 code 8
2. Annotated output data file (outputl.dat) for TXYZ20 program 9
3. CPU times used in TXYZ20 calculation using inputl.dat data 11
4. Annotated input data file (inputla.dat) for TXYZ20 code 11
5. Annotated input data file (input2.dat) for TXYZ20 code 12
6. Annotated output data file (output2.dat) for TXYZ20 program 13
7. Annotated input data file (input3.dat) for TXYZ20 code 14
8. Annotated output data file (output3.dat) for TXYZ20 program 15
iii
Semiconductor Measurement Technology:
Version 2.0 of the TXYZ Thermal Analysis Program: TXYZ20
John Albers
Semiconductor Electronics Division
National Institute of Standards and Technology
Gaithersburg, Maryland 20899
Abstract
The TXYZ computer program has been used for a number ofyears for the thermal analysis
ofsemiconductor devices and packages. This programmakes use ofthe closed form, Fourier
series solution of the steady-state heat flow equation for the general case of a rectangular
three-layer structure with multiple heat sources on the top surface. TXYZ provides for the
calculation of the temperature at any set of points in this structure and has proven useful
for the determination of the steady-state temperature distribution of semiconductor chips
and packages. This report presents TXYZ Version 2.0 (TXYZ20) which is a revised and
updated version of the original TXYZ program. The TXYZ20 program incorporates more
flexible handling of input data, assignment of positive or negative noninteger weights to
the various heat sources or heat sinks, and improved evaluation of limiting forms in the
code.
The first part of this report consists of a discussion of the general elements in the TXYZ
code and the particular changes which have been made to it to obtain TXYZ20. The
second part of the report contains a discussion of several examples of the running of the
code. Several annotated input data files are presented and discussed to show both the
increased flexibility of the input data and the actual use of the updated code. Running the
TXYZ20 code for one of the input files provides a benchmark for several machines. The
user may wish to run this example for the purpose of comparing the CPU times involved.
The appendix contains an annotated, internally documented listing of the FORTRAN
source code for TXYZ20.
The FORTRAN source code (total of about 21 kbytes) and sample input and output data
files are available in ASCII format using a number of transfer vehicles. These include:
= = =
standard 8-track magnetic tape (ASCII, density 1600, record 80, block 1600), 5.25-
in. (360-kbyte and 1.2-Mbyte) DOS-formatted floppy disks, and electronic mail over the
Internet. The sample input and output data files are included so that the user can check
the program for proper operation as well as to become acquainted with the setup and use
of the code. Users of the TXYZ code will find the updated TXYZ20 code easy to use and
should benefit from the more flexible input and the more general treatment ofheat sources
and heat sinks.
Key words: FORTRAN; Fourier analysis; integrated circuit; semiconductor devices; semi-
conductor materials; steady-state heat flow; thermal analysis; thermal conductivity.
1
L L
Introduction
The operation of a semiconductor device relies on the passage of current through portions
of the structure. This is accompanied by power dissipation and heating which gives rise
to a temperature distribution. In time, thermal stresses may arise and lead to device
degradation and possible failure. In addition to the effects of thermal stresses on device
reliability, the modeling of the steady-state thermal response of a system is also useful
since the temperature distribution may have an effect upon the local mechanical, thermal,
electrical, or optical properties of the system. Consequently, an accurate physical model
of the temperature distribution under the power condition of actual operation is of great
importance. It is also possible to ascertain the relative effects of composition (thermal
conductivity) and geometry (layer thickness). By carefully optimizing composition and
geometry, it may be possible to minimize the thermal stress and hence ensure optimal
device lifetime.
The physical and mathematical model used here is taken from the work ofKokkas [1]. This
has been used previously to construct the TXYZ program. The Kokkas model and the
TXYZ code have been described in considerable detail in NBS Special Publication 400-76
[2]. This report is available and should be consulted for the details of the analysis which
are still current and valid.
Review of the Model, the TXYZ Code, and the TXYZ20 Code
For the purposes of providing a background for the TXYZ20 code, a brief outline of the
salient features of the earlier report on TXYZ is presented here. It is assumed that the
reader has a copy of reference [2]. The symbols used here are denned in this reference.
The outline presented here should provide the reader with an overview of the analysis and
a quick guide as to where to find the various portions of the calculation.
The starting point of the calculation is the steady-state heat flow equation for a single
rectangular layer. In Cartesian coordinates, this takes on the form
r Q2 Q2 Q2 ^|
This equation is solved in this coordinate system using Fourier analysis. The requirement
that there be no heat transport out of the lateral boundaries of the structure leaves only
"quantized" cosine functions in the Fourier representation of the temperature,
T{x,V,z) = V^^1 4r(n,m, z) cos(mvx/ x) cos(rmry/ y)
n=0 m2=-,0 (v*n0 + i)'(x*m0 + 1')LxL9 '
where 8nn> is the Kronecker delta and is equal to unity if n = n' and zero otherwise.
The Fourier coefficients, r(n,m,z), are given by
r(n,m, z) = a cosh(72) + sinh(7-z),
j3
2
L
'
where the coefficients a and /?, which may be functions of 7, are determined from the two
z-dependent boundary conditions.
