Abstract:
A system and method to fully characterize the thermal behavior of complex 3D submicron electronic devices. The system replaces and/or supplements laser-based surface temperature scanning with a CCD camera-based approach. A CCD camera records multiple points of light energy reflected from an integrated circuit to obtain a temperature measurement. The system is used to non-invasively measure with submicron resolution the 2D surface temperature field of an activated device. The measured 2D temperature field is used as input for an ultra-fast inverse computational solver. The system couples measured results and computations in a novel approach, making it possible to extract geometric and thermal features of a device, and to obtain critically needed temperature distributions over the entire 3D volume of that device, including regions that are physically or optically inaccessible. The obtained distributions reflect the real, and not merely theoretical, physical construction and thermal behavior of that device.

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
   This application claims the benefit of provisional Application No. 60/847,170 filed Sep. 25, 2006. 

   STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT 
   Not Applicable 
   THE NAMES OF THE PARTIES TO A JOINT RESEARCH AGREEMENT 
   Not Applicable 
   INCORPORATION-BY-REFERENCE OF MATERIAL SUBMITTED ON A COMPACT DISC 
   Not Applicable 
   BACKGROUND OF THE INVENTION 
   1. Field of the Invention 
   The present invention relates generally to semiconductor thermography and more specifically to a system and method for measuring the thermal behavior of integrated circuits. 
   2. Description of Related Art 
   The present invention references Applicant&#39;s U.S. Pat. No. 6,064,810 (the “&#39;810 patent”), the contents of which are incorporated herein by reference. 
   As semiconductor devices become physically more complex with shrinking feature sizes and higher local power densities, cooling problems multiply, which can lead to a decrease in performance and reliability. Thus, understanding and determining the thermal behavior of modern electronics has become a key issue in their design. As a result, there is a critical demand for methods that can be used to determine the temperature of features at the submicron level, particularly when most important features are physically inaccessible. Computational approaches can provide insight into the internal thermal behavior of such complex devices, but can be limited by the inherent necessity of modeling the heat sources, which in the case of self-heating microelectronic devices, are the result of electrical fields whose exact shapes and locations are difficult to specify with reasonable certainty. Moreover, such devices can actually experience irreversible changes in thermo-physical properties and/or geometries that cannot be otherwise predicted from theory or monitored. Experimental approaches can also be helpful in determining thermal behavior, but require either physical access or a visual path to the region of interest. Contact methods, for example, present the difficulties of having to access features of interest with an external probe, or in the case of embedded features, fabricate a measuring probe into the device, and then having to isolate and exclude the influence of the probe itself. Non-contact methods, on the other hand, can provide surface temperature profiles, but in and of themselves cannot impart information on internal behavior. In other words, these methods provide a two-dimensional perspective on what otherwise is, in the case of stacked complex devices, an intricate three-dimensional thermal behavior. 
   Therefore, a need exists for an improved and more comprehensive method of conducting three-dimensional thermal characterization of semiconductor devices with shrinking characteristics and higher operating speeds. Further, a need exists for a non-invasive method of conducting thermal characterization that accurately measures the thermal characteristics without imparting its own energy into the system. Further, a need exists for a method of conducting thermal characterization in situ and at the sub-micron level of today&#39;s devices. 
   BRIEF SUMMARY OF THE INVENTION 
   The present invention overcomes most of the disadvantages and problems associated with thermal characterization of submicron semiconductor devices. It combines a system capable of mapping the surface temperature of a complex device with high spatial and temporal accuracy, a computational engine capable of rapidly and accurately resolving the geometric and material complexities of a full three-dimensional microelectronic device, and a method for utilizing the system to accurately and efficiently obtain a fully three-dimensional thermal characterization of an active micro-device. With the independent information from the experimental measurements it becomes possible to mitigate the lack of knowledge in the source model parameters which directly affect the usefulness of the computational results. 
   In accordance with one feature of the invention, the system uses a variable light source to illuminate the surface of an active micro-device. A CCD camera system then takes a high resolution image of the reflected light and produces a measure of the change in thermoreflectance for a given input power across the surface of the device. This change in reflectivity translates to a two-dimensional temperature distribution across the visible surface of the device. The two-dimensional data is then used by the computational engine to compute a three-dimensional temperature field of the entire device. 
