Abstract:
Dynamic Digital Modulation obtains thermal image data on active semiconductor devices with sufficient sensitivity to be used in situ with packaged devices. These techniques can be applied to dynamic failures, but can also produce quantitative data of actual power dissipation as the device is placed into different operational modes. The thermal image results can be analyzed to assist in thermal management and assessing reliability and failure analysis issues in semiconductor devices.

Description:
BACKGROUND 
       [0001]    Thermal imaging is a traditional technique for thermal management, failure analysis and reliability studies of semiconductor devices (see for example, G. C. Albright, J. A. Stump, J. D. McDonald, H. Kaplan, “True” Temperature Measurements on Microscopic Semiconductor Targets”, SPIE Conference on Thermosense ( SPIE  Vol. 3700) 1999). However, limitations occur in thermal sensitivity, especially in the face of complex backgrounds and the need to exercise complex circuit structures in order to stimulate the desired site within a device. 
         [0002]    Enhanced signal-acquisition techniques, such as binary sampling or quadrature sampling (e.g., Lock-in Thermography S. Kiefer, et al., “Infrared Microthermography for Integrated Circuit Fault Location; Sensitivity and Limitations”, Proceedings of the 28th International Symposium for Testing and Failure Analysis (ISTFA) 2002) that rely on modulating a power supply have extended basic hotspot detection for failure analysis to extremes of sensitivity and show some ability to determine defective depth in simple structures (C. Schmidt, F. Altmann, “Non-Destructive Defect Depth Determination at Fully Packaged and Stacked Die Devices using Lock-in Thermography”, 17th IEEE International Symposium on the Physical and Failure Analysis of Integrated Circuits (IPFA), 2010). 
         [0003]    Binary signal enhancement techniques have been commercially available for many years. Simple binary pulse modulation consists of a single image sample being obtained at each power-on and power-off state, using a synchronously pulse-modulated power supply that is connected to the semiconductor device under test (DUT). This pair of images is digitally subtracted to produce a differenced image in which the common background is removed and only the thermal difference between the on and off power states remains. Averaging multiple pairs of signal samples over tens of minutes allows detection of shorts that are dissipating only a few microwatts. 
         [0004]    Recent efforts have focused on quadrature sampling in which two images are taken during the device&#39;s power-on state and two during the power-off state. The four images are then combined to produce in-phase and out-of-phase images. An inverse tangent of their ratio produces an additional phase angle image. 
         [0005]    These pulse sampling thermography (PST) techniques, described above, have been used for a variety of failure localization applications on static failures. However, there are many dynamic thermal issues in either failure analysis or reliability that require the DUT to be fully on and then placed into a specific state in order for the issue to be become manifest. Desirable thermal measurements are not currently realized. 
       SUMMARY 
       [0006]    This disclosure provides new techniques of dynamic digital modulation for obtaining thermal image data on active semiconductor devices with sufficient sensitivity to be used in situ with packaged devices. These techniques can be applied to dynamic failures but can also produce quantitative data of actual power dissipation as the device is placed into different operational modes. The thermal image results can be analyzed to assist in thermal management and assessing reliability and failure analysis issues in semiconductor devices. 
         [0007]    An exemplary system includes a device under test (DUT), a thermal camera that produces a plurality of thermal images of the DUT, a computer device and a control unit. The DUT is placed in a first operational state and the processing device acquires at least one first thermal image of the DUT from the thermal camera, based on a first signal. Then, the DUT is placed in a second operational state and the processing device acquires at least one second thermal image of the DUT from the thermal camera based on a second signal. The processing device generates at least one output image, based on a difference between the at least one first thermal image and the at least one second thermal image and the output image is presented by the output device. 
         [0008]    In one embodiment, the output image includes a thermal difference image, a thermal time-constant map or a power dissipation map. 
         [0009]    In another embodiment, the first operational state or second operational state includes an active operational state. Exemplary operational states include a state of a pulse width modulation cell or a multiplier cell. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]    The detailed description is described with reference to the accompanying figures. The same reference numbers in different figures indicate similar or identical items. 
