Abstract:
Disclosed is a method and apparatus for measuring semiconductor substrate temperature using a differential acoustic time of flight measurement technique. The measurement is based on measuring the time of flight of acoustic (ultrasonic) waves across the substrate, and calculating a substrate temperature from the measured time of flight and the known temperature dependence of the speed of sound for the substrate material. The differential acoustic time of flight method eliminates most sources of interference and error, for example due to varying coupling between an ultrasonic transducer and the substrate. To further increase the accuracy of the differential acoustic time of flight measurement, a correlation waveform processing algorithm is utilized to obtain a differential acoustic time of flight measurement from two measured ultrasonic waveforms. To facilitate signal recognition and processing, a symmetric Lamb mode may be used as mode of excitation of the substrate.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application is based on and claims the benefit of and priority to U.S. Provisional Patent Application No. 61/879,552, entitled “DIFFERENTIAL TIME OF FLIGHT MEASUREMENT OF ACOUSTIC WAVES FOR SEMICONDUCTOR WAFERS”, filed on Sep. 18, 2013, the entire contents of which are herein incorporated by reference. 
    
    
     BACKGROUND OF THE INVENTION 
     Field of the Invention 
     The present invention relates to a method and apparatus for measuring the temperature of a semiconductor substrate, particularly in-situ during semiconductor processing. 
     Description of Related Art 
     Production of semiconductor devices, displays, photovoltaics, etc., proceeds in a sequence of steps, each step having parameters optimized for maximum device yield. Among the controlled parameters strongly affecting yield is the temperature of the substrate from which devices are formed, because temperature strongly affects the rate of and outcome of a processing step. While ensuring that the temperature of the substrate is within limits for each processing step, it is also equally important to maintain temperature steady over time, i.e. from substrate to substrate, and substrate lot to substrate lot, to prevent process drift. It is also very important to maintain uniformity of temperature across the substrate during each processing step, such that properties of devices do not vary considerably from one region of the substrate to another. 
     The goal of maintaining control of substrate temperature, and its uniformity across the substrate and over multiple substrates requires monitoring of substrate temperature during processing, preferably across multiple locations on the substrate. Active monitoring of substrate temperature is frequently complicated by the fact that processing occurs in harsh and unfavorable environments. For example, in situ temperature measurement devices need to be unaffected by the aggressive chemistries and environments (e.g. plasma) sometimes used in semiconductor processing. In plasma processing environments, strong RF coupling from the RF excitation method used to drive a plasma in the plasma processing system can lead to noisy and erroneous temperature measurements due to induced currents in unshielded or poorly-shielded temperature sensor circuits. Some temperature measurement methods have sought to solve these issues by placing temperature sensors inside the substrate support, but such a measurement is further complicated by the fact that the substrate is seldom in good thermal contact with the substrate support, so the reading of the temperature sensors embedded within the substrate support is rarely accurate due to the temperature difference (i.e. “jump”) between the substrate and the substrate support. Attempts to directly measure the substrate temperature have typically involved some sort of single or multi-point optical temperature measurement system installed inside the processing chamber, but such system also have their shortcomings, such as the tendency of optical components to get coated with processing byproducts adhering to the wall of the processing chamber, thus affecting measurement accuracy; and also high cost. 
     Therefore, there still exists a need for a robust and inexpensive system and associated method for measuring a temperature and temperature distribution of a substrate itself, during processing. Direct ultrasonic measurement of substrate temperature, particularly as described hereinafter, addresses most of the aforementioned concerns. 