   In accordance with another feature of the invention, the system uses a laser light to measure the change in thermoreflectance for a given input power across the surface of an active micro-device. This change in reflectivity translates to a two-dimensional temperature distribution across the visible surface of the device. The two-dimensional data is then used by the computational engine to compute a three-dimensional temperature field of the entire device. 
   Accordingly, it is one general object of the invention to provide a means for rapidly measuring the two-dimensional surface temperature of an active micro-device. It is another general object of the invention to provide a means for conducting a full three-dimensional thermal characterization of an active micro-device. It is another general object of the invention to perform this characterization on an active micro-device in situ. It is another general object of the invention to perform this characterization with sub-micron spatial resolution. It is another general object of the invention to perform this characterization non-invasively. The invention accordingly comprises the features described more fully in the remainder of the specification, and the scope of the invention will be indicated in the claims. Further objects of the present invention will become apparent in the following detailed description read in light of the drawings. 

   
     BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S) 
     The present invention will be more fully understood by reference to the following detailed description of the preferred embodiments of the present invention when read in conjunction with the accompanying drawings, in which like reference numbers refer to like parts throughout the several views, wherein: 
       FIG. 1  is a high-level flow diagram of the Thermography Measurement system operation as a whole; 
       FIG. 2  is a schematic representation of one embodiment of the Thermoreflectance Thermography system (TRTG) of the present invention, having only a CCD camera assembly for static and slow transient measurements; 
       FIG. 3  is a schematic representation of another embodiment of the TRTG of the present invention, having both a CCD camera assembly for static and slow transient thermoreflectance measurements and a CW laser assembly for transient thermoreflectance measurements; 
       FIG. 4  is a flow diagram indicating the steps required to compute a three-dimensional temperature field for a device under test using the CCD camera of the TRTG; 
       FIG. 5  is a flow diagram indicating the steps required to compute a three-dimensional temperature field for a device under test using the CW laser of the TRTG; 
       FIG. 6  is a flow diagram indicating the steps required to compute the thermoreflectance coefficient (C TR ) for a device under test using the TRTG; 
       FIG. 7  is a schematic representation of a micro-resistor device (“test device”) used to prove the concept and validate the present invention. The geometric parameters of the device are indicated in the drawing; 
       FIG. 8  is a graph comparing the experimental data achieved with the corresponding numerically computed temperature distributions in the test device of  FIG. 7 . The bottom oxide thickness of the device was varied while the heat source length was held constant; 
       FIG. 9  is a graph comparing the experimental data achieved with the corresponding numerically computed temperature distributions in the test device. The bottom oxide thickness of the device was held constant while the heat source length was varied; 
       FIG. 10  is a graph indicating the path taken in optimizing the oxide thickness (h) and heater length (L) parameters of the test device so as to converge on the optimum values of h and L which yield a numerical solution that agrees with the measured device surface signature to within a minimum RMS error; 
       FIG. 11  is a contour of one slice of the optimal numerical solution for the surface temperature rise extracted from the full three-dimensional thermal solution for the test device; 
       FIG. 12  is a contour plot of the experimentally measured temperature signature on the surface of the test device; 
       FIG. 13  provides a top view ( FIG. 13A ) and bottom view ( FIG. 13B ) of a contour of the optimal three-dimensional numerical thermal solution of the test device. For clarity, only two slices are shown and the spatial domain is restricted to the region surrounding the test device heater; and 
       FIG. 14  is a schematic representation of another embodiment of a system for determining the thermoreflectance coefficient of a device under test by utilizing a variable wavelength light source and two photodetectors. 
   

   Where used in the various figures of the drawing, the same reference numbers designate the same or similar parts. Furthermore, when the terms “top,” “bottom,” “first,” “second,” “upper,” “lower,” “height,” “width,” “length,” “end,” “side,” “horizontal,” “vertical,” and similar terms are used herein, it should be understood that these terms have reference only to the structure shown in the drawing and are utilized only to facilitate describing the invention. 
   All figures are drawn for ease of explanation of the basic teachings of the present invention only; the extensions of the figures with respect to number, position, relationship, and dimensions of the parts to form the preferred embodiment will be explained or will be within the skill of the art after the following teachings of the present invention have been read and understood. Further, the exact dimensions and dimensional proportions to conform to specific force, weight, strength, and similar requirements will likewise be within the skill of the art after the following teachings of the present invention have been read and understood. 
