Patent Publication Number: US-9405290-B1

Title: Model for optical dispersion of high-K dielectrics including defects

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     The present application for patent claims priority under 35 U.S.C. §119 from U.S. provisional patent application Ser. No. 61/753,855, entitled “Model For Optical Dispersion of High-K Dielectrics Including Defects,” filed Jan. 17, 2013, the subject matter of which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The described embodiments relate to systems for wafer inspection, and more particularly to characterization and defect detection of thin films used in semiconductor manufacturing. 
     BACKGROUND INFORMATION 
     Semiconductor devices such as logic and memory devices are typically fabricated by a sequence of processing steps applied to a substrate or wafer. The various features and multiple structural levels of the semiconductor devices are formed by these processing steps. For example, lithography among others is one semiconductor fabrication process that involves generating a pattern on a semiconductor wafer. Additional examples of semiconductor fabrication processes include, but are not limited to, chemical-mechanical polishing, etch, deposition, and ion implantation. Multiple semiconductor devices may be fabricated on a single semiconductor wafer and then separated into individual semiconductor devices. 
     Inspection processes are used at various steps during a semiconductor manufacturing process to detect defects on wafers to promote higher yield. As design rules and process windows continue to shrink in size, inspection systems are required to capture a wider range of physical defects on wafer surfaces while maintaining high throughput. 
     Semiconductor devices are increasingly valued based on their energy efficiency, rather than speed alone. For example, energy efficient consumer products are more valuable because they operate at lower temperatures and for longer periods of time on a fixed battery power supply. In another example, energy efficient data servers are in demand to reduce their operating costs. As a result, there is a strong interest to reduce the energy consumption of semiconductor devices. 
     Leakage current through insulator layers is a major energy loss mechanism of semiconductor devices manufactured at the 65 nm technology node and below. In response, electronic designers and manufacturers are adopting new materials (e.g., hafnium silicate (HfSiO4), nitrided hafnium silicates (HfSiON), hafnium dioxide (HfO2), zirconium silicate (ZrSiO4), etc.) with higher dielectric constants than traditional materials (e.g., silicon dioxide). These “high-k” materials reduce leakage current and enable the manufacture of smaller sized transistors. 
     Along with the adoption of new dielectric materials, the need has arisen for measurement tools to characterize the dielectric properties and band structures of high-k materials early in the manufacturing process. More specifically, high throughput monitoring tools are required to monitor and control the deposition of high-k materials during wafer manufacture to ensure a high yield of finished wafers. Early detection of deposition problems is important because the deposition of high-k materials is an early process step of a lengthy and expensive manufacturing process. In some examples, a high-k material is deposited on a wafer at the beginning of a manufacturing process that takes over one month to complete. 
     In some examples, device performance over time is influenced by charge trapping centers located in high-K dielectric layers of gate dielectric stacks. Charge trapping centers are caused by structural or material imperfections, interface states, and other defect states. Spectroscopic ellipsometry (SE) is a non-invasive characterization technique suitable for identifying process-induced defects, such as charge trapping centers, during device fabrication. In high-throughput measurement applications, the SE measurement technique includes a parametric representation of a measured optical dispersion. The particular parameterization is selected to reduce the number of unknown parameters and decrease correlations among parameters. 
     In some examples, the modeled optical response of one or more high-K dielectric layers is based on a harmonic oscillator model. In principle, the harmonic oscillator model is capable of representing defect states. However, this model does not work for amorphous materials including high-K dielectrics. Moreover due to an indirect connection between model parameters and meaningful physical values (e.g., defect activation energy, number of defects etc.) the harmonic oscillator model is limited in its ability to effectively represent defect states in high-K layers. 
     In some other examples, a Tauc-Lorentz model or a Cody-Lorentz model is employed as described by way of example in A. S. Ferlauto et al., “Analytical model for the optical functions of amorphous semiconductors from the near-infrared to ultraviolet: Application in thin film photovoltaics,” J. Appl. Phys. 92, 2424 (2002), the subject matter of which is incorporated herein in its entirety. In these models, the imaginary part of the dielectric function is represented by a parameterized dispersion function, and the real part of the dielectric function is determined based on enforcement of Kramers-Kronig consistency. Model parameters (e.g., optical function parameters and thicknesses) are evaluated by fitting modeled spectra to measured spectra by numerical regression. The validity and limitations of the models are assessed by statistical evaluation of fitting quality and confidence limits of model parameters. 