The above analysis provides the basis for the general multiple-layer problem in terms of
the Fourier coefficients. For this case, it is assumed that the temperature satisfies the
steady-state equation in each of the layers. Consequently, in each of the layers, the Fourier
expansion of the temperature takes the form
T{(n,m, z) = oti cosh(7z) + sinh(7z), (i = 1, ...,n).
For each layer, there are two coefficients which arise from the fact that the basic tem-
perature equation is second order. These are determined from the z-dependent boundary
conditions. The general set of z-dependent boundary conditions for the multiple layer
problem is provided by the conditions that: heat enters (or leaves) the structure through
the portions of the top layer where power is applied, the temperature and the heat flow
are continuous across the internal interfaces, and the temperature is continuous across the
interface between the bottom layer and the heat sink. The above equations and boundary
conditions provide the mathematical framework in which the multiple layer problem may
be solved. For the purposes of numerical modeling, the multiple layer problem is special-
ized to one with three layers. In this case, the Kokkas analysis leads to the solution in
[1]
the form
lt{,x,y,z)\ _ fp0 2. 4U(n,Tn)Tj(n,m+, z) cos(nirx/Lx) cos(mny/Ly)
n=0 m=0 (*„o l)(*mo
The symbols, analysis and particular forms of Ti(n,m, z) are discussed in some detail
in reference [2]. There are several parts of the above equation which can be identified.
First, there is the z-dependent part of the solution in each layer, Ti(n,m, z). The two
cosine functions, cos(mvx/Lx) cos(rmvy/ y), arise from the x- and y-dependent parts of
the steady-state equation. Their specific form arises from the requirement of the lateral
boundary conditions. Finally, the requirement that heat enters or leaves the top surface
only through the heating elements introduces the function PoU(n,m). It should be noted
that Pq appears only in front of the sum. It is a scale or multiplicative factor which for
convenience may be set equal to unity. Its effect is to scale the temperature in a direct
linear fashion. Performing the calculation for PQ = 1 for a given structure then provides
the temperature distribution for arbitrary P which can then be obtained by multiplying
0
by the Po which is used.
In the above, the function, ?7(n,m), is related to the heat sources (sinks) on the top surface
of the structure. This function is defined as the double cosine transform of the geometric
part of the surface power function, U(x,y),
U(n,m) = U(x,y)cos(mvx/L )cos(mny/L )dxdy.
/ / x y
Jo Jo
The detailed form of this function is considered in reference [2]. There, the relation of
this function to the location and size of the heat sources was presented. In addition, the
m
limiting form ofthe function as either n or approach zero was considered, and the correct
limit was established for the purposes of numerical analysis. The reader is referred to that
report for details.
In the original version ofTXYZ, all heat sources had equal unit weights. Ifthe user wanted
to construct a more general heat source, this could only be done in unit increments by
superimposing sources. Hence, to perform the calculation for a heating element which had
five times the effect as another, the element would have to constructed five times. Only
integer increments were possible. The first change made in TXYZ is the generalization
of the heating elements. It should be noted that this does change the basic model. The
updated TXYZ20 program allows for noninteger weights to be assigned to each heating
element. This weight may be positive or negative and hence represent the effects of either
a heat source or a heat sink. The weight is entered through the input data file and appears
with the location and size of the element. The reader should refer to the last lines in
the input data files presented in tables 1, 4, 5, and 7 for the use of weighting of the heat
elements. The more general information is then passed from the main program to the
UZERO function through a COMMON statement. The weighting of the heating element
was used originally in a calculation of the thermal interactions between electromigration
test structures [3]. The careful and consistent evaluation ofthe temperature is important in
this case as the test structures are used in accelerated testing and the temperature profile
generated by one of the elements may increase the temperature at adjoining elements
which may accelerate the failure rates. As electromigration is temperature dependent, it
is of central importance to evaluate the temperature. This is especially the case where the
thermal interaction of neighboring heating elements comes into play.
The weight is entered on the same line of the input data file where the position and lengths
along x and y are entered. Instead of having to enter the same element five times, a weight
of 5 may simply be used. In addition, noninteger weights which are positive or negative
may be entered to represent sources or sinks. This feature should increase the versatility
of the code to a more general class of problems.
Returning to the overview of the original TXYZ code and the basis for the evolution of
TXYZ20, the behavior of the Fourier coefficients for both small and large values of the
arguments is next on the agenda. The importance of this aspect of the analysis and
the resulting numerical code is found in the fact that the equations involved contain the
possibility of numerical infinities or overflow situations. These occur for n = 0 and m = 0
as well as for large values of these summation variables. The reason for the relatively
compact nature of the original TXYZ code and its numerical efficiency is that great care
was exercised in the investigation ofthe functions for the small and large argument regime.
For small values ofthe argument, the evaluation oflimiting forms led to expressions which
were correct and free of artificial numerical singularities. For large values of the argument,
special care was required. Many ofthe functions involved contained the hyperbolic sinh and
cosh functions. These approach the exponential function for large values of the argument.
These provide the possibility for numerical infinities or overflows. Careful investigation
of the forms of the functions showed that these numerical infinities were removable by
4