           [0011]      FIG. 1  illustrates an exemplary system formed in accordance with an exemplary embodiment; 
           [0012]      FIG. 2  is an exemplary timing diagram used by the components shown in  FIG. 1 ; 
           [0013]      FIG. 3  illustrates an exemplary temperature-response graph in response to square wave modulation; 
           [0014]      FIG. 4  is a flowchart of an exemplary master-simplex mode process; 
           [0015]      FIG. 5  is a flowchart of an exemplary slave-simplex mode process; 
           [0016]      FIG. 6  is a flowchart of an exemplary master-duplex mode process; 
           [0017]      FIG. 7  is a flowchart of an exemplary slave-duplex mode process; 
           [0018]      FIGS. 8   a  through  8   c  illustrate exemplary images generated by the system of  FIG. 1  during a first exemplary process; 
           [0019]      FIGS. 9   a  and  9   b  illustrate exemplary images generated by the system of  FIG. 1  during a second exemplary process; and 
           [0020]      FIGS. 10   a  through  10   c  illustrate exemplary images generated by the system of  FIG. 1  during a third exemplary process. 
       
    
    
     DETAILED DESCRIPTION 
       [0021]      FIG. 1  is an exemplary dynamic differential thermal measurement (DDTM) system  20 . The system  20  includes a semiconductor measurement unit  21  shown in dot/dashed line, which includes two components (a user control unit (UCU)  32  and a device under test (DUT)  30 . These two components  32  and  30  can be separate items as in a development/controller board connected to an integrated circuit to be tested. Alternatively, devices such as microcontrollers and microprocessors that have internal programming capability and external control lines can act as both the UCU  32  and the DUT  30 . In either case, two digital control lines  23  and  25  are used to communicate between the UCU  32  and a pulse-sampling thermography control unit (PSTCU)  24 . The line  23  allows the PSTCU  24  to signal a start/ready condition to the UCU  32 . The line  25  allows the UCU  32  to signal a start/ready condition to the PSTCU  24 . These signals are used to coordinate the state condition of the DUT  30  with respect to the other parts of the DDTM system  20 . 
         [0022]    A thermal camera  26  uses a lens  28  to produce an optical image of the DUT  30 . In one embodiment, the optical emissions from the DUT  30  are due to thermal photons produced by heating of electrical elements of the DUT as currents flow through them. However, this same technique can in principle be applied to any source of emission from the DUT  30 , such as electron-hole recombination radiation. 
         [0023]    The camera  26  produces two electrical signals of interest, an image signal and a frame trigger (or framing signal) on lines  34  and  36 , respectively. The image signal can be in the form of analog or digital image data that are sent to a computing device  40  for digital processing and display. Many types of camera-to-computer interfaces are commercially available. The framing signal on line  36  is typically a short digital pulse from the camera  26  that signals the beginning of a frame sequence in the camera  26 . The framing signal is sent to the PSTCU  24 , which uses it to coordinate the state condition of the DUT  30  with respect to the other components of the DDTM system  20 . 
         [0024]    A digital communications signal sent on line  42  (e.g., USB or other two-way digital communication lines) located between the computing device  40  and the PSTCU  24  allows the computing device  40  to send instructions to the PSTCU  24 . These instructions include “setup in duplex mode,” “start data acquisition,” or comparable instructions. In one embodiment, the instructions include numerical parameters, such as delay times. The line  42  also allows the PSTCU  24  to send responses to the computing device  40  such as “duplex mode set” and to send messages to the computing device  40  such as “acquire current camera  26  frame”. 
         [0025]    The PSTCU  24  acts as timing coordinator between the other components in  FIG. 1 . The timing for the case of the PSTCU  24  acting as the master controller to the slave UCU  32  with simplex communications (only digital control line  23  is in operation) is shown in  FIG. 2 . The camera  26  is assumed to be in a free-running mode in which it takes image frames at a repeated fixed interval. At the beginning of a frame, a pulse is sent out by the camera  26  on the frame trigger line  36  (first time sequence in  FIG. 2 ). Internally, the camera  26  integrates incoming photons on each pixel for a period of time (second time sequence in  FIG. 2 ). The pixel values are then read out and sent via the image signal line  34  to the computing device  40  (third time sequence in  FIG. 2 ). The PSTCU  24  uses the pulse on the frame trigger line  36  to initiate two nearly simultaneous events. First, the PSTCU  24  sends out a signal on the digital control line  23  that indicates to the UCU  32  to switch the DUT  30  between a first state A to a second state B (see fourth and fifth time sequences in  FIG. 2 ). Second, the PSTCU  24  sends a predefined binary sequence or command signal on line  42  to the computing device  40  that indicates to the computing device  40  that it should digitally acquire the next available camera  26  frame data on line  34  and tag it as belonging to DUT state B (see sixth time sequence in  FIG. 2 ). This same timing sequence is performed again except that control line  23  goes low, the DUT  30  is placed into state A, and the acquired camera  26  frame data is tagged as being from state A. This whole process can then be repeated as many times as desired with multiple frames being averaged to improve the signal-to-noise ratio (SNR). 