     The Tauc-Lorenz and Cody-Lorentz models have been successfully applied to measurements of defect-free, high-K dielectric layers. However, these models are limited in their ability to characterize defects such as charge trapping centers. Defect states are evident in optical and transport measurements of high-K dielectric layers. However, the Tauc-Lorentz model and the Cody-Lorentz model, as presently constructed, do not sufficiently represent such states. Moreover, the Tauc-Lorentz model is unable to account for low energy absorption tails characteristic of the amorphous materials. In one example, described in N. V. Nguyen et al., “Sub-bandgap defect states in polycrystalline hafnium oxide and their suppression by admixture of silicon,” APL 87, 192903 (2005) and N. V. Nguyen et al., “Optical properties of Jet-Vapor-Deposited TiAlO and HfAlO determined by Vacuum Ultraviolet Spectroscopic Ellipsometry,” AIP Conf. Proc. 683, 181 (2003), the sum of three Tauc-Lorentz functions is used to describe near band-edge defects in HfO2 layers. However, these functions do not describe sharp middle gap peaks noticeable in the absorption spectra of high-K film stacks. 
     In some other examples, the optical response of one or more high-K dielectric layers is predicted based on a direct inversion method. These methods are described by way of example in J. Price et al., “Identification of interfacial defects in high-k gate stack films by spectroscopic ellipsometry,” J. Vac. Sci. Technol. B 27 (1), 310 (2009) and J. Price et al., “Identification of sub-band-gap absorption features at the HfO2/Si(100) interface via spectroscopic ellipsometry,” APL 91, 061925 (2007), the subject matter of each is incorporated herein in its entirety. Such methods have traditionally been employed when defects have a noticeable contribution to the optical response of the high-K layers. However, direct inversion methods are computationally burdensome, very sensitive to statistical measurement errors, and do not provide a physically based model of the measured structure (i.e., the optical functions do not satisfy the Kramers-Kronig consistency condition). As a result, the utility of direct inversion methods for high-throughput inspection and process control is limited. 
     Accordingly, it would be advantageous to develop high throughput methods and/or systems for characterizing high-k dielectric layers early in the manufacturing process. In particular, it would be advantageous to develop a robust, reliable, and stable approach to in-line SE metrology of gate stacks including high-K dielectrics. 
     SUMMARY 
     Methods and systems for determining band structure characteristics of high-k dielectric films deposited over a substrate based on spectral response data are presented. Optical models of semiconductor structures such as gate dielectric stacks are presented. In particular, models capable of accurate characterization of defects in high-K dielectric layers and nanostructures based on high-K dielectric materials are described. These models quickly and accurately represent experimental results in a physically meaningful manner that can be subsequently used to gain insight and control over a manufacturing process. 
     In one aspect, the selected dispersion model includes a generalized Cody-Lorentz model to describe the complex bulk band structure of high-K dielectric layer augmented with additional Lorentz peaks to describe defects, interface states, or excitonic states. In one further aspect, the dielectric function includes one or more Lorentz functions to account for one or more defects across the energy range. By way of non-limiting example, middle gap defects, interface states, or sharp band-edge excitonic transitions may each be described by one or more Lorentz functions. In another further aspect, a Lorentz function modulated by a band gap function includes a summation of multiple Lorentz oscillator functions. 
     In a further aspect, a band structure characteristic indicative of an electrical performance of the measured layer, or stack of layers, is determined based at least in part on the parameter values of the optical dispersion model of the multi-layer semiconductor wafer. 
     In another further aspect, device performance is improved by controlling a process of manufacture of the semiconductor wafer based at least in part on the identified band structure characteristic. In one example, charge trapping centers may be controlled based on band structure characteristics identified from the parameter values of the optical dispersion model. 
     The foregoing is a summary and thus contains, by necessity, simplifications, generalizations, and omissions of detail; consequently, those skilled in the art will appreciate that the summary is illustrative only and is not limiting in any way. Other aspects, inventive features, and advantages of the devices and/or processes described herein will become apparent in the non-limiting detailed description set forth herein. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a simplified diagram illustrative of a wafer inspection system  100  including thin film characterization functionality. 
         FIG. 2  is a simplified diagram illustrative of a semiconductor substrate  112  with attached thin film layers  114 A and  114 B that may be characterized by methods and systems as described herein. 
         FIG. 3  is a flowchart illustrative of a method  200  of determining parameter values of an augmented Cody-Lorentz model from spectral response data. 
         FIG. 4  is a plot illustrative of an exemplary representation of the imaginary part of the dielectric function of a thin film sample having defects. 
         FIG. 5  is a plot illustrative of an exemplary representation of the real part of the dielectric function of a thin film sample having defects. 
     