   DETAILED DESCRIPTION OF THE INVENTION 
     FIG. 1  represents the high-level flow diagram of the basic operation of the present invention as it is used to measure the three-dimensional temperature field distribution of an active semiconductor micro-device. To conduct a full three-dimensional temperature characterization of a device requires essentially three steps  100 . First, because different equipment is required, a decision must be made as to whether the static or transient thermal distribution is sought  102 . Second, a two-dimensional thermal measurement is conducted. If static data is required, then CCD camera-based Thermoreflectance Thermography system (TRTG) is used  104 . If transient data is required, then a laser-based TRTG is used  106 . Third, a three-dimensional thermal characterization is computed from the two-dimensional data and various device parameters  108 . 
   The present invention is based on the thermoreflectance (TR) method, where the change in surface temperature is measured by detecting the change in the reflectivity of the sample. Since the change in reflectivity per unit temperature (i.e., thermo-reflectance coefficient, C TR ) is of the order of 10 −3 -10 −5  K −1  for most electronic materials, the system had to be designed, built, and fine-tuned to achieve the levels of certainty required for successful measurements. 
   The measurement methodology requires two steps. First, the coefficient of thermal reflectance (C TR ) must be determined for each of the surface materials to be scanned (calibration). Second, the changes in the surface reflectivity as a function of changes in temperature are measured with submicron spatial resolution using either a CCD camera or a photodetector. 
     FIG. 2  shows a schematic of one embodiment of the TRTG  200 . This embodiment uses only a CCD camera assembly  202  for conducting measurements. As mentioned previously, this configuration is primarily useful for static and slow transient measurements only.  FIG. 3  shows a schematic of another embodiment of the TRTG  300 . This embodiment combines two different techniques for acquiring the reflectivity change induced by the temperature change on the surface of the activated device, namely, (i) a multipoint CCD camera  202  approach, and (ii) a single-point laser-based  308  approach. With the addition of the CW laser  308 , this embodiment provides for both static and transient thermo-reflectance measurements. 
     FIGS. 2 and 3  provide similar schematic representations of the CCD camera-based TRTG. In the case of the multipoint approach, the change in reflectivity is captured as the change in the intensity of the reflected light on each element (pixel) of the CCD camera  202 . The advantage of this approach is that it is simpler to use, easier to vary the wavelength of the probing light  214  (to maximize the C TR  coefficient), has excellent spatial resolution (as low as 200 nm with a 100× objective lens, and potentially lower with higher magnification lenses) and is orders of magnitude faster than the single-point approach. Current sensor technology limitations include that the CCD approach captures slower transients than possible with a photodetector based approach. As CCD camera technology improves, the present time resolution gap between CCD and photodetector approaches will close. 
   The device under test  216  is mounted on a platform  208  such as those used in wafer fabrication. The present invention utilizes a ThermoChuck® temperature inducing vacuum platform for mounting and testing semiconductor wafers on a probing station. However, one skilled in the art will appreciate that any wafer heating element will suffice. After mounting, the DUT  216  is connected to a pulsed current source  212  using 4-wire probes  210 . The current source  212  and probes  210  allow for the DUT  216  to be powered during testing. 
   A variable wavelength light source  214  directs its light through a beam splitter  204 , where it is reflected through the objective lens  206  and onto the DUT  216 . The light reflected from the DUT  216  travels upward to the CCD camera  202  where it is processed and measured. A computer  218  controls the data acquisition and system operation. 
     FIG. 3  provides a schematic representation of the laser-based TRTG. The probing laser beam from a CW laser  308  is groomed with a collimator and then directed through the CCD camera&#39;s optics (beamsplitters  312  and  204 ) utilizing beamsplitter  316  and a mirror  318 . The beam is projected through the objective lens  206  perpendicularly to the heated surface of the device under test (DUT)  216  from which it reflects back along the optical path to the sensitive area of the photodiode  310 . 
   The intensity of the reflected light depends on the reflectivity of the DUT&#39;s surface  216 , which in turn depends upon the surface temperature. To overcome the inherently low signal to noise ratio, the activation voltage of the DUT  216  is modulated  212 , resulting in a modulated photodetector signal that can more easily yield the useful signal from the raw photodetector signal output. The photodetector signal, containing the change in surface reflectivity caused by the temperature variations of the DUT  216 , is pre-amplified  306  and then acquired with a lock-in amplifier  302  (or an oscilloscope  304 ) and is then scaled according to the calibrated data (C TR ). The key difference between the two techniques is that the lock-in approach cannot be used to measure transient temperate fields, while the oscilloscope technique makes it possible to measure transient temperature with microsecond or better temporal resolution. However, the oscilloscope technique is less accurate than the lock-in technique. The temperature field over a desired area of an active device  216  can still be mapped by repeating the procedure at multiple physical locations, which is achieved by precisely moving the laser beam with submicron resolution. 