         [0026]    In this binary sampling (two-image) mode of operation, images from state A are subtracted from images from state B to produce a difference image that is displayed by the computer. This subtraction process removes the common background with only the thermal difference between the two states present. Summing up repeated samples of the difference image adds to the signal strength, while averaging out the noise component of the background, thus improving the SNR and allowing very weak difference signals to be detected. 
         [0027]    In one embodiment, the master/slave relationship between the PSTCU  24  and the UCU  32  is reversed with line  25  now acting as the digital control line (transmitting the digital control signal). Also, a full duplex operation in which both lines  23  and  25  are used, with the master controller sending the slave a signal to Start and the slave sending back a Done indication. Timing diagrams for these other three modes of operation can be created based on those shown  FIG. 2 . 
         [0028]    The above operational mode performs similar data acquisition and image analysis as do prior-art pulse-sampling thermography (PST) systems except that, instead of controlling a voltage level, digital signaling is used to allow the DUT  30  to change states between image samples. As a result, the computing device  40  displays an image that is the difference in thermal signature between a first operational state A and a second operational state B. Possible operational states include baseline operation (e.g., a microprocessor at idle not doing any heavy processing), pulse-width modulation (PWM) cell of the DUT  30  in operation, multiplier cell of the DUT  30  in operation, etc. Importantly, these cells are components that are operated dynamically. For example, a multiplier cell functions only when the internal code operation of c=a*b is invoked and does nothing upon turning the device&#39;s power on and off Thus, study of the thermal behavior of the multiplier cell can be studied only by switching between a state in which the processor is idle, state A, and a state in which a sequence of multiplies is performed, state B. 
         [0029]    The functionality of the DDTM system  20  can be further extended by allowing more than one image to be acquired during each state condition.  FIG. 3  shows a typical temporal thermal response  200  due to a square wave (On/Off) stimulus  210  (volts) in a semiconductor device (the DUT  30 ). As expected, the response  200  follows an exponential rise during the On stimulus and an exponential fall during the Off stimulus. A single pair of images, one taken during the On condition (state B) and one taken during the Off condition (state A) can be used to determine the thermal difference between the two conditions, as described previously in the binary mode of operation. However, more than one image can be acquired during each state condition.  FIG. 3  shows four such frames of image acquisition  220 . The high state of line  220  indicates image integration and the low state indicates no integration in direct correlation to the second line of  FIG. 2 . As shown, two image frames are acquired during each device state. This arrangement is often referred to as “quadrature sampling”. Clearly, higher-order sampling with three or more frames taken in each state can also be performed. 
         [0030]    For quadrature sampling, the image intensity in each frame  1  through  4  will be different, due to the temporal variation in the thermal response  200 . The intensity in image  1 , S 1 , will be less bright than the intensity in image  2 , S 2 . Similarly, the intensity in image  4 , S 4 , will be less bright than the intensity in image  3 , S 3 . Various linear combinations and ratios of these four signals can produce interesting results. Past efforts have utilized a Fourier transform approach in which images are combined to produce both an in-phase and an out-of-phase image, indicating the power in the first-order Fourier component of the square wave modulation. These two images can in turn be mathematically combined to extract the relative phase of the measured signal at each pixel in the image. Although mathematically interesting, a phase image has little meaning in thermal analysis. A more interesting result would be an image of the thermal rise time that is observed in the thermal response  200  in  FIG. 3 . The mathematical combination of image intensities below, which can be related to the hyperbolic tangent of the ratio of the camera  26  frame period, τ, and the thermal rise time, γ, 
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         [0031]    Thermal rise time is a very important parameter when analyzing integrated circuits. It is basically an indication of how well the circuit is being cooled, poor cooling producing a long thermal rise time. In one embodiment, the quadrature-sampling mode and above analysis are used to produce images in which the image intensities are a direct map of the thermal rise time within the circuit. 