    
    
     DETAILED DESCRIPTION 
     Reference will now be made in detail to background examples and some embodiments of the invention, examples of which are illustrated in the accompanying drawings. 
     In one aspect, optical models of semiconductor structures such as gate dielectric stacks are presented. In particular, models capable of accurate characterization of defects in high-K dielectric layers and nanostructures based on high-K dielectric materials are presented. These models quickly and accurately represent experimental results in a physically meaningful manner that can be subsequently used to gain insight and control over a manufacturing process. 
       FIG. 1  illustrates a system  100  for measuring a spectral response of a thin film of a semiconductor wafer, in accordance with one embodiment of the present invention. As shown in  FIG. 1 , the system  100  may be used to perform spectroscopic ellipsometry on one or more films  114  of a semiconductor wafer  112  disposed on a translation stage  110 . In this aspect, the system  100  may include a spectroscopic ellipsometer equipped with an illuminator  102  and a spectrometer  104 . The illuminator  102  of the system  100  is configured to generate and direct illumination of a selected wavelength range (e.g., 150-850 nm) to the thin film (e.g., HfSiON thin film) disposed on the surface of the semiconductor wafer  112 . In turn, the spectrometer  104  is configured to receive illumination reflected from the surface of the semiconductor wafer  112 . It is further noted that the light emerging from the illuminator  102  is polarized using polarizer  107  to produce a polarized illumination beam  106 . The radiation reflected by the thin film  114  disposed on the wafer  112  is passed through an analyzer  109  and to the spectrometer  104 . In this regard, the radiation received by the spectrometer  104  in the collection beam  108  is compared to the incident radiation of the illumination beam  106 , allowing for spectral analysis of the thin film  114 . 
     In a further embodiment, the system  100  may include one or more computing systems  116 . The one or more computing systems  116  may be communicatively coupled to the spectrometer  104 . In one aspect, the one or more computing systems  116  may be configured to receive a set of spectral measurements performed by the spectrometer  104  on one or more wafers. Upon receiving results of the one or more sampling process from the spectrometer, the one or more computing systems  116  may then calculate parameters of an optical dispersion model. In this regard, the computing system  116  may extract the real component (n) and the imaginary component (k) of the complex index of refraction of the thin film across the selected spectral range (e.g., 150-850 nm) for the acquired spectrum from the spectrometer  104 . Further, the computing system  116  may extract the n- and k-curves utilizing a regression process (e.g., ordinary least squares regression) applied to a selected dispersion model. In a preferred embodiment, the selected dispersion model is an augmented Cody-Lorentz model as described herein. 
     In a further embodiment, the computing system  116  may determine a band structure characteristic indicative of a defect of the film  114  based on parameter values of the optical dispersion model. For example, the computing system  116  may be configured to automatically identify defects from parameter values of the optical dispersion model. 
     In another further embodiment, the computing system  116  may control a process of manufacture a semiconductor wafer based at least in part on a band structure characteristic determined from parameter values of the optical dispersion model. For example, computing system  116  may be configured to communicate process control parameter values to one or more manufacturing tools responsible for the manufacture of the semiconductor wafers being measured. 
     As illustrated in  FIG. 2 , in some embodiments, an intermediate layer  114 B is located between a semiconductor substrate  112  (e.g., silicon) and a high-k insulative layer  114 A to promote adhesion between the high-k material and the semiconductor substrate. Typically, the intermediate layer  114 B is very thin (e.g., ten Angstroms). In some examples, the high-k insulative layer  114 A and the intermediate layer  114 B are modeled together as one layer for purposes of analysis employing the methods and systems as described herein. In this example, the one or more computing systems  116  may determine one or more parameters of an optical dispersion model of the film layer  114  including both the intermediate layer  114 B and high-k insulative layer  114 A. However, in some other examples, each layer may be modeled separately. In this example, the one or more computing systems  116  may determine one or more parameters of an optical dispersion model of the high-k insulative layer  114 A and one or more parameters of an optical dispersion model of the intermediate layer  114 B film layer. 
     It should be recognized that the various steps described throughout the present disclosure may be carried out by a single computer system  116  or, alternatively, a multiple computer system  116 . Moreover, different subsystems of the system  100 , such as the spectroscopic ellipsometer  101 , may include a computer system suitable for carrying out at least a portion of the steps described above. Therefore, the above description should not be interpreted as a limitation on the present invention but merely an illustration. Further, the one or more computing systems  116  may be configured to perform any other step(s) of any of the method embodiments described herein. 
     In another embodiment, the computer system  116  may be communicatively coupled to the spectrometer  104  or the illuminator subsystem  102  of the ellipsometer  101  in any manner known in the art. For example, the one or more computing systems  116  may be coupled to a computing system of the spectrometer  104  of the ellipsometer  101  and a computing system of the illuminator subsystem  102 . In another example, the spectrometer  104  and the illuminator  102  may be controlled by a single computer system. In this manner, the computer system  116  of the system  100  may be coupled to a single ellipsometer computer system. 
     The computer system  116  of the system  100  may be configured to receive and/or acquire data or information from the subsystems of the system (e.g., spectrometer  104 , illuminator  102 , and the like) by a transmission medium that may include wireline and/or wireless portions. In this manner, the transmission medium may serve as a data link between the computer system  116  and other subsystems of the system  100 . Further, the computing system  116  may be configured to receive spectral results via a storage medium (i.e., memory). For instance, the spectral results obtained using a spectrometer of an ellipsometer may be stored in a permanent or semi-permanent memory device. In this regard, the spectral results may be imported from an external system. 