     FIG. 6  is a flow-diagram of the steps required to determine the C TR  using the TRTG CCD method  600 . The first step involves determining the temperature coefficient of resistance (α) for the device under test by heating the latter to known temperature elevation levels (ΔT) and measuring the resulting relative changes in electrical resistance (ΔΩ/Ω) by the use of a standard four-probe method  602 . The resulting data are used to determine α, which is the slope of the linear equation relating ΔΩ/Ω to ΔT  604 . The TRTG system  300 , wafer chuck  208 , and probes  210  are situated on a probing station designed so as to render the measurements insensitive to ambient vibrations and to the small movements caused by the thermal expansions experienced in the system. 
   The second step is similar to the first one, except that the DUT  216  is heated by pulse activation and the wafer chuck temperature is held at room temperature. As the DUT  216  is activated with various specified levels of DC current, the four-probe approach is used to simultaneously record both the resistance of the microresistor strip as well as the Joule activation power  606 . Thus, the relative change in electrical resistance, ΔΩ/Ω as a function of applied electrical power (P) is obtained, where the slope of the linear dependence is hereafter referred to as β  608 . 
   Given α and β, the relationship between power, P, and resistor average temperature rise, ΔT, can be easily calculated. Next, the TRTG system  300  is used to measure the changes in reflectance, ΔR/R, over the micro-resistor surface at prescribed electrical power levels, P, by recording the relative light intensity for each pixel of the CCD  610 . To obtain C TR  as a property of the surface material of the DUT  216  at each light wavelength, the recorded fields of intensity change are averaged over the active area of the heater to obtain an average (ΔR/R) avg . Finally, the actual C TR  for the DUT  216  is calculated from the overall relationship C TR =(ΔR/R) avg ·P −1 ·(α/β)  612 . 
   The resulting value of C TR  for the DUT  216  at a given light wavelength (λ) can then be compared to previously obtained values  614 . Different values of λ produce different values of C TR . Therefore, the λ yielding the highest C TR  is the best wavelength to use during the temperature scans. In order to maximize the signal to noise ratio, it is important to use a light source whose wavelength produces the maximum value of C TR  for the material layer being examined. 
   The primary impact of the methodology described is that it provides an approach for measuring C TR  in situ, at any wavelength of the illumination light  214 , and for surface materials of real microelectronic devices, as opposed to using generic data from the literature, which are limited and tend to be inaccurate for a specific device configuration of interest. Of potentially greater significance is the fact that by combining the experimental TRTG method and the computational system, it also becomes possible to determine the C TR  of surface materials on the device other than the specific material that was directly involved in the calibration stage. 
   An additional method of determining CTR involves using a variable wavelength light source and two photodetectors.  FIG. 14  shows such a device  1400 . The multi-wavelength light source  1402  illuminates the DUT surface at a specific wavelength, generated by passing a Xenon white light through a monochromator grating  1412  adjusted to produce a 5-nm line width. The wavelength can be specified with a precision of 0.5 nm. The probing light is divided into two beams, one collected on a reference photodetector  1406  (PD 2 ) and the other collected on the primary photodetector  1404  (PD 1 ) after being reflected from the sample surface. The light intensities of the two branches are balanced to within 1% by the use of a pair of polarization devices  1416  and  1418 . 
   The DUT  1408  is heated by a thermoelectric element  1410  and a K-type thermocouple is used to measure the surface temperature of the sample as closely as possible to the lighted area. Since the multi-wavelength light source  1402  produces a non-planar light front, the smallest possible probing surface area is on the order of 3-5 mm. The base level of reflected energy (E) at room temperature is first recorded, and then the sample is heated to a stabilized prescribed temperature rise, ΔT. A lock-in-amplifier  1414 , locked on the frequency and phase of the modulated light source, is used to measure the change in reflected energy (ΔE) resulting from the rise in surface temperature. Once ΔE and E are known, the ratio of the change in reflectivity (ΔR) to the base reflectivity (R) becomes known since ΔE/E=ΔR/R. Finally, the C TR  coefficient can be calculated from its definition, C TR ≡(ΔR/R)/ΔT. 