         [0032]    As with binary mode operation, quadrature mode operation can be analyzed with timing diagrams, however, the only difference is to have multiple image acquisitions between each change in device state. Higher-order sampling (i.e., three or more samples per device state) can also be diagrammed.  FIGS. 4 through 7  illustrate exemplary flow charts for each of the acquisition modes related to Master/Slave and Simplex/Duplex operation. 
         [0033]    These processes are illustrated as logical flow graphs, each operation of which represents a sequence of operations that can be implemented in hardware, software, or a combination thereof. In the context of software, the operations represent computer-executable instructions stored on one or more computer-readable storage media that, when executed by one or more processors, perform the recited operations. Generally, computer-executable instructions include routines, programs, objects, components, data structures, and the like that perform particular functions or implement particular abstract data types. The order in which the operations are described is not intended to be construed as a limitation, and any number of the described operations can be combined in any order and/or in parallel to implement the processes. 
         [0034]      FIG. 4  shows a flowchart for a Master-Simplex mode of operation of the DDTM system  20  in  FIG. 1 . First, at a block  302 , the PSTCU  24  sends a ‘Start A’ signal to the UCU  32  by setting a signal to low on the control line  23 . Next, at a block  304 , the UCU  32  places the DUT  30  into a first state A, based on the low signal of the control line  23 . At a block  306 , the computing device  40  is instructed by the PSTCU  24  via the line  42  to acquire one or more thermal images of the DUT  30  in the first state A, as the images are obtained by the thermal camera  26 . Then, at a block  308 , the PSTCU  24  sends a ‘Start B’ signal to the UCU  32  by setting a signal to high on the line  23 . Next, at a block  310 , the UCU  32  places the DUT  30  into a second state B, based on the high signal on the line  23 . At a block  312 , the computing device  40  is instructed by the PSTCU  24  via the line  42  to acquire one or more thermal images of the DUT  30  in the second state B, as the images are obtained by the thermal camera  26 . Then, at a block  314 , the computing device  40  analyzes the thermal images at the two states to determine one of thermal difference images, thermal time-constant maps, power dissipation maps, or other characteristics. At a block  316 , the results of the analysis are displayed on a display device associated with the computing device  40 , thus allowing a user to extract quantitative data from regions of interest on the displayed images/maps, at block  318 . 
         [0035]      FIG. 5  shows a flowchart for a Slave-Simplex mode of operation of the DDTM system  20  in  FIG. 1 . First, at a block  320 , the UCU  32  places the DUT  30  into a first state A and sends a ‘Start A’ signal to the PSTCU  24  by setting the signal on the line  25  to low. At a block  322 , the computing device  40  is instructed via the line  42  by the PSTCU  24  based on the low signal on the line  25  to acquire one or more thermal images of the DUT  30  in the first state A, as the images are obtained by the thermal camera  26 . Then, at a block  324 , the UCU  32  places the DUT  30  into a second state B and sends a ‘Start B’ signal to the PSTCU  24  by setting the signal on the line  25  to high. At a block  326 , the computing device  40  is instructed via the line  42  by the PSTCU  24  based on the high signal on the line  25  to acquire one or more thermal images of the DUT  30  in the second state B, as the images are obtained by the thermal camera  26 . Then, at a block  328 , the computing device  40  analyzes the thermal images at the two states to determine one of thermal difference images, thermal time-constant maps, power dissipation maps, or other characteristics. At a block  330 , the results of the analysis are displayed on a display device associated with the computing device  40 , thus allowing a user to extract quantitative data from regions of interest on the displayed images/maps, at block  332 . 
         [0036]      FIG. 6  shows a flowchart for a Master-Duplex mode of operation of the DDTM system  20  in  FIG. 1 . First, at a block  340 , the PSTCU  24  sends a ‘Start A’ signal to the UCU  32  by setting the signal on the line  23  to low. Next, at a block  342 , the UCU  32  places the DUT  30  into a first state A, based on the low signal on the line  23 , and sends a ‘Done/Ready’ signal to the PSTCU  24  by sending a first pulse (low-high-low) signal on the line  25 . At a block  344 , the computing device  40  is instructed by the PSTCU  24  via line  42  to acquire one or more thermal images of the DUT  30  in the first state A, as the images are obtained by the thermal camera  26 , based on the first pulse signal on the line  25 . Then, at a block  346 , the PSTCU  24  sends a ‘Start B’ signal to the UCU  32  by setting the signal on the line  23  to high. Next, at a block  348 , the UCU  32  places the DUT  30  into a second state B, based on the high signal on the line  23  and sends a ‘Done/Ready’ signal to the PSTCU  24  by sending a second pulse (low-high-low) signal on the line  25 . At a block  350 , the computing device  40  is instructed by the PSTCU  24  via line  42  to acquire one or more thermal images of the DUT  30  in the second state B, as the images are obtained by the thermal camera  26 , based on the second pulse signal on the line  25 . Then, at a block  352 , the computing device  40  analyzes the thermal images at the two states to determine one of thermal difference images, thermal time-constant maps, power dissipation maps, or other characteristics. At a block  354 , the results of the analysis are displayed on a display device associated with the computing device  40 , thus allowing a user to extract quantitative data from regions of interest on the displayed images/maps, at block  356 . 