     Moreover, the computer system  116  may send data to external systems via a transmission medium. Moreover, the computer system  116  of the system  100  may be configured to receive and/or acquire data or information from other systems (e.g., inspection results from an inspection system or metrology results from a metrology system) by a transmission medium that may include wireline and/or wireless portions. In this manner, the transmission medium may serve as a data link between the computer system  116  and other subsystems of the system  100 . Moreover, the computer system  116  may send data to external systems via a transmission medium. 
     The computing system  116  may include, but is not limited to, a personal computer system, mainframe computer system, workstation, image computer, parallel processor, or any other device known in the art. In general, the term “computing system” may be broadly defined to encompass any device having one or more processors, which execute instructions from a memory medium. 
     Program instructions  120  implementing methods such as those described herein may be transmitted over or stored on carrier medium  118 . The carrier medium may be a transmission medium such as a wire, cable, or wireless transmission link. The carrier medium may also include a computer-readable medium such as a read-only memory, a random access memory, a magnetic or optical disk, or a magnetic tape. 
     The embodiments of the system  100  illustrated in  FIG. 1  may be further configured as described herein. In addition, the system  100  may be configured to perform any other step(s) of any of the method embodiment(s) described herein. 
       FIG. 3  illustrates a process flow  200  suitable for implementation by the system  100  of the present invention. In one aspect, it is recognized that data processing steps of the process flow  200  may be carried out via a pre-programmed algorithm executed by one or more processors of computing system  116 . While the following description is presented in the context of system  100 , it is recognized herein that the particular structural aspects of system  100  do not represent limitations and should be interpreted as illustrative only. 
     In block  201 , a spectral response of an unfinished, multi-layer semiconductor wafer across a broad spectral range is received after a high-k thin film is deposited on the wafer. For example, spectra may be received from an ellipsometer  101 . In another example, spectra may be received from a reflectometer (not shown). The spectral data may be acquired from each of the thin films  114  deposited on the wafer  112  utilizing the spectroscopic ellipsometer  101 . For instance, the ellipsometer  101  may include an illuminator  102  and a spectrometer  104 , as discussed previously herein. The spectrometer  104  may transmit results associated with a spectroscopic measurement of the thin films of the wafer to one or more computing systems  116  for analysis. In another example, the spectra for multiple thin films  114  may be acquired by importing previously obtained spectral data. In this regard, there is no requirement that the spectral acquisition and the subsequent analysis of the spectral data need be contemporaneous or performed in spatial proximity. For instance, spectral data may be stored in memory for analysis at a later time. In another instance, spectral results may be obtained and transmitted to an analysis computing system located at a remote location. 
     In block  202 , a plurality of parameter values of an optical dispersion model of one or more layers of the multi-layer semiconductor wafer are determined based at least in part on the spectral response. In one example, the optical dispersion model includes a Cody-Lorentz model augmented by one or more oscillator functions sensitive to one or more defects of the unfinished, multi-layer semiconductor wafer. 
     In general, the optical dispersion model as described herein may be configured to characterize any useful optical dispersion metric. For example, any of the real (n) and imaginary (k) components of the complex index of refraction may be characterized by the optical dispersion model. In another example, any of the real (∈ 1 ) and imaginary (∈ 2 ) components of the complex dielectric constant may be characterized by the optical dispersion model. In other examples, any of the square root of ∈ 2 , absorption constant α=4πk/λ, conductivity (σ), skin depth (δ), and attenuation constant (σ/2)*sqrt(μ/∈), where μ is the free space permeability, may be characterized by the optical dispersion model. In other examples, any combination of the aforementioned optical dispersion metrics may be characterized by the optical dispersion model. The aforementioned optical dispersion metrics are provided by way of non-limiting example. Other optical dispersion metrics or combinations of metrics may be contemplated. 
     In one example, the parameter values of an optical dispersion model of the real (∈ 1 ) and imaginary (∈ 2 ) components of the complex dielectric constant across the selected spectral range are determined utilizing a regression process. In this regard, a regression method may be applied to the measured spectral data using a selected dispersion model. 
     In one aspect, the selected dispersion model includes a generalized Cody-Lorentz model to describe the complex bulk band structure of high-K dielectric layer augmented with additional Lorentz peaks to describe defects, interface states, or excitonic states. In one example, the imaginary part of the dielectric function, ∈ 2 (E), is defined by Equation (1). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
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     In one further aspect, the dielectric function, ∈ 2 (E), includes one or more Lorentz functions to account for one or more defects across the energy range. By way of non-limiting example, middle gap defects, interface states, or sharp band-edge excitonic transitions may each be described by one or more Lorentz functions. As described with reference to the second terms of both expressions of equation (1), a summation of N d  Lorentz functions are employed both above and below the demarcation energy, E t , to describe N d  possible defects. An exemplary Lorentz function describing a particular defect, d i , is described by equation (2). 
                     L   ⁡     (       E   ;     A   di       ,     E     0   ⁢           ⁢   di       ,     Γ   di       )       =         A   di     ⁢     E     0   ⁢           ⁢   di       ⁢     Γ   di     ⁢   E           (       E     0   ⁢           ⁢   di     2     -     E   2       )     2     +       Γ   di   2     ⁢     E   2                   (   2   )               
where A di  describes an amplitude of the contribution of the particular defect to the optical response, E 0di , describes a resonance energy of the contribution of the particular defect to the optical response, and Γ di  describes a width of the contribution of the particular defect to the optical response.
 