   The above approach provides very good results and the flexibility of prescribing different light wavelength, but as noted above, the spot size is too large to measure the C TR  of actual integrated circuit devices in situ. Also, the use of external heating (e.g., thermoelectric element) introduces systematic uncertainties associated with thermal expansion of the device and the need for measuring the temperature rise with an external probe (e.g., thermocouple). In combination, the major disadvantage is connected to the differences in scales between the device used for calibration and the significantly smaller device in whose surface temperature is of interest. 
   The flow diagram of  FIG. 4  provides the steps  400  necessary to take a static two-dimensional temperature measurement of the surface of the DUT  216 . The first step is to determine if the C TR  is known for the device  402 . If not, then a C TR  measurement is conducted  404  using the previous TRTG procedure ( FIG. 6 ). Next, the DUT is activated  406  and allowed to stabilize in temperature. Next, the wavelength of the variable light source  214  is set to the optimum wavelength (λ) determined in the C TR  calibration  408 . One skilled in the art will recognize that a fixed wavelength light source may also be utilized so long as the wavelength provided is one that yields a significantly high C TR  which improves the signal-to-noise ratio. Next, a “snapshot” of the light energy reflected from the surface of the DUT  216  is then taken by the CCD camera  202  for processing  410 . This pixel-by-pixel energy recorded by the CCD camera  202  is then processed to compute a two-dimensional thermal characterization of the device&#39;s surface  412 . This is possible because the change in temperature for a given spot on the device&#39;s surface is directly related to the change in the spot&#39;s reflectivity (ΔT=1/C TR *ΔR/R). Once the two-dimensional surface characterization is obtained, the results are processed to compute a three-dimensional temperature solution for the overall device  414 . 
   The flow diagram of  FIG. 5  provides the steps  500  necessary to take a transient two-dimensional temperature measurement of the surface of a device. This process is more involved as it requires movement of either the DUT  216  or the CCD camera assembly in the XY coordinate system. The first step is to properly position the DUT  216  beneath the objective lens  206  such that the probing laser beam will contact the DUT  216  at the origin of the XY coordinate system  502 . Next, determine if the C TR  is known for the device  504 . If not, then a C TR  measurement is conducted  404  using the previous TRTG procedure  506 . Next, the DUT is activated with a drain pulser at a chosen modulation frequency  508 . Modulating the signal driving the DUT increases the signal to noise ratio at the photodetector  310 . Next, the laser wavelength (λ) is chosen to that which yields an optimum C TR    510 . The probing laser beam is then directed through the objective lens  206  and onto the surface of the DUT  216 . Next, the light energy reflected from a particular spot on the device&#39;s surface is collected by the photodetector  310  and measured  512 . Either a lock-in amplifier or an oscilloscope is then used to measure the modulation amplitude of the photodetector signal to determine ΔE  516 . Next, the photodetector signal is averaged (E)  518 . If there are any other spots on the DUT to be measured  520 , then either the DUT  216  or the CCD camera assembly  202  is moved in the XY coordinate space  514 . If the entire surface of interest of the DUT has been measured, then the two-dimensional thermal characterization of the device&#39;s surface is computed (ΔT=1/C TR *ΔR/R, where ΔR/R equals ΔE/E)  522 . Once the two-dimensional surface characterization is obtained, the results are processed to compute a three-dimensional temperature solution for the overall device  524 . 
   The numerical computing thermal modeling engine is capable of simulating the transient thermal behavior of active multi-layered devices whose dimensions vary over several orders of magnitude and where the thermophysical properties of the materials used may not be isotropic. The thermal modeling engine is used for solving the required three-dimensional heat transfer problem of the corresponding physical device. The surface temperature field (previously measured using the TRTG system) is used as an input signature for an optimization scheme that varies control parameters (e.g., source power, length) until the RMS error between the computed solution and the input signature is minimized. 