     SUMMARY OF THE INVENTION 
     An aspect of the invention includes a method for determining the temperature of a substrate, comprising disposing the substrate on a substrate support; contacting the substrate with a first ultrasonic transducer; contacting the substrate with a second ultrasonic transducer; exciting a Lamb wave in the substrate by energizing the first ultrasonic transducer and the second ultrasonic transducer; measuring at the second ultrasonic transducer a first ultrasonic waveform of the Lamb wave originating at the first ultrasonic transducer and propagating along a first path from the first ultrasonic transducer to the second ultrasonic transducer; measuring at the second ultrasonic transducer a second ultrasonic waveform of the Lamb wave originating from the first ultrasonic transducer and propagating along a second path from the first ultrasonic transducer to the second ultrasonic transducer, wherein the second path is different than the first path; calculating a difference of times of flight of the second ultrasonic waveform and the first ultrasonic waveform, and determining the substrate temperature from the calculated difference of times of flight and from a known temperature dependence of the speed of sound for the substrate material. The second path is generally longer than the first path, and can include at least one reflection from the substrate edge, while the ultrasonic transducers can contact the substrate in many different configurations, including diametrally-opposed, next to each other but not touching each other, etc. Another aspect of the invention is that the Lamb wave can be symmetric. 
     Yet another aspect of the invention is related to the calculation of the difference of times of flight comprising converting the first ultrasonic waveform into a first wavelet representation; converting the second ultrasonic waveform into a second wavelet representation, and determining the difference of times of flight from a calculated correlation of the first and second wavelet representations. 
     Further aspects of the invention include a controller for controlling and powering the ultrasonic transducers, measuring ultrasonic waveforms, calculating the difference of times of flight of ultrasonic waveforms, and further calculating the substrate temperature from the difference of times of flight. Another aspect includes actuators for selectably bringing into contact with, or withdrawing from the substrate, the ultrasonic transducers. 
     Yet another aspect of the invention includes a method for determining the temperature distribution across a substrate, comprising disposing the substrate on a substrate support; contacting the substrate with a plurality of ultrasonic transducers; measuring a plurality of substrate temperatures in a plurality of substrate zones, each temperature measurement in each substrate zone comprising: selecting a substrate zone in which the substrate temperature is to be measured; exciting a Lamb wave in the substrate by energizing a selected pair of ultrasonic transducers from the plurality of ultrasonic transducers; measuring at a second ultrasonic transducer of the selected pair of ultrasonic transducers a first ultrasonic waveform of the Lamb wave originating at a first ultrasonic transducer of the selected pair of ultrasonic transducers and propagating along a first path from the first ultrasonic transducer of the selected pair of ultrasonic transducers to the second ultrasonic transducer of the selected pair of ultrasonic transducers; measuring at the second ultrasonic transducer of the selected pair of ultrasonic transducers a second ultrasonic waveform of the Lamb wave originating from the first ultrasonic transducer of the selected pair of ultrasonic transducers and propagating along a second path from the first ultrasonic transducer of the selected pair of ultrasonic transducers to the second ultrasonic transducer of the selected pair of ultrasonic transducers, wherein the second path is different than the first path; calculating a difference of times of flight of the second ultrasonic waveform and the first ultrasonic waveform; determining a temperature of the selected substrate zone from the calculated difference of times of flight and from a known temperature dependence of the speed of sound for the substrate material. Another aspect of the invention is that the Lamb wave can be symmetric. Yet another related aspect includes selecting a functional form for the temperature distribution of the substrate, the selected functional form having a plurality of floating parameters; and calculating the plurality floating parameters by fitting the plurality of substrate temperatures measured in the plurality of substrate zones, so a temperature distribution of the substrate is obtained. The fitting can be done using algorithms such as regression, or using tomographic inversion algorithms. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       A more complete appreciation of the invention and many of the attendant advantages thereof will become readily apparent with reference to the following detailed description, particularly when considered in conjunction with the accompanying drawings, in which: 
         FIG. 1  is a schematic of the displacement field induced in a substrate in which a symmetric Lamb wave is excited, for example, by ultrasonic transducers. 
         FIG. 2  is a schematic of the substrate temperature measurement system in accordance with an embodiment of the invention. 
         FIG. 3A  is a schematic of locations of ultrasonic transducers relative to the substrate, and the resultant ultrasonic wave propagation pattern, in accordance with an embodiment of the invention. 