     Below the demarcation energy, E t , between the Urbach tail transitions and the band-band transitions, the dielectric function, ∈ 2 (E), includes two terms. The first term describes the Urbach tails with an exponential function as conventionally described in a Cody-Lorentz model. E u  describes the rate of attenuation of the Urbach function and E 1  is an amplitude defined by the continuity of the dielectric function at the demarcation energy, E t , as described by equation (3).
 
 E   1   =E   t   G   C ( E   t ) L ( E   t )  (3)
 
where L(E t )=L(E t ;A,E 0 ,Γ)+L(E t ;A d ,E 0d ,Γ d ).
 
     Above the demarcation energy, E t , between the Urbach tail transitions and the band-band transitions, the dielectric function, ∈ 2 (E), also includes two terms. The first term is a Lorentz function, L(E), modulated by a gap function used to describe the band-edge of amorphous dielectrics. The gap function is described by equation (4) 
                       G   c     ⁡     (       E   ;     E   g       ,     E   p       )       =         (     E   -     E   g       )     2           (     E   -     E   g       )     2     +     E   p   2                 (   4   )               
where E g  is the band gap and E p  is the transition energy.
 
     In another further aspect, the Lorentz function, L(E), that is modulated by the gap function, G c (E), includes a summation of multiple Lorentz oscillator functions. Equation (5) describes a summation of two Lorentz oscillator functions,
 