         [0037]      FIG. 7  shows a flowchart for a Slave-Duplex mode of operation of the DDTM system  20  of  FIG. 1 . First, at a block  360 , the UCU  32  places the DUT  30  into a first state A and sends a ‘Start A’ signal to the PSTCU  24  by setting a signal on the line  25  to low. At a block  362 , the computing device  40  is instructed by the PSTCU  24  via line  42  to acquire one or more thermal images of the DUT  30  in the first state A, as the images are obtained by the thermal camera  26 , based on the low signal on the line  25 . Then, at a block  364 , the PSTCU  24  sends a ‘Done/Ready’ signal to the UCU  32  by sending a first pulse (low-high-low) signal on the line  23 . Next, at a block  366 , the UCU  32  places the DUT  30  into a second state B, based on the first pulse signal on the line  23 , and sends a second ‘Start B’ signal to the PSTCU  24  by setting the signal on the line  25  to high. At a block  368 , the computing device  40  is instructed by the PSTCU  24  via line  42  to acquire one or more thermal images of the DUT  30  in the second state B, as the images are obtained by the thermal camera  26 , based on the high signal on the line  25 . At a block  370 , the PSTCU  24  sends a ‘Done/Ready’ signal to the UCU  32  by sending a second pulse (low-high-low) signal on the line  23 . Then, at a block  372 , the computing device  40  analyzes the thermal images at the two states to determine one of thermal difference images, thermal time-constant maps, power dissipation maps, or other characteristics. At a block  374 , the results of the analysis are displayed on a display device associated with the computing device  40 , thus allowing a user to extract quantitative data from regions of interest on the displayed images/maps, at block  376 . 
         [0038]    In all four modes of operation, the sequence can loop in order to average many images, in order to obtain increased signal-to-noise performance, and detect weak signals. This looping is shown in  FIGS. 4 through 7  via the line  390 . 
         [0039]    This new modulation approach opens up a wide range of investigations into dynamic processes in both analog and digital devices, such as heat loading as a function of process parameters and reverse engineering. 
         [0040]    The following examples represent a small subset of the potential application of these techniques. 
         [0041]    An exemplary DUT  30  is a MicroChip PIC32MX460F512L microcontroller with a standard-100 pin glass-epoxy package with a typical thickness of ˜0.4 mm between the semiconductor die&#39;s active layer and the top side of the package. Several measurements were made through the package via direct observation of the heat signature propagated to the package&#39;s top using a 2× lens and thermal camera with a field of view of 6 by 6 millimeters. This field of view is roughly the same size as the underlying die, which has the active side facing upward. A reference thermal image of the device with power supplied to the DUT is shown in  FIG. 8   a . As can be seen by the darkening of the image toward the bottom of the image, there is a significant thermal background, simply due to its baseline dynamic operation. The goal of the differential thermal measurements is to separate out heating, due to specific active components, from this general thermal background. 
         [0042]    A program was introduced into the microcontroller that allowed a USB communication interface. Instructions were sent to the device via the USB interface to place it into various test states, as described below. In this setup, the UCU  32  and the DUT  30 , as seen in  FIG. 1 , are one and the same object. 
         [0043]    In an initial demonstration, a program was loaded into the microcontroller that caused a digital output pin to switch states in response to the digital signal from the PSTCU  24 —the simplest of dynamic conditions in which only the background heating is due to dynamic processes. In this example, one can discriminate between the dynamic heat sources and a single modulated component. 