         FIG. 3B  is a schematic of locations of ultrasonic transducers relative to the substrate, and the resultant ultrasonic wave propagation pattern, in accordance with another embodiment of the invention. 
         FIG. 4  is a graph of ultrasonic waveforms measured by a method in accordance with an embodiment of the invention. 
         FIG. 5  is a graph of the correlation of wavelet representations of two measured ultrasonic waveforms, used to determine the difference of times of flight of ultrasonic waveforms in accordance with an embodiment of the invention. 
         FIG. 6  is a graph comparing substrate temperature measured using an embodiment of the invention compared to measurements using thermocouples. 
         FIG. 7A  is an exemplary flowchart of a method for substrate temperature measurement in accordance with an embodiment of the invention. 
         FIG. 7B  is an exemplary flowchart of a method for determining the difference of times of flight of ultrasonic waveforms in accordance with an embodiment of the invention. 
         FIG. 8  is a schematic of locations of a plurality of ultrasonic transducers to be used in conjunction with a method for determination of the substrate temperature distribution, in accordance with another embodiment of the invention. 
         FIG. 9  is an exemplary flowchart of a method for determining the substrate temperature distribution in accordance with another embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     In the following description, in order to facilitate a thorough understanding of the invention and for purposes of explanation and not limitation, specific details are set forth, such as particular geometries of a substrate temperature measurement system, and descriptions of various components and processes. However, it should be understood that the invention may be practiced in other embodiments that depart from these specific details. 
     In the description to follow, the terms radiation-sensitive material and photoresist may be used interchangeably, photoresist being only one of many suitable radiation-sensitive materials for use in photolithography. Similarly, hereinafter the term substrate, which represents the workpiece being processed, may be used interchangeably with terms such as semiconductor wafer, LCD panel, light-emitting diode (LED), photovoltaic (PV) device panel, etc., the processing of all of which falls within the scope of the claimed invention. 
     Reference throughout this specification to “one embodiment” or “an embodiment” means that a particular feature, structure, material, or characteristic described in connection with the embodiment is included in at least one embodiment of the invention, but do not denote that they are present in every embodiment. Thus, the appearances of the phrases “in one embodiment” or “in an embodiment” in various places throughout this specification are not necessarily referring to the same embodiment of the invention. Furthermore, the particular features, structures, materials, or characteristics may be combined in any suitable manner in one or more embodiments. 
     Various operations will be described as multiple discrete operations in turn, in a manner that is most helpful in understanding the invention. However, the order of description should not be construed as to imply that these operations are necessarily order dependent. In particular, these operations need not be performed in the order of presentation. Operations described may be performed in a different order than the described embodiment. Various additional operations may be performed and/or described operations may be omitted in additional embodiments. 
     Ultrasonic substrate temperature measurement systems operate on the principle of measuring the speed of propagation of sound in the substrate, which is depends on the substrate temperature. A change of substrate temperature thus causes a measurable change of speed of propagation of sound. The coefficient of temperature dependence of speed of sound in a metallic or semiconductor substrate is of the order of 10 −5 . This is a small value, but advances in signal processing, both in terms of hardware and software algorithms have made it feasible to measure the small change of speed of sound induced by temperature change in a substrate. The ability for a system and method to accurately measure speed of sound, and thus substrate temperature, depends on the ability to accurately determine the time of flight that an ultrasonic waveform takes to traverse the substrate from one transducer to another. 
     Acoustic waves induced in a thin plate-like substrate whose thickness is much smaller than the acoustic wavelength are called Lamb waves. One of the problems encountered in prior art ultrasonic temperature measurement systems stems from the strongly dispersive nature of Lamb waves induced in a thin substrate by a typical prior art ultrasonic excitation and measurement configuration wherein one transducer is used to excite the Lamb wave, and another transducer, generally located on the opposite side of the substrate is used to measure the acquired waveform and determine the time of flight of the ultrasonic wave, from which the speed of sound, and thus temperature can both be determined. The type of Lamb wave induced by this configuration is called a antisymmetric Lamb wave, and its strongly dispersive nature, i.e. tendency for various frequency components of the wave to propagate at different velocities means that by the time the waveform reaches the measuring transducer, the waveform is itself modified by dispersion, sometimes beyond recognition, which causes a lot of difficulty for accurate determination of time of flight, which is key to accurate temperature measurement. 