 L ( E )= L ( E;A,E   0 ,Γ)+ L ( E;A   d   ,E   0d ,Γ d )  (5)
 
     where A and A d , E d  and E 0d , and Γ and Γ d  represent the amplitude, frequency and the width of the two oscillator peaks, respectively. The parameters of the aforementioned model, {E g , E p , E t , E u , A (d) , E 0(d) , Γ (d) , A di , E 0di , Γ di }, are fitting parameters in terms of energy. 
     Although one or more Lorentz functions may be employed to describe particular defects, in general, any suitable functional description may be contemplated. In this manner, a variety of physical features of studied structures may be better represented. For example, one or more Gaussian functions may be employed to capture possible defects, and may better represent certain effects, including excitonic and chaos effects. 
     The real part of the dielectric function, ∈ 1 (E), is determined by enforcing Kramers-Kronig consistency to arrive at the following expression for ∈ 1 (E): 
                       ɛ   1     ⁡     (     E   ,     b   i       )       =         ɛ   1     ⁡     (   ∞   )       +       2   π     ⁢   P   ⁢       ∫   o   ∞     ⁢         ξ   ⁢           ⁢       ɛ   2     ⁡     (     ξ   ,     b   i       )             ξ   2     -     E   2         ⁢           ⁢     ⅆ   ξ                     (   6   )               
where ∈ 1 (∞) is the high frequency electron component of the dielectric constant and P is the principal value of the integral. Equation (6) can be reformulated from its integral form to an analytical expression as follows:
 
                       ɛ   1     ⁡     (     E   ,     b   i       )       =         ɛ   1     ⁡     (   ∞   )       +       I   U     ⁡     (     E   ,     b   i       )       +       ∑     n   =     1   :   8                 ⁢     (           X   n     ⁡     (     b   i     )       ⁢       φ   n     ⁡     (     b   i     )         +         X   n   d     ⁡     (     b   i     )       ⁢       φ   n   d     ⁡     (     b   i     )           )       +       ∑     i   =     1   :   Nd                 ⁢       F   di     ⁡     (     E   ;     b   i       )                   (   7   )               
where I u (E,b i ) is the Urbach integral, X n(d)  and P n(d)  are well-defined functions of the model parameters, and F di (E,b i ) are the Kramers-Kronig integrals of Lorentz oscillator peaks given by Equation (8).
 
     
       
         
           
             
               
                 
                   