   The novel approach begins by solving the corresponding steady-state problem by the use of a grid nesting technique as described in the inventor&#39;s &#39;810 patent. Since the physical dimensions of the various materials used in modeling high performance electronic devices vary greatly, a uniform mesh that resolves all of the details in three dimensions results in a prohibitively large computational grid. A common method for dealing with dimensional variation is to skew the mesh and concentrate more grid points in areas where higher resolutions are needed. The shortcoming of using a biased-mesh approach to resolve the geometry is that the problem geometry, and not the temperature gradients, will end up dictating the meshing. The meshing strategy used in the development of the numerical engine was set on ensuring that it is (i) automatic and adaptive, (ii) independent of user expertise, and (iii) independent of materials (including air), geometry features, embedded vias, and heat source locations. The approach makes it possible to start with the full 3D geometry of a real device in all of its complexity, and then uses a physics-based automatic error predictor to focus the entire available computational power on only those regions that require further refinement in order to achieve the level of acceptable error prescribed by the analyst. The power of this method is that it uses effective thermal properties that are consistent with the local grid spacing at the particular grid level in use. As a result, dealing with air, embedded vias, and ultra-thin multi-layered structures requires no special treatment. 
   The fidelity of a computational solution depends directly on the accuracy of the material properties and geometric characteristics of the features making up the device of interest. Since thermal properties of thin-films vary from those of bulk materials, it is necessary for the numerical simulation to use the real values of the thermal conductivity for all of the materials making up the system under study. The previously described TTR measurement system can also be used to determine any unknown properties of thin-film materials and their interface resistances. 
   In order to carefully characterize the geometry of a device of interest, an ellipsometer was used to measure the thicknesses of transparent layers as well as to confirm the optical properties of surface metals. In addition, a profiler was used to measure the thicknesses of various opaque and transparent layers making up the device. 
   With reference to  FIG. 7 , to prove the concept and validate the described method, basic reference aluminum micro-resistor devices  700  buried in a layer of silicon dioxide  714  were constructed. In the micro-resistor device, shown schematically in  FIG. 7 , an aluminum (Al) strip heater  708  is sandwiched between top  702  and bottom  716  oxide layers in the vertical direction, and between two Al activation pads ( 704  and  710 ) in the horizontal direction. When activated at known electrical power levels, the known heat source will generate a three-dimensional temperature field throughout the device. Since the approach calls for measuring the temperature signature on the surface of the device, a gold (Au) layer  706 ,  712  was deposited over the anticipated area of interest, which includes the heat source  708  and large surrounding regions. Gold was chosen since when used in conjunction with an LED light source at 485 nm, it will maximize the C TR  value in the TRTG technique. However, one skilled in the arts will appreciate that other combinations of surface materials and probing light wavelengths can be effectively used without straying from the present invention. 
   The simple construction of this micro-resistor device makes it possible to specify and measure all essential heat transfer problem parameters. Specifically, (i) the geometry of the different layers can be controlled in the fabrication process and later measured for confirmation; (ii) the oxide layers and Al strip provide a Joule heat source with known uniform power distribution; and (iii) the large pads make it possible to use a four-wire scheme to simultaneously activate the strip heater and measure its electrical power. The validation device has a heater with a width of 14 μm and a length of 200 μm. All other pertinent geometric parameters are provided in  FIG. 7 . 
   By using the experimental system and combining it with the numerical approach presented above it is possible to solve the inverse conduction problem associated with a complex, multi-layered, deep submicron electronic device in order to infer the thermal behavior of the embedded features that cannot be otherwise accessed. The inverse solution is obtained by varying key parameters that define the heat transfer problem under consideration. For the aforementioned test micro-resistor, these parameters include the size and location of the heater strip, its power and distribution, the thicknesses of the top and bottom layers, and their thermal properties. One skilled in the art would appreciate that a solution that comprehends all of these variables would be impractical. Furthermore, many of these parameters can be either directly measured (e.g., thermal properties, applied power, thicknesses of layers) and/or have smaller influences on the final temperature distribution in the device (e.g., layers of SiO 2  and Au above the heater). 
   Since heat flows primarily toward the substrate, the thickness of the bottom oxide layer  716  is expected to strongly affect the temperature distribution because of the resistance that this layer presents. Therefore this parameter was chosen for its sensitivity on the final result. A second parameter was chosen to be the length of the heat source because (i) it cannot be directly measured and (ii) it is expected to be longer than the length of the heater strip itself because of the end effects at the junction between the aluminum heater strip and its activation pads. In fact, one would expect that the end effects would extend beyond the strip by a distance that scales with the heater&#39;s width. 
     FIGS. 8 and 9  compare the experimental data along the mid-plane of the heater strip to the corresponding numerically computed temperature distributions for different values of the bottom oxide thickness and for a fixed heat source length of 200 μm. As expected, thicker layers of bottom oxide result in higher strip temperatures ( FIG. 8 ). As indicated in  FIG. 8 , the numerical temperature distribution for the thickness of 1,500 Å fits nicely the measured data in the center of the heater, Xε[400,600], but misses inside the pad areas. The discrepancy is due to the end effect previously mentioned. 
   To examine the end effects, the bottom oxide thickness was held fixed at 2000 Å, and the heat source length was varied as depicted by the results in  FIG. 9 . Extending the heat source length beyond the ends of the strip heater pulls the temperature curves outwardly toward the experimental data at the lower temperatures. However, the agreement between the numerical and experimental distributions becomes worse toward the center of the domain. Simply extending the rectangular heat source, as done here, captures the spherical nature of the heat distribution at the pad-heater junction to a first order approximation. In order to more precisely simulate the end effects, one would have to introduce a more sophisticated heat power distribution model in this region. 
   Nevertheless, taken together, the results of  FIGS. 8 and 9  highlight the importance of both physical parameters and hence the need to optimize over both of them simultaneously. The optimization method used in this work is a variant of the “steepest descent” method. The search begins with a numerical solution of the heat transfer problem at nominal values of the two parameters being considered, i.e., bottom oxide thickness (h) and heat source length (L). This numerical solution is compared with the measured signature field  708  on the gold pad  706  at every common location to compute an RMS error. Then, additional trial solutions and associated errors are obtained at neighboring pair values of h and L in the two-dimensional parameter space. Evaluation of the errors at the nominal and eight neighboring pairs provides the direction for modifying h and L in order to decrease the error. This process continues until the changes in both h and L are acceptably small. 
   For the specific device under consideration,  FIG. 10  shows the path taken in the (h, L) parameter space to converge Onto the final values of h and L which yield a numerical solution that matches the measured surface signature to within the minimum RMS error. Of particular interest here is that the process of reaching the final result required 57 solutions of the full three-dimensional heat transfer problem, each of which was converged to less than 1% numerical error at the specified h and L values. It is obvious that such an approach would be impractical with traditional numerical solvers. The self-adaptive, ultra-fast computational engine used here and described in the inventor&#39;s &#39;810 patent required approximately 20 minutes to solve this particular problem using a 3.4 GHz Pentium(R) 4 desktop PC. 
   To compare the experimental and numerical surface temperature fields, a surface slice is extracted from the full 3D solution that corresponds to the size, location, and resolution of the experimental area.  FIG. 11  shows the surface temperature slice at stage  9  ( FIG. 10 ), which represents the “optimal” solution that is closest to the experimental signature shown in  FIG. 12 . While the agreement is very favorable, it is clear that the end effects give the experimental contours a more “rectangular” shape on the heater edges. A more sophisticated model of the power distribution at the pad-heater junction may be applied to further improve the agreement. 
   While the present invention discloses primarily a system for measuring the thermal field on a surface and computing the corresponding 3D internal thermal characterization, the same technique can be used for various other fields. For example, the electrical field across the surface of the heart is a function of its internal boundary conditions. If this 2D electrical surface field were measured with sufficient accuracy and an accurate representation of the internal structure of the heart cavity were obtained (CT-Scan, MRI, etc.), then the complete 3D characterization of the electrical field throughout the volume of the heart could be computed. 
   It will now be evident to those skilled in the art that there has been described herein an improved system and method for measuring the three-dimensional thermal characteristics of a semiconductor material. The ultimate benefit of the coupled experimental-numerical system is that it takes an inherently 2D experimental approach and provides a full 3D thermal characterization of the complete device. The optimization-based coupling ensures that the 3D solution is consistent with the 2D experimental signature within the chosen heat transfer model as defined by the combination of known and unknown (i.e., optimizable) parameters.  FIG. 13  provides an example of the optimal 3D numerical solution. For clarity, only two slices are shown and the spatial domain is restricted to the region surrounding the heater, even though the computational domain is 1000×500×1000 μm. The horizontal slice is on the surface of the device and the vertical slice cuts across the mid-plane of the heater. 
   Although the invention hereof has been described by way of a preferred embodiment, it will be evident that other adaptations and modifications can be employed without departing from the spirit and scope thereof. For example, multiple lasers might be employed to conduct multiple simultaneous measurements in order to improve the speed and efficiency of the measurement process. The terms and expressions employed herein have been used as terms of description and not of limitation; and thus, there is no intent of excluding equivalents, but on the contrary it is intended to cover any and all equivalents that may be employed without departing from the spirit and scope of the invention.