     The inventors have overcome this limitation by inducing another form of Lamb wave, i.e. a symmetric Lamb wave, whose displacement field is schematically depicted in  FIG. 1 . As can be seen from  FIG. 1 , the displacement field of the symmetric Lamb wave in a substrate is symmetric with respect to the central plane of the thin substrate, the central plane being parallel to the flat substrate sides. Moreover, the dispersion of a symmetric Lamb wave is significantly lower than the dispersion of an antisymmetric Lamb wave, making it particularly suitable for ultrasonic temperature measurement, because propagating waveforms tend to maintain their shape better, making it easier to recognize them at the receiving transducer and thus making the time of flight measurement easier. The symmetric nature of the wave further implies that two ultrasonic transducers are needed to excite a symmetric Lamb wave in the substrate, unlike the prior art where only a single transducer is used to excite the wave. 
       FIG. 2  shows a schematic of an substrate temperature measurement system  200  in accordance with an embodiment of the invention. A substrate  100  is mounted generally atop a substrate support  205 , such as an electrostatic chuck. In other embodiments, the substrate may be supported by edge supports, or by a plurality of lift pins, depending on the configuration of the processing system in which the substrate temperature measurement system  200  is used. At least two transducers  220  contact the edge of the substrate  100  via buffer rods  210 , to excite a symmetric Lamb wave in substrate  100 . Buffer rods  210  can be made or quartz, for example. To allow placement and removal of substrate  100  from substrate support  205 , actuators  230  can be optionally used to selectably bring into contact and withdraw from the edge of substrate  100 , the transducers  220  and buffer rods  210 . In other embodiments, transducers  220  can be mounted in a position to contact other portions of the substrate  100 , such as the lower side of substrate  100  (not shown). Transducers  220  are also used to measure ultrasonic waveforms. To facilitate signal recognition and processing, transducers  220  may be operated in pulsed mode. 
     A controller  250  is connected to the transducers  220  and optional actuators  230  for controlling and powering the transducers  220  and actuators  230 , and also to acquire and measure ultrasonic waveforms from transducers  220 . The controller  250  is also configured to do signal processing on all measured ultrasonic waveforms, to determine the times of flight, and to determine the substrate temperature from the time of flight data. 
       FIGS. 3A and 3B  show two different configurations of ultrasonic transducers contacting the edge of the substrate  100 . The configuration in  FIG. 3A  will be used now to explain another important aspect of the invention, i.e. differential time of flight measurement. 
     In prior art systems, time of flight measurements are typically made from the instant of introducing excitation at a first ultrasonic transducer to the time when the acoustic wave has reached the second ultrasonic transducer where the signal is received. This configuration has the disadvantage that there is no way to factor out the effects on the time of flight measurement introduced by time of flight along buffer rods  210  and due to any variations in coupling of ultrasonic transducers to the substrate, for example. Temperature variations during substrate processing can themselves induce significant variations in the coupling and the time of flight along the coupling hardware, which in the aforementioned setup are impossible to remove from the acquired signal. Other factors that influence the time of flight measurement include mechanical effects such as the repeatability of contact of transducers and the substrate, and the effects of substrate chucking, as when electrostatic chucks  205  are used. 