                     
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     Reformulating the optical function enforcing Kramers-Kronig consistency (e.g., Equation 6) into a closed form analytical expression (e.g., Equation 7) allows for more rapid computation. In addition, the parametric derivatives of the optical function can be found in closed form. The availability of closed form expressions for the optical function and its derivatives is required to perform efficient, effective regression calculations. In addition, maintaining Kramers-Kronig consistency between the real and imaginary parts of the optical dispersion model ensures that the functions defined by the model are physically meaningful. Thus, parameter values of the optical dispersion model resolved based on measured data can be directly used to determine structural and optical features of high-K dielectric layers, and can also be used to control charge trapping centers. 
       FIG. 4  is a plot illustrative of an exemplary representation of the imaginary part of the dielectric function, ∈ 2 (E), for a thin film sample having defects. Plotline  130  is a plot of the imaginary part of the dielectric function, ∈ 2 (E), determined in accordance with the optical dispersion model presented in Equation (1). Plotline  131  is a plot of the imaginary part of the dielectric function, ∈ 2 (E), determined in accordance with the conventional Cody-Lorentz model (i.e., the optical dispersion model of Equation (1) with all defect parameters set to zero). 
     From plotline  130  it is clear that the optical dispersion model of Equation (1) captures more features of the band structure of the high-K layers than the conventional Cody-Lorentz model. For example, the complex band structure of amorphous films (illustrated by the camel-shaped portion  132  of plotline  130 ) is captured by the gap modulated Lorentz function of Equation (5) that includes a summation of two Lorentz oscillator functions. In another example, Equation (1) includes three Lorentz oscillator terms (i.e., N d =3) that effectively capture the three defect states visible as peaks  133 ,  134 , and  135  in the middle of the band gap. These defects are effectively ignored by the conventional Cody-Lorentz model. 
       FIG. 5  is a plot illustrative of an exemplary representation of the real part of the dielectric function, ∈ 1 (E), for the same thin film sample as described with reference to  FIG. 4 . Plotline  140  is a plot of the real part of the dielectric function, ∈ 1 (E), determined in accordance with the optical dispersion model presented in Equation (1). Plotline  141  is a plot of the real part of the dielectric function, ∈ 1 (E), determined in accordance with the conventional Cody-Lorentz model (i.e., the optical dispersion model of Equation (1) with all defect parameters set to zero). 
     From plotline  140  it is clear that the complex band structure of the high-K dielectric layer including the three defect states is evident in the real part of dielectric function. 
     In one embodiment, the optical dispersion model described with reference to Equation (1) has been implemented in the Film Thickness Measurement Library (FTML) of the Off-line Spectral Analysis (OLSA) stand-alone software designed to complement thin film measurement systems such as the Aleris 8510 available from KLA-Tencor Corporation, Milpitas, Calif. (USA). Measurements performed on test samples including high-K gate dielectric stacks showed high precision and reliability at high throughput. In some examples, improved fitting of ellipsometric data with a 2-3 times improvement in throughput was observed in comparison to existing methods. 
     In block  203 , the plurality of parameter values of the optical dispersion model determined from a fitting of measurement data to the augmented Cody-Lorentz model are stored in a memory. The stored values may be used, for example, to perform further analysis of the specimen, or to control manufacturing process parameters. 
     In a further aspect, a band structure characteristic indicative of an electrical performance of the measured layer, or stack of layers, is determined based at least in part on the parameter values of the optical dispersion model of the multi-layer semiconductor wafer. As discussed hereinbefore, the parameters of the model, e.g., {E g , E p , E t , E u , A (d) , E 0(d) , Γ (d) , A i , E 0i , Γ i }, are not just fitting parameters, but are related to physically measurable values. For example, E g  represents the band gap energy. In some examples, a band structure characteristic is a defect identified based on one or more parameter values of the optical dispersion model. For example, A di , E 0di , and Γ di  are related to exciton or defect states. In one example, A di  is indicative of the strength of an exciton transition or the concentration of defects, E 0di  is indicative of the corresponding defect or transition energies, and Γ di  is indicative of defect/exciton trap lifetime. In yet another example, the product of A di  and Γ di  is indicative of a defect concentration. In yet another example, the camel-shape of the absorption band is indicative of the quantum size effect due to decreasing film thickness. In the case of quantized band spectra the values of E 0  and E 0d  are related to the two lowest optical transitions between quantized conduction and valence bands. In this manner, the derived optical dispersion functions represent structure and/or optical features of the high-K layers. 
     The aforementioned examples are provided for illustration purposes and do not limit the type of band structure characteristics that may be contemplated. Many other band structure characteristics that correlate with the electrical properties, and thus act as effective indicators of the electrical performance of a finished wafer, may be contemplated. 
     In another further aspect, device performance is improved by controlling a process of manufacture of the semiconductor wafer based at least in part on the identified band structure characteristic. In one example, charge trapping centers may be controlled based on band structure characteristics identified from the parameter values of the optical dispersion model illustrated in Equation (1). 
     Although, the generalized Cody-Lorentz model augmented with additional Lorentz peaks is described with reference to modeling of high-K dielectric layers, the model can be applied to other materials. In some examples, the model can be configured to describe the band structure of a variety of nanostructures (e.g., nanowires, quantum dots and quantum wells), including any number of bands of any origin, such as excitonic states. The model can be generalized to include any number of defect levels. In another example, the model can be applied to nanostructures (e.g., quantum wells, quantum dots and nanowires) embedded in another amorphous dielectric slab or layer. 
     In another further aspect, separate determinations of optical dispersion metrics and band structure characteristics associated with different layers of a wafer can be made based on the same spectral response data. For example, a wafer under measurement may include a semiconductor substrate  112 , an intermediate layer  114 B, a high-k insulative layer  114 A, and an additional film layer (not shown). The spectral response data received from spectrometer  104  includes contributions from all of these layers. A stack layer model that captures the contributions of each of these layers can be used to separately determine band structure characteristics associated with each different physical layer or group of physical layers under analysis. 