         [0044]    In one embodiment, an external resistor included in the DUT  30  circuitry controls the internal power dissipation of an output driver of the microcontroller. For the initial imagery, the internal power dissipation was set at maximum power level of ˜25 mW. A differential image,  FIG. 8   b , was obtained via the process shown in  FIG. 4 . False shading (e.g., coloring) of the image is used to indicate the strength of the heating in the image. White coloring indicates very hot with lessening heat signature as the color moves through red, orange, green, then blue. This is shown as different shades in the figures. A source of peak heating  402  is readily apparent in  FIG. 8   b.    
         [0045]    In order to visualize the position of the peak  402  with respect to the DUT image seen in  FIG. 8   a , the image in  FIG. 8   b  is passed through a digital filter in which heating less than a certain value is made transparent. The user can set the transparency level by setting a threshold value. This filtered image is then overlaid onto the reference image in  FIG. 8   a , the image being used as a spatial reference image. The result is shown in  FIG. 8   c  wherein the peak heat signature from  FIG. 8   b  is clearly located on the DUT  30 . 
         [0046]    Further, as the internal power dissipation is controlled by the external resistor, a sequence of differential images can be taken with differing power levels. The measured intensity of the peak heating  402  at these different power levels can then be used as a calibration curve for determining the power dissipation of other components of the DUT  30 . 
         [0047]    In another embodiment, the signal and the background were made to be dynamic processes by setting the control program to switch between running a sequence of no-ops in state A and either a repeating sequence of multiplies or a sequence of memory swaps (read variable  1 , swap to variable  2 , save variable  2 ) in state B. The overlaid results are shown in  FIG. 9   a  for the multiplies and  FIG. 9   b  for the memory swaps. Both operations have a common area heating up, peak  412 , and a second heat source on the upper right for the multiply, peak  410 , and on the upper left for the memory swap, peak  414 . The location of these hotspots can be correlated to operational circuitry. 
         [0048]    Moreover, the calibration generated by exercising the output pin described above, can be used as a calibration source to estimate the power dissipation at these hotspots. In  FIG. 9   a , the peak  410  is estimated to have internal power dissipation of 3.2 milliwatts and the peak  412  is estimated at 2.7 milliwatts. In  FIG. 9   b , the estimated power dissipation for both of the peaks  412  and  414  is 4.2 milliwatts. These results present a technique for in situ verification of expected location and magnitude of internal heating from these two dynamic processes. 
         [0049]    In a further embodiment, the signal and the background are again made to be dynamic processes by setting the control program to switch between running a sequence of no-ops in state A and a repeating sequence of multiplies in state B. However, unlike the previous binary sampling, quadrature sampling is used to produce four images, two taken during state A and two taken during state B, as indicated previously in  FIG. 3 . Many different mathematical combinations of these four images can be produced. Three images of particular interest are shown in  FIGS. 10   a  through  10   c .  FIG. 10   a  is produced by displaying the mathematical combination of the four image intensities given by S 1 +S 2 −S 3 −S 4 , which produces an indication of the heating early in time.  FIG. 10   b  is the mathematical combination of the four image intensities given by −S 1 +S 2 +S 3 −S 4  and produces an indication of the heating later in time. The two thermal peaks  410  and  412  seen in  FIG. 9   a  (using binary sampling) are apparent in both images. However, the location of the peaks in  FIG. 10   a  are more clearly defined, as the heat has had less time to spread. Moreover, examination of the  FIGS. 10   a  and  10   b  shows how the heat is spreading through the DUT  30  in time. This heat flow is defined by producing an image based on inverting the Equation (1) above to extract the thermal rise time g. An image of the thermal rise time is shown  FIG. 10   b . As with the previous images, false color/shade is used to show the different time constants, where red indicates very short time constants, with time constant increasing to 200 milliseconds as the color moves through red, orange, green, then blue. Note that the overall white background is this image is an artifact of insufficient signal and not an indication of a thermal rise time. As shown, the time constant is shortest near the heat source peaks  410  and  412  and becomes longer as the heat takes time to flow away from the sources. The heat flow path is clearly defined from this image. The combination of peak heating and heat flow derived from these images can be utilized in designing efficient heat extraction and in locating overheating sites. 
         [0050]    While the preferred embodiment of the invention has been illustrated and described, as noted above, many changes can be made without departing from the spirit and scope of the invention. Accordingly, the scope of the invention is not limited by the disclosure of the preferred embodiment. Instead, the invention should be determined entirely by reference to the claims that follow.