     To overcome these issues and make the time of flight measurement as insensitive to these external factors, as possible, the inventors utilize differential time of flight measurements. In a differential time of flight measurement, at least two actual time of flight measurements are made. In the exemplary configuration in  FIG. 3A , where the two ultrasonic transducers T and R are spaced apart in diametrally-opposing locations with respect to the substrate  100 , the first time of flight measurement would be taken of the time it takes the acoustic wave originating at ultrasonic transducer T to reach ultrasonic transducer R along the shortest, diametral path  310 . In ultrasonic measurements, multiple reflections of the acoustic wave are typically present depending on the geometric configuration of the sample, i.e. the substrate  100 . For example, one particularly strong reflected acoustic wave is associated with propagation sideways from ultrasonic transducer T, reflecting off of the substrate edge, and approaching the receiving ultrasonic transducer R sideways, which is depicted in  FIG. 3A  as path  320 . Since the same ultrasonic transducers T and R are used to determine the time of flight along both paths, at the same instant and using the same excitation, the two time of flight measurements along both paths  310  and  320  are equally affected by coupling variations, temperature variations, substrate chucking, etc. Therefore, if the time of flight along path  310  is subtracted from the longer time of flight along path  320 , the resultant differential time of flight measurement has all coupling variations, temperature variations, and other effects subtracted out, and the resultant differential time of flight measurement is only dependent on the known difference in lengths of paths  320  and  310 , which is a known geometrical quantity, and the speed of sound in the material of which the substrate  100  is made, which is also known, and is directly related to the temperature of the substrate along paths  310  and  320 . 
       FIG. 3B  shows an alternate embodiment in which ultrasonic transducers T and R are located in close proximity to each other and contact the edge of substrate  100  without touching each other. In this case, the shortest path  310  is directly from ultrasonic transducer T to ultrasonic transducer R, while the longer reflected path  320  involves traversing the substrate  100  twice along a substantially diametral path. Again, as in  FIG. 3A , the effects of coupling variations, temperature variations, chucking, etc., are subtracted out, while the differential time of flight measurement substantially depends only on the length of path  320  and the speed of sound, and thus substrate temperature along path  320 . 
       FIG. 4  shows an typical oscilloscope trace  400  of the measured pulsed ultrasonic waveform acquired at e.g. ultrasonic transducer T of  FIG. 3A or 3B . The first waveform  410  corresponds to propagation of the symmetric Lamb wave along the shorter path  310  of  FIG. 3A or 3B , while the second waveform  420  corresponds to propagation of the symmetric Lamb wave along the longer reflected path  320  of  FIG. 3A or 3B . The difference of times of flight t R  is directly related to the difference of lengths of paths  320  and  310 , and the speed of sound, and thus temperature along paths  320  and  310 , as explained before. 
     An accurate measurement of the difference of times of flight t R  represents a challenge in prior art ultrasonic temperature measurement systems, because the use of an antisymmetric Lamb wave excitation, with the highly dispersive nature of the propagating acoustic wave causes the longer-propagating waveform  420  to lose all resemblance to waveform  410 , making signal recognition difficult, and rendering any measurements of the difference of times of flight t R  inaccurate. 
     However, with the symmetric Lamb wave excitation in accordance with an embodiment of the present invention, the low dispersion of the symmetric Lamb wave ensures that waveforms  410  and  420  are substantially similar and thus easily recognizable, which eases signal acquisition and processing. For example, the difference of times of flight t R  can be determined, in one embodiment, by simply subtracting the times of first zero crossings (i.e. first zero amplitudes) of the two waveforms  420  and  410 . However, the inventors have found a more accurate way of determining the difference of times of flight t R , using wavelet transformation. Because the waveforms  410  and  420  are similar due to the substantially nondispersive nature of the symmetric Lamb wave, they can be converted to a wavelet representation using a same previously acquired seed waveform. Once converted to wavelet representations, the process of determining the difference of times of flight t R  involves calculating the correlation of the wavelet representations of the first waveform  410  and the second waveform  420 , wherein the peak of the correlation corresponds to the time “shift”, i.e. the difference of times of flight t R  of the two waveforms  410  and  420 . An exemplary correlation  500  is depicted in the graph in  FIG. 5 , where the peak  510  is located at the difference of times of flight t R , which is the quantity that needs to be determined so the speed of sound and thus substrate temperature can be calculated from the known geometry of the propagation paths and the known relationship between the speed of sound and temperature for the material of which the substrate is made. 