     In another further aspect, the stack model includes a model of the intrinsic absorption peaks of the semiconductor substrate  112  (e.g., silicon). In one example, the intrinsic absorption peaks are accounted for in the spectral measurement of the high-k film. In this manner, the absorption peaks of the semiconductor substrate may be effectively removed from the spectral response of the high-k film. By isolating the spectral response of the high-k film from the semiconductor substrate, a more accurate determination of defects and band structure characteristics associated with the high-k film layer is achieved. 
     In another further aspect, band structure characteristics (e.g., band gap and defects) are used to grade wafers and microchips early in the production process based on the quality of the gate insulator. This may avoid the need to grade wafers and microchips at the end of the production process using expensive and time consuming electrical test equipment. 
     In one or more exemplary embodiments, the functions described may be implemented in hardware, software, firmware, or any combination thereof. If implemented in software, the functions may be stored on or transmitted over as one or more instructions or code on a computer-readable medium. Computer-readable media includes both computer storage media and communication media including any medium that facilitates transfer of a computer program from one place to another. A storage media may be any available media that can be accessed by a general purpose or special purpose computer. By way of example, and not limitation, such computer-readable media can comprise RAM, ROM, EEPROM, CD-ROM or other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium that can be used to carry or store desired program code means in the form of instructions or data structures and that can be accessed by a general-purpose or special-purpose computer, or a general-purpose or special-purpose processor. Also, any connection is properly termed a computer-readable medium. For example, if the software is transmitted from a website, server, or other remote source using a coaxial cable, fiber optic cable, twisted pair, digital subscriber line (DSL), or wireless technologies such as infrared, radio, and microwave, then the coaxial cable, fiber optic cable, twisted pair, DSL, or wireless technologies such as infrared, radio, and microwave are included in the definition of medium. Disk and disc, as used herein, includes compact disc (CD), laser disc, optical disc, digital versatile disc (DVD), floppy disk and blu-ray disc where disks usually reproduce data magnetically, while discs reproduce data optically with lasers. Combinations of the above should also be included within the scope of computer-readable media. 
     As used herein, the term “wafer” generally refers to substrates formed of a semiconductor or non-semiconductor material. Examples of such a semiconductor or non-semiconductor material include, but are not limited to, monocrystalline silicon, gallium arsenide, and indium phosphide. Such substrates may be commonly found and/or processed in semiconductor fabrication facilities. 
     One or more layers may be formed upon a wafer. For example, such layers may include, but are not limited to, a resist, a dielectric material, a conductive material, and a semiconductive material. Many different types of such layers are known in the art, and the term wafer as used herein is intended to encompass a wafer on which all types of such layers may be formed. 
     One or more layers formed on a wafer may be patterned or unpatterned. For example, a wafer may include a plurality of dies, each having repeatable patterned features. Formation and processing of such layers of material may ultimately result in completed devices. Many different types of devices may be formed on a wafer, and the term wafer as used herein is intended to encompass a wafer on which any type of device known in the art is being fabricated. 
     A typical semiconductor process includes wafer processing by lot. As used herein a “lot” is a group of wafers (e.g., group of 25 wafers) which are processed together. Each wafer in the lot is comprised of many exposure fields from the lithography processing tools (e.g. steppers, scanners, etc.). Within each field may exist multiple die. A die is the functional unit which eventually becomes a single chip. One or more layers formed on a wafer may be patterned or unpatterned. For example, a wafer may include a plurality of dies, each having repeatable patterned features. Formation and processing of such layers of material may ultimately result in completed devices. Many different types of devices may be formed on a wafer, and the term wafer as used herein is intended to encompass a wafer on which any type of device known in the art is being fabricated. 
     Although embodiments are described herein with respect to wafers, it is to be understood that the embodiments may be used for characterizing thin films of another specimen such as a reticle, which may also be commonly referred to as a mask or a photomask. Many different types of reticles are known in the art, and the terms “reticle,” “mask,” and “photomask” as used herein are intended to encompass all types of reticles known in the art. 
     The embodiments described herein generally relate to methods for determining band structure characteristics of multi-layer thin films based on optical model parameter values at high throughput. For example, one embodiment relates to a computer-implemented method for determining band structure characteristics of multi-layer thin films based on optical model parameter values derived from spectroscopic ellipsometer data. However, the methods described herein are not limited in the types of inspection systems from which optical model parameter values may be derived. For example, in one embodiment, the inspection system includes a reflectometer for thin film inspection of the wafer. In general, the optical dispersion models described herein may be applied to the analysis of measurement data received from a variety of broadband and narrowband metrology tools. 
     In addition, the inspection system may be configured for inspection of patterned wafers and/or unpatterned wafers. The inspection system may be configured as a LED inspection tool, edge inspection tool, backside inspection tool, macro-inspection tool, or multi-mode inspection tool (involving data from one or more platforms simultaneously), and any other metrology or inspection tool that benefits from the determination of band structure characteristics of multi-layer thin films based on optical model parameter values at high throughput. 
     Although certain specific embodiments are described above for instructional purposes, the teachings of this patent document have general applicability and are not limited to the specific embodiments described above. Accordingly, various modifications, adaptations, and combinations of various features of the described embodiments can be practiced without departing from the scope of the invention as set forth in the claims.