       FIG. 6  shows a correlation of substrate temperature measured with the differential time of flight method T TOF  and substrate temperature measured using a thermocouple T TC  (trace  610 ). A straight line fit of the correlation, trace  620 , is also included to demonstrate the linearity of the correlation over a relatively large span of substrate temperatures, from 20 to 160° C. 
     The differential acoustic time of flight method of substrate temperature measurement can be used with both antisymmetric and symmetric Lamb wave excitation, though obviously the latter is preferred due to low dispersion and thus easier signal recognition, but the former may be used as well if the waveforms can be recognized. Furthermore, the method can also be used in mixed-mode excitation cases, where for example, the difference of times of flight of a symmetric Lamb wave and an antisymmetric Lamb wave, both propagating along the same or different acoustic paths, is determined, and from that quantity the substrate temperature can be determined as well. 
     All of the functions required for determination of the difference of times of flight and the substrate temperature can be implemented in software embedded in controlled  250  of the substrate temperature measurement system  200 , of  FIG. 2 , for example. 
       FIG. 7A  now summarizes an exemplary process of  700  of measuring the substrate temperature in accordance with an embodiment of the invention. In step  710 , a substrate is disposed on a substrate support, such as an electrostatic chuck  205  of  FIG. 2 , a mechanical chuck, or alternatively a plurality of substrate edge supports or a plurality of lift pins. 
     In steps  715  and  720 , the substrate  100  is contacted with a first and second ultrasonic transducers, such as transducers  220  of  FIG. 2 , and T and R of  FIGS. 3A and 3B . 
     In step  725 , a symmetric Lamb wave is excited in the substrate  100  using ultrasonic transducers in contact with the substrate. 
     In steps  730  and  735 , the first and second ultrasonic waveforms, such as waveforms  410  and  420  of  FIG. 4 , are measured using ultrasonic transducers in contact with the substrate. The first and second ultrasonic waveforms represent the symmetric Lamb propagating along different paths in substrate  100 , as is shown in exemplary configurations depicted in  FIGS. 3A and 3B . 
     In step  740 , the difference of times of flight of the first and second waveforms is calculated from the measured waveforms. 
     The method concludes at step  745 , in which the difference of time of flight is converted to a temperature measurement, using known wave propagation lengths along paths  310  and  320 , of  FIGS. 3A and 3B , for example, and using the known temperature dependence of the speed of sound for the material of the substrate. 
       FIG. 7B  expands the step  740  of calculating the difference of times of flight, in accordance with an embodiment of the invention, to include the steps  741  and  742 , of converting the first and second waveform into respective wavelet transformations, followed by calculation of the correlation of the first and second wavelets, from which the difference of times of flight t R  is determined by peak detection 
     As was previously mentioned, determining the temperature distribution in a substrate is of particular importance because it allows an assessment to be made of the uniformity of substrate processing, and allows corrective action to be taken, increasing device yield from the same substrate. 
     The previously-described embodiments for measuring a single substrate temperature can be readily extended to measure substrate temperature across a plurality of substrate zones, which measurements can in turn be fitted to a predetermined functional form for the temperature distribution, for example, yielding a full temperature distribution across the entire substrate (or portion thereof). 
       FIG. 8  shows a schematic of an exemplary embodiment of a temperature distribution measurement system and method. The substrate  100  is contacted by a plurality of ultrasonic transducers  810 A,  810 B,  810 C, . . .  810 J, at various locations along the circumference of the substrate  100 . Locations of contact for ultrasonic transducer are selected to allow multiple temperature measurements to be made along paths  820 A,  820 B,  820 C,  820 D, and  820 E, traversing the substrate  100  which is subdivided into zones A, B, C, D, and E. Zones A, B, C, and D roughly correspond to paths  820 B,  820 C,  820 D, and  820 E. The path  820 A traverses multiple zones D, E, and B, so the temperature measurement in zone A can be determined from the temperature measurement along path  820 A, using previously-determined temperatures of zones B and D in a weighted average, or similar scheme. Once the temperatures of all individual zones A, B, C, D, and E are known, the values of the temperatures can be fitted to a previously selected functional form for the temperature, which functional form involves floating parameters, i.e. coefficients, which can be fitted using a method such as regression, yielding a temperature distribution function for substrate  100 , which can be evaluated at any location, i.e. coordinate pair, across substrate  100 . Suitable functions include various polynomials and function suitable for fitting to a circularly-symmetric geometry, such as Zernike polynomials. 
     In the exemplary embodiment in  FIG. 8 , ten ultrasonic transducers and five zones are used. In other embodiments, a larger or smaller number of ultrasonic transducers and a larger or smaller number of zones can be used, depending on the needed accuracy of the measurements. 
     In a further embodiment of the invention, tomographic inversion can be used to establish the temperature distribution from known temperatures measured along paths A, B, C, D, and E of the exemplary embodiment of  FIG. 8 . Tomographic inversion may involve the use of the Radon transform, or other similar algorithm to establish the full temperature distribution. 
       FIG. 9  now summarizes an exemplary process of  900  of measuring the substrate temperature distribution in accordance with an embodiment of the invention. In step  910 , a substrate is disposed on a substrate support, such as an electrostatic chuck  205  of  FIG. 2 , a mechanical chuck, or alternatively a plurality of substrate edge supports or a plurality of lift pins. 
     In step  915 , the substrate  100  is contacted by a plurality of ultrasonic transducers, as shown in the exemplary embodiment of  FIG. 8 . 
     In step  920 , a pair of ultrasonic transducers is selected, for example, a pair of ultrasonic transducers corresponding to a desired path A, B, C, D, or E of the exemplary embodiment of  FIG. 8 . 
     In step  925 , a symmetric Lamb wave is excited in the substrate  100  using only the selected pair of ultrasonic transducers. 
     In steps  930  and  935 , the first and second ultrasonic waveforms, such as waveforms  410  and  420  of  FIG. 4 , are measured using the selected pair of ultrasonic transducers. The first and second ultrasonic waveforms represent the symmetric Lamb propagating along different paths in substrate  100 , which paths connect the selected pair of transducers. 
     In step  940 , the difference of times of flight of the first and second waveforms is calculated from the measured waveforms. Optionally, wavelet representations and correlation of wavelet representation can be used as part of this step, in accordance with the flowchart of  FIG. 7B . 
     In step  945 , the difference of time of flight is converted to a temperature measurement for a zone defined by the path between the selected pair of ultrasonic transducers, using known wave propagation lengths, and using the known temperature dependence of the speed of sound for the material of the substrate. Given that the temperature measurement is made in a zoned manner, this allows the possibility of variation of the temperature dependence of the speed of sound across various zones to be accounted for, if such a variation exists. In most case, the temperature dependence of the speed of sound will be the same across the entire substrate  100 . 
     In step  950 , a determination is made if all zonal temperature measurements have been made, and if so, the method proceeds to step  955 . If not, the method proceeds again with step  920  in which a new pair of ultrasonic transducers is selected, and a new zonal temperature measurement is made. 
     In step  955 , the method proceeds with optionally selecting a functional forms for the substrate temperature distribution. This, and subsequent steps are optional because there are situations where individual zonal temperature measurements suffice to define the temperature distribution in of substrate  100 , and a full functional form is unnecessary. 
     In step  960 , following the selection of the functional form  955 , the functional form is fitted to the zonal temperature measurements, and floating parameters or coefficients of the selected functional form are determined by e.g. regression, to determine the final functional form of the substrate temperature distribution. 
     An alternative to steps  955  and  960  would involve the use of tomographic inversion to determine the substrate temperature distribution. 
     Persons skilled in the relevant art can appreciate that many modifications and variations are possible in light of the above teaching. Persons skilled in the art will recognize various equivalent combinations and substitutions for various components shown in the figures. It is therefore intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto.