# Patent

## Title
Method to transiently detect samples in atomic force microscopes

## Abstract

An approach to determine cantilever movement is presented. An observer based state estimation and statistical signal detection and estimation techniques are applied to Atomic Force Microscopes. A first mode approximation model of the cantilever is considered and a Kalman filter is designed to estimate the dynamic states. The tip-sample interaction is modeled as an impulsive force applied to the cantilever in order to detect the presence of sample. A generalized likelihood ratio test is performed to obtain the decision rule and the maximum likelihood estimation of the unknown arrival time of the sample profile and unknown magnitude of it. The use of the transient data results in sample detection at least ten times faster than using the steady state characteristics.

## Background
CROSS-REFERENCE TO RELATED PATENT APPLICATIONS
This patent application claims the benefit of U.S. Provisional Patent Application No. 60/507,409, filed Sep. 30, 2003.


STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH AND DEVELOPMENT
This invention was made in part with Government support under Grant Number 9733802 awarded by the National Science Foundation. The Government may have certain rights in this invention.


FIELD OF THE INVENTION
This invention pertains to surface image measurements and more particularly relates to imaging of surfaces in atomic force microscopes.
BACKGROUND OF THE INVENTION
Scanned-proximity Probe Microscopes (SPM's) are instruments that have been in use in universities and industrial research laboratories since the early 1980's. These instruments allow for various imaging of surfaces as well as measurement of the intermolecular forces between two surfaces (or a small tip and a flat surface) in vapors or liquids with a distance resolution of 1 Å. This means that images and forces can be obtained at the atomic level. Over the years, the technique has been improved and its scope extended so that it is now capable of measuring many different surface properties and phenomena.
One type of SPM is an atomic force microscope (AFM), which generally consists of a sample surface and a probe that is supported at the end of a force-measuring cantilever spring. The AFM works measures a local property such as height, optical absorption, or magnetism, with a probe or “tip” placed very close to a sample. It operates by first positioning the tip near the surface and then moving the tip or surface vertically (contact or tapping mode) or laterally (scanning mode) while measuring the force produced on the tip by the surface. The force is calculated by measuring the deflection of the cantilever spring supporting the tip.
The most common method of measuring deflection in an AFM is the optical or beam deflection method where vertical deflection can generally be measured with picometer resolution. The method works by reflecting a laser beam off end of the cantilever. Angular deflection of the cantilever causes a twofold larger angular deflection of the laser beam. The reflected beam strikes a split photodiode (i.e., two side-by-side photodiodes) and the difference between the two photodiode signals indicates the position of the laser beam on the split photodiode and thus the angular deflection of the cantilever.
FIGS. 23a and b illustrate a typical setup of an AFM. The laser 1000 outputs a laser beam 1002 that is pointed at the cantilever 1004. A piezoelectric scanner 1006 is used to position the sample. The laser beam 1002 deflects off the cantilever 1004 and is reflected into the split photo-diode 1008 via mirror 1010. The output of the split photo-diode is conditioned via module 1012 and is input into feedback control module 1014 that is used to control the position of the sample movement of piezoelectric scanner 1006. In static force spectroscopy the cantilever deflection is solely due to the tip-sample inter-action. The piezoelectric scanner 1006 is rastered in the lateral directions and the deflection of the tip is used to interpret sample properties. In the dynamic mode, the cantilever support 1016 is forced sinusoidally using a dither piezo 1018. The changes in the oscillations caused by the sample are interpreted to obtain its properties.
The cantilever has low stiffness and high resonant frequency that allows it to probe interatomic forces. Micro-cantilevers, which are cantilevers having lengths typically ranging from 100 to 200 μm with tips of 5 nm, have been utilized in biological sciences to perform feats such as cutting DNA strands and monitoring RNA activity. Another application of the micro-cantilever is in the detection of single electron spin that has significant ramifications for quantum computing technology.
In spite of the underlying promise, considerable challenges remain. Pivotal to harnessing the vast potential of micro-cantilever based technology is ultra-fast interrogation capabilities. This is apparent as the manipulation, interrogation and control of atoms or spins of electrons needs to be accomplished for material that has macroscopic dimensions. To achieve high throughput, fast interrogation is imperative. It is becoming increasingly evident that for many nanotechnological studies, high bandwidth is a necessity. For example, in the field of cell biology, proposals on using nanotechnology have been presented where nano-probes track events in the cell. These events often have time-scales in the micro-second or nano-second regimes. Current measurement techniques do not meet the aforementioned high precision and bandwidth requirements. For example, the micro-cantilever is often operated in the dynamic mode where it is forced sinusoidally by a dither-piezo. This mode of operation has advantages of better signal to noise ratio and being gentle on the sample. Most dynamic imaging methods employing micro-cantilevers currently use variables such as the amplitude and phase, or the equivalent frequency of the micro-cantilever to infer sample characteristics. These are steady state characteristics and do not hold much significance during the transient of the cantilever oscillation. The present methods are therefore inherently slow owing to the large settling times of the cantilever oscillations.
BRIEF SUMMARY OF THE INVENTION
The invention provides an ultra-fast method to obtain imaging data using transient data of an atomic force microscope (AFM). The tip-sample interaction is modeled via a Kalman observer that is used to estimate the state of the model. Corresponding data during the transient state of the probe is analyzed using a likelihood ratio test to detect and probe the sample profile. The use of the transient data results in sample detection at least ten times faster than using the steady state characteristics. The time taken to detect the sample is proportional to the natural frequency of the cantilever and independent of the quality factor of the cantilever.
These and other advantages of the invention, as well as additional inventive features, will become more apparent from the following detailed description when taken in conjunction with the accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram generally illustrating an exemplary atomic force microscope environment on which the present invention resides;
FIG. 2 is a block diagram generally illustrating the control blocks of the present invention;
FIG. 3 is a block diagram of a spring-damper-mass model of cantilever dynamics;
FIG. 4 is a plot illustrating the effect of tip-sample interaction on the cantilever;
FIG. 5 is a block diagram of the Kalman observer based detection method in accordance with the teachings of the present invention;
FIG. 6 is a block diagram of a tip-sample interaction model in accordance with the teachings of the present invention;
FIG. 7 is diagram illustrating the tracking of the Kalman observer of FIG. 5 with respect to the cantilever deflection signal;
FIG. 8 is a graph of the deflection signal of a cantilever when the tip hits a sample;
FIG. 9 is a plot of the error in estimation of the Kalman observer of FIG. 5 and a demodulated cantilever deflection signal of FIG. 8;
FIG. 10 is a plot showing the tip-sample interaction force of the cantilever of FIG. 1 when the cantilever hits a sample of 4 nm height;
FIG. 11 is a plot showing innovations from the Kalman observer of FIG. 5 and an estimated dynamic profile in accordance with the teachings of the present invention;
FIG. 12 is a plot showing innovations from the Kalman observer of FIG. 5 in accordance with the teachings of the present invention when the cantilever of FIG. 1 has a single hit with a sample of 2 nm and 4 nm height;
FIG. 13 is a plot showing innovations from the Kalman observer of FIG. 5 in accordance with the teachings of the present invention when the cantilever of FIG. 1 has multiple hits with a sample of 2 nm and 4 nm height;
FIG. 14 is a plot showing innovations from the Kalman observer of FIG. 5 when the cantilever of FIG. 1 has a single hit with a sample of 4 nm height and a corresponding generalized likelihood ratio and decision rule in accordance with the teachings of the instant invention;
FIG. 15 is a plot showing innovations from the Kalman observer of FIG. 5 when the cantilever of FIG. 1 has multiple hits with a sample of 4 nm height and a corresponding generalized likelihood ratio and decision rule in accordance with the teachings of the instant invention;
FIG. 16 is a plot showing innovation sequence data from the Kalman observer of FIG. 5 and an estimated dynamic profile in accordance with the teachings of the instant invention with a sample of 2 nm height;
FIG. 17 is a plot showing innovation sequence data from the Kalman observer of FIG. 5 and an estimated dynamic profile in accordance with the teachings of the instant invention with a sample of 4 nm height;
FIGS. 18a and 18b are graphs illustrating the model response and the piezo response of an atomic force microscope in accordance with the invention;
FIGS. 19a and b are graphs illustrating an input voltage and the piezo response to the input voltage;
FIGS. 20a and 20b are graphs illustrating the cantilever position and innovation process in response to a pulse shape generated using the piezo dynamics;
FIGS. 21a and 21b are graphs illustrating the innovation response for a piezo oscillation;
FIGS. 22a–c are graphs illustrating the innovation and likelihood ratio with respect to a piezo movement;
FIG. 23a is a block diagram of a prior art atomic force microscope; and
FIG. 23b is a diagram of the cantilever of FIG. 18a in contact with a sample.



DETAILED DESCRIPTION OF THE INVENTION
The invention provides a method to utilize the transient behavior of the cantilever oscillations for sample imaging and detection in atomic force microscopy. The transient based methods of the invention improve performance of interrogation speeds by approximately two orders of magnitude over steady state based methods. Fundamental limitations due to high quality factors are removed using the transient part of the cantilever motion. A model of the cantilever is used to build a Kalman Observer that is used to estimate the state of the cantilever dynamics. The tip-sample interaction is modeled and corresponding data during the transient state of the probe is analyzed using generalized likelihood ratio test to detect and probe the sample profile. The use of the transient data results in sample detection at least ten times faster than using the steady state data based methods.
Turning to the drawings, wherein like reference numerals refer to like elements, the invention is illustrated as being implemented in a suitable environment. FIG. 1 illustrates an AFM 100. The laser 102 outputs a laser beam 104 that is pointed at the cantilever 106. A piezoelectric scanner 108 is used to position the sample in one embodiment. The laser beam 104 deflects off the cantilever 106 and is reflected into the split photo-diode 110 via mirror 112. The output of the split photo-diode is conditioned via module 114 and is input into control module 116 that is used to control the position of the sample by movement of piezoelectric scanner 108. The piezoelectric scanner 108 is rastered in the lateral directions and the deflection of the tip is used to interpret sample properties. In the dynamic mode, the dither piezo 118 is used to force the cantilever support 120 to move sinusoidally. The changes in the oscillations caused by the sample are interpreted to obtain its properties.
Turning now to FIG. 2, a block diagram of the system model that the control module 116 implements is shown. The cantilever module 200 is the model of the cantilever. The cantilever module 200 has inputs g(t) and u(t) where g(t) is the external forcing on the cantilever and u(t) is the thermal noise. The model of the cantilever when it is forced sinusoidally at its first resonant is accurately described by

{umlaut over (p)}+2ξω0{dot over (p)}+ω02p=g(t)  (1)

where p(t), g(t), ω0 and ξ denote the micro-cantilever deflection as measured by the photo-diode 110, the external forcing on the cantilever, the first resonant frequency of the cantilever and the damping factor in free medium respectively. This model can be viewed as a spring mass damper system as illustrated in FIG. 3. Turning briefly to FIG. 3, the spring-damper-mass model depicts the cantilever dynamics oscillating freely (Φ=0) at its resonance frequency as described by Equation (1) with





  
    
      
        
          
            ω
            0
          
          =
          
            
              k
              m
            
          
        
        ,
        
          
            2
            ⁢
            
              
            
            ⁢
            ξ
            ⁢
            
              
            
            ⁢
            
              ω
              0
            
          
          =
          
            c
            m
          
        
        ,
        
          
            and
            ⁢
            
              
            
            ⁢
            
              g
              ⁡
              
                (
                t
                )
              
            
          
          =
          
            
              k
              m
            
            ⁢
            
              b
              ⁡
              
                (
                t
                )
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




where k is the spring constant, c is the damping constant, m is the mass, and b(t) is the sinusoidal forcing due to the dither piezo.

Note that




  ξ
  =
  
    1
    
      2
      ⁢
      
        
      
      ⁢
      Q
    
  




with Q being the quality factor of the micro-cantilever. The parameters of the cantilever model can be accurately obtained by thermal noise analysis. From Equation 1, the continuous time state-space model of the micro-cantilever dynamics can be described as,





  
    
      
        
          
            
              
                (
                
                  
                    
                      
                        
                          x
                          .
                        
                        1
                      
                    
                  
                  
                    
                      
                        
                          x
                          .
                        
                        2
                      
                    
                  
                
                )
              
              =
              
                
                  
                    (
                    
                      
                        
                          0
                        
                        
                          1
                        
                      
                      
                        
                          
                            -
                            
                              ω
                              0
                              2
                            
                          
                        
                        
                          
                            
                              -
                              2
                            
                            ⁢
                            
                              
                            
                            ⁢
                            ξ
                            ⁢
                            
                              
                            
                            ⁢
                            
                              ω
                              0
                            
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            1
                          
                        
                      
                      
                        
                          
                            x
                            2
                          
                        
                      
                    
                    )
                  
                
                +
                
                  
                    (
                    
                      
                        
                          0
                        
                      
                      
                        
                          1
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      u
                      +
                      
                        f
                        s
                      
                    
                    )
                  
                
              
            
          
        
        
          
            
              y
              =
              
                
                  
                    (
                    
                      
                        
                          1
                        
                        
                          0
                        
                      
                    
                    )
                  
                  ⁢
                  
                    (
                    
                      
                        
                          
                            x
                            1
                          
                        
                      
                      
                        
                          
                            x
                            2
                          
                        
                      
                    
                    )
                  
                
                +
                υ
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




where state x1 denotes the cantilever-tip position p(t), state x2 denotes the cantilever velocity {dot over (p)}(t), and ν denotes the noise affecting the photodiode sensor. u=nth+g(t) represents the combined effect of the thermal noise nth, and the sinusoidal forcing b(t) due the dither piezo. The function ƒs, which in terms of FIG. 3 is





  
    
      f
      s
    
    =
    
      
        k
        m
      
      ⁢
      
        
      
      ⁢
      Φ
    
  
  ,




represents the equivalent force on the cantilever due to tip-sample interaction. The discretized version of the above model when the sample is not present can be denoted by

xi+1=Fxi+Gui

yi=Hxi+νi,i≧0  (4)

where





  
    
      
        
          E
          ⁢
          
            {
            
              
                
                  [
                  
                    
                      
                        
                          u
                          i
                        
                      
                    
                    
                      
                        
                          υ
                          i
                        
                      
                    
                    
                      
                        
                          x
                          i
                        
                      
                    
                  
                  ]
                
                ⁡
                
                  [
                  
                    
                      
                        
                          u
                          j
                        
                      
                    
                    
                      
                        
                          υ
                          j
                        
                      
                    
                    
                      
                        
                          x
                          0
                        
                      
                    
                    
                      
                        1
                      
                    
                  
                  ]
                
              
              T
            
            }
          
        
        =
        
          [
          
            
              
                
                  
                    Q
                    i
                  
                  ⁢
                  
                    δ
                    ij
                  
                
              
              
                0
              
              
                0
              
              
                0
              
            
            
              
                0
              
              
                
                  
                    R
                    i
                  
                  ⁢
                  
                    δ
                    ij
                  
                
              
              
                0
              
              
                0
              
            
            
              
                0
              
              
                0
              
              
                
                  Π
                  0
                
              
              
                0
              
            
          
          ]
        
      
    
    
      
        (
        5
        )
      
    
  




It is assumed that the input noise and the noise in the output are uncorrelated. Note that when the sample is not present, the cantilever settles into a sinusoidal trajectory buried in noise.

When a sample is present, there is an attractive or repulsive force on the cantilever depending on the regime of interaction. Typically this tip-sample interaction is characterized by short range repulsive forces and long range attractive forces as shown in FIG. 4.
In many AFM applications when the sample is introduced after the cantilever has settled into a sinusoidal orbit, the effect of the tip-sample interaction force of block 202 is modeled as an impulsive force. Intuitively this also follows because the time spent by the cantilever tip under the sample's influence is negligible compared to the time it spends outside the sample's influence. This assumption is particularly true for samples that have a small attractive regimes. Under such an assumption the cantilever dynamics is described by

xi+1=Fxi+Gui+δθ,i+1ν

yi=Hxi+νi, i≧0  (6)

where θ denotes the time instant when the impact occurs, δi,j denotes the dirac delta function, and ν signifies the magnitude of the impact. Essentially, the impact causes an instantaneous change in the state by ν at time instant θ. In this setting, the time of impact and the resulting change in the state are unknown quantities. The instant specification determines when the cantilever is “hitting” the sample and when it is not, that is to detect whether the change in state is occurring or not.

In order to detect whether the change in state is occurring, an observer based state estimation techniques and related tools for AFM is used. Note that in the AFM setup, the position of the cantilever-tip (deflection signal from the photo-diode sensors) is measurable, not the velocity. Returning to FIG. 2, cantilever 200 is the linear time invariant system depicting the model of the cantilever. The interaction with the sample is modeled and shown as a non-linear tip-sample interaction system 202 appearing in the feedback path. The Kalman observer 204 is designed based on the thermal noise and measurement noise characteristics and experimentally determined values of the parameters of the model as given in equation 3.
The Kalman filter 204 estimates the state of the cantilever by observing the known input signal (sinusoidal forcing from the dither piezo) and available output signal (cantilever tip deflection data from photo diode sensor). When the sample is present as given by the model in equation 6, the estimated and filtered states from the Kalman filter 204 are given by,

{circumflex over (x)}i+1|i=F{circumflex over (x)}i|i,

{circumflex over (x)}i|i={circumflex over (x)}i|i−1+Kiγi  (7)

with the measurement residual γi, Kalman filter gain Ki and error covariance matrices given by γi=yi−H{circumflex over (x)}i|i−1, Ki=Pi|i−1HTVi−1, Pi+1|i=FPi|iFT+GQGT, Pi|i−1−KiHPi|i−1, Vi=HPi|i−1HT+R . . . In the steady state the Kalman gain and the error covariance matrices become constant matrices as K=limi→∞Ki, P=limi→∞Pi|i−1 and V=limi→∞Vi=HPHT+R. For convenience of implementation, steady state Kalman filter parameters are used.

The measurement residual γi, is given by,

γi=Yi;θν+γi1  (8)

where Yi;θ is a known dynamic signal profile with unknown magnitude ν defined by the following recursive formulae,

Yi;θ=H[Φ(i,θ)−FX(i−1;θ)],

X(i;θ)=KiYi;θ+FX(i−1;θ),  (9)

with Φ(i,θ)=Πj=θiF and X(i;i)=KiH. Here Yi;θ and X(i;θ) are the additive parts in the innovation and the state estimate respectively. It can be shown that {γi1} is a zero mean white noise sequence with covariance Vi and is the measurement residual had the jump not occurred.

Note that the damping present in a typical cantilever is very low with quality factor as high as 100 or above. This results in the cantilever taking considerable time to settle to the steady state periodic orbit. Whenever there is an impulsive input to the cantilever (possibly due to interaction with sample), the state changes to a new value. Since this input is not applied to the observer 204, the estimated state does not change instantly. This change is fed to the observer through the output data. The observer takes noticeable time to correct the estimates and track the output again. This gives rise to transients in the estimation error. Note that the observer is capable of tracking the transient response of the cantilever much before the system settles to the steady state periodic orbit by appropriately choosing the gain Ki. However, at the starting of the transients, there is a mismatch between the actual output and the estimated output. This results in the dynamic profile Yi;θ. This observation motivated us to develop a faster detection technique. Present detection schemes use the steady state deflection data, whereas the approach described herein uses the residual in innovation to probe the sample.
We pose the detection problem in the hypothesis testing framework as following.

H0:Yi=γi1, i=1,2, . . . ,n

versus

H1:Yi=Yi;θν+γi1, i=1,2, . . . ,n  (10)

where γi1 is a zero mean white gaussian process





  
    p
    ⁢
    
      
    
    ⁢
    
      (
      
        γ
        i
        1
      
      )
    
  
  =
  
    
      1
      
        
          
            (
            
              2
              ⁢
              
                
              
              ⁢
              π
            
            )
          
          
            n
            2
          
        
        ⁢
        
          
            (
            
              
              
                V
                i
              
              
            
            )
          
          
            n
            2
          
        
      
    
    ⁢
    
      
    
    ⁢
    
      exp
      ⁡
      
        (
        
          
            -
            
              1
              2
            
          
          ⁢
          
            
          
          ⁢
          
            
              γ
              i
              1
            
            T
          
          ⁢
          
            V
            i
            
              -
              1
            
          
          ⁢
          
            γ
            i
            1
          
        
        )
      
    
  




and Yi;θ is a known dynamic profile with unknown arrival time θ and unknown magnitude ν as given in equation 8. The estimation problem is to compute the maximum likelihood estimate (MLE's) {circumflex over (θ)}(n) and {circumflex over (ν)}(n) based on the residual γ1, . . . , γn. In one embodiment, Willsky's generalized likelihood ratio test is used to solve both the detection and estimation problem. Willsky's generalize ratio test is described in the publications “A Generalized Likelihood Ratio Approach To Estimation In Linear Systems Subject To Abrupt Changes”, Alan S. Willsky and Harold L. Jones, Proc. IEEE Decision and Control, pp. 108–112, November 1974 and “A Generalized Likelihood Ratio Approach To The Detection And Estimation Of Jumps In Linear Systems”, Alan S. Willsky and Harold L. Jones, IEEE Transactions on Automatic Control, February 1976, which are hereby incorporated by reference.

The likelihood ratio when the jump occurs at time instant θ={tilde over (θ)} of magnitude ν={tilde over (ν)} is given by




  
    
      Λ
      n
    
    ⁡
    
      (
      
        
          θ
          ~
        
        ,
        
          v
          ~
        
      
      )
    
  
  =
  
    
      
        p
        ⁡
        
          (
          
            
              γ
              1
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                γ
                n
              
              |
              
                H
                1
              
            
            ,
            
              θ
              =
              
                θ
                ~
              
            
            ,
            
              v
              =
              
                v
                ~
              
            
          
          )
        
      
      
        P
        ⁡
        
          (
          
            
              γ
              1
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                γ
                n
              
              |
              
                H
                0
              
            
          
          )
        
      
    
    .
  




The generalized likelihood ratio computes the likelihood ratio





  
    
      Λ
      n
    
    =
    
      
        p
        ⁡
        
          (
          
            
              γ
              1
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                γ
                n
              
              |
              
                H
                1
              
            
            ,
            
              θ
              =
              
                θ
                ^
              
            
            ,
            
              v
              =
              
                v
                ^
              
            
          
          )
        
      
      
        P
        ⁡
        
          (
          
            
              γ
              1
            
            ,
            ...
            ⁢
            
              
            
            ,
            
              
                γ
                n
              
              |
              
                H
                0
              
            
          
          )
        
      
    
  
  ,




where θ and ν are the maximum likelihood estimates of θ and ν under the hypothesis H1 (i.e. ({circumflex over (θ)},{circumflex over (ν)})=arg max({tilde over (θ)},{circumflex over (ν)})Λn({tilde over (θ)},{tilde over (ν)})).

The decision function gn defined as the double maximization of Λn over the parameters θ and ν is given by, gn=max1≦θ≦ndT(n;θ)C−1(n;θ)d(n;θ) where




  
    C
    ⁡
    
      (
      
        n
        ;
        θ
      
      )
    
  
  =
  
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          
            i
            ;
            θ
          
        
        ⁢
        
          
        
        ⁢
        and
        ⁢
        
          
        
        ⁢
        
          d
          ⁡
          
            (
            
              n
              ;
              θ
            
            )
          
        
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          
            Y
            i
          
          .
        
      
    
  




The likelihood ratio is defined as l(n;θ)=dT(i;θ)C−1(n;θ)d(n;θ). The maximum likelihood ratio estimate (MLE) of θ is given by the value θ≦n such that {circumflex over (θ)}n=arg max1≦θ≦nl(n;θ). The decision rule is




  
    
      
        
          
            g
            n
          
          =
          
            
              l
              ⁡
              
                (
                
                  n
                  ;
                  
                    
                      θ
                      ^
                    
                    n
                  
                
                )
              
            
            ⁢
            
              
                
                  
                    
                      H
                      1
                    
                    >
                  
                
              
              
                
                  
                    
                      H
                      0
                    
                    <
                  
                
              
            
            ⁢
            ɛ
          
        
        ,
      
    
    
      
        (
        11
        )
      
    
  




where the threshold ε is chosen to provide a suitable trade off between false alarm and missed alarms. The MLE of ν is given by {circumflex over (ν)}n(θ)=C−1(n;θ)d(n;θ). This method requires that the search for θn be on the entire data set between 1 and n, which requires a bank of filters with increasing length. In practice, the search may be carried out on a data window of finite length M. Note that the length M also affects the missed probability. The false alarm and detection probabilities are calculated as, PF=P0(Γ1)=∫ε∞p(l=L|H0)dL and PD(ν,θ)=P1(Γ1)=∫ε∞p(l=L|H1,ν,θ)dL respectively. Since





  
    p
    ⁡
    
      (
      
        l
        =
        
          L
          ❘
          
            H
            0
          
        
      
      )
    
  
  =
  
    
      -
      
        1
        2
      
    
    ⁢
    
      
        ∑
        
          i
          =
          1
        
        n
      
      ⁢
      
        
      
      ⁢
      
        
          Y
          i
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          i
        
      
    
  




under hypothesis H0 and Yi are independent identically distributed gaussian random variables, p(l=L|H0) is Chi-squared (χ2) density with n degrees of freedom. Similarly





  
    p
    ⁡
    
      (
      
        l
        =
        
          L
          ❘
          
            H
            1
          
        
      
      )
    
  
  =
  
    
      
        -
        
          1
          2
        
      
      ⁢
      
        
          ∑
          
            i
            =
            1
          
          n
        
        ⁢
        
          
            Y
            i
            T
          
          ⁢
          
            V
            
              -
              1
            
          
          ⁢
          
            Y
            i
          
        
      
    
    -
    
      
        1
        2
      
      ⁢
      
        
          v
          n
        
        ⁡
        
          (
          θ
          )
        
      
      ⁢
      
        
          ∑
          
            i
            =
            θ
          
          n
        
        ⁢
        
          
            Y
            
              i
              ;
              θ
            
            T
          
          ⁢
          
            V
            
              -
              1
            
          
          ⁢
          
            Y
            i
          
        
      
    
  




is a non-central χ2 density with noncentrality parameter νTC(n;θ)ν. This shows that PD is dependent upon values of θ and ν.

For specified PF or PD, the threshold value ε can be computed from the tables in “Handbook of Statistical Tables”, D. B. Owen, Addison Wesley, Reading MA, 1922. Given ε, the values PF or PD(ν,θ) can be computed similarly. In one embodiment, ν is used as the minimum jump that is required to be detected and θ as the size of the data window M to compute PD.
Once a jump has been detected by a GLR detector, the MLEs {circumflex over (θ)} and {circumflex over (ν)} can be used to update the state estimate as {circumflex over (x)}i|i,new={circumflex over (x)}i|i,old+{Φ(i,{circumflex over (θ)}−F[i;{circumflex over (θ)}]}{circumflex over (ν)} and the covariance as Pi|i,new=Pi|i,old+{Φ(i,{circumflex over (θ)}−F[i;{circumflex over (θ)}]}C−1[i,{circumflex over (θ)}]{Φ(i,{circumflex over (θ)})−F[i;{circumflex over (θ)}]}. This adaptive filtering scheme can be used to detect successive jumps. The size of the data window has to be chosen carefully to ensure the stability of the scheme.
Turning now to FIG. 5 the method of the present invention was simulated using Simulink in Matlab. The linear cantilever model 500 is given by equation 3, with natural frequency f0=73,881 Hz, quality factor Q=130 and a sinusoidal forcing b(t) resulting in a free oscillation amplitude of pmax=24 nm. From testing, the mean deflection of the cantilever is 0.3 nm due to thermal noise during normal test conditions, which corresponds to input noise power to the model of Q=0.001 nmHz2. The resolution of the photo-diode sensor in the AFM testing set up is 1 Å, which corresponds to output noise power R=0.1 nm. The discrete time Kalman filter 504 is designed using the Matlab function ‘dkalman’. For simplicity in the simulation, a purely repulsive tip-sample interaction model 502 as given in FIG. 6 is used. The tip interacts with the sample only when p(t)>1. The force from the sample is then given by fs(t)=4h(t) pN. For simulation purposes, ks=4 pN/nm was chosen. This model corresponds to a static nonlinear block having a dead zone region of length 1. It is noted that a model that includes only the repulsive part of the tip-sample interaction is appropriate for many samples where the attractive regime is small.
Note that the observer based state estimation is sensitive to plant model uncertainties and experimental conditions that depend on thermal noise and sensor noise characteristics. In the simulations, the Kalman filter 504 is designed based on the actual model 204 and noise characteristics while the perturbed models of the cantilever and noise characteristics are employed as the real system.
Turning now to FIG. 7, the estimated deflection (curve 700) of the cantilever from the Kalman filter 504 is plotted with the actual deflection (curve 702). The initial estimation error seen in the first ten seconds is due to the mismatch in the initial state of the cantilever model 500 and the Kalman filter 504.
In FIG. 8, the cantilever deflection signal 800 is plotted when tip hits the sample of 4 nm height at time instant θ=4000 μsecond. It takes approximately 1200 μseconds for the cantilever to reach the steady state. In FIG. 9, the innovation sequence (curve 900) and the error in estimation (curve 902) are plotted. The dashed line in curve 900 is a 18 nm threshold level for the detection scheme of the invention. There is an error in estimation as soon as the cantilever hits the sample. The change in the deflection is not immediately tracked by the Kalman observer 504. However, it can be seen that the observer 504 is fast enough to track the deflection signal during the transient state of the cantilever. It should be noted that in reality, the estimation error approaches a zero mean white noise much before the system stabilizes to a periodic orbit.
FIG. 10 shows a typical tip-sample interaction force 1000 observed when the cantilever encounters a step sample profile of 4 nm high during simulation. This repulsive force is high in magnitude and exists for a smaller time compared to the time-period of oscillation of the cantilever. It is observed that the time spent by the cantilever away from sample is 12 times of what is spent near the sample. Note that for analysis purposes, this force is modeled as an impulsive force (i.e. instantaneous jump in state) as previously described and is not being directly applied during simulation.
In FIG. 11, the estimation error (dashed curve 1100) is plotted with the calculated dynamic profile (solid curve 1102) when the arrival time and the magnitude are assumed to be known. It can be seen from FIG. 11 that the residual innovations from the Kalman filter 504 can be modeled as a known dynamic profile with an unknown magnitude as implied by equation 8 with additive white Gaussian noise.
In FIG. 12 the innovation sequence is plotted when the cantilever has a single hit with the sample of heights 2 nm (dashed curve 1200) and 4 nm (solid curve 1202). FIG. 13 plots the innovation sequence when the cantilever has multiple hits with the sample of heights 2 nm (dashed curve 1300) and 4 nm (solid curve 1302). For the single hit case, the dynamic profiles are proportional to each other. However, this is not true for the multiple hits case as the dynamic profiles overlap with each other and the arrival time of them are not uniformly separated in time.
We derived that when the cantilever hits the sample, the innovation sequence from the Kalman filter 504 consists of a dynamic profile of unknown arrival time and magnitude with additive white gaussian noise. Such detection and estimation problem is solved using the generalized likelihood ratio test (GLRT) as described above. Thus for simulation we used GLRT to detect the sample and estimate the size of impact. The unknown arrival time and the unknown magnitude of impact is estimated in maximum likelihood sense. The threshold ε is chosen to keep the false alarm rate below 1%. In FIGS. 14 and 15 the generalized likelihood ratio (solid curves 1400, 1500) and the decision rule (dash-dot curves 1402, 1502) are shown when the cantilever hits a repulsive sample once and multiple times respectively with the innovation sequence (dotted curves 1404, 1504). The generalized likelihood ratio plots are scaled down by a factor of 2*10−4 times to fit into the graph. The decision rule for detection is scaled by a factor of 5 for the same reason. Note that there is a delay between the occurrence of the jump and its detection. The delay depends upon the threshold value ε and the size of the data window. From simulations, it can be seen that the occurrence is accurately estimated after the delay.
In FIGS. 16 and 17, the estimated dynamic profiles (curves 1602, 1702) are plotted against the innovation sequence data (curves 1600, 1700) with a sample of 4 nm height. FIG. 16 shows the single hit case and FIG. 17 shows the multiple hit case. Note that for FIG. 17, adaptive filtering is used to update the innovation sequence after successive detection and estimation. The good match between the two plots suggests the efficient performance of the GLRT algorithm even for the multiple hits case when the successive hits (jumps) are not widely separated in time. Note that for the multiple hit case, one has to be careful in choosing the finite data window size to keep the false alarm and missed alarm rates low.
Conventional detection schemes rely on steady state amplitude and phase data. In the simulation model it takes approximate 1200μ seconds to reach a steady state periodic orbit. In conventional detection schemes that depend on demodulated amplitude profile, a threshold is chosen as 0.8 times the steady state amplitude. The invention takes around 200 μseconds to detect the jump considered for simulation. The transient data exists for 60 μseconds and the decision based on GLRT is made within 20 μseconds. Thus, transient data based detection is at least 10 times faster than the conventional case. For a single hit case, the estimation of jump is therefore 600 times faster compared to steady state estimation.
Experiments were performed to further verify the efficacy of the new transient signal based approach. The slowness of the piezo (the Z component of the X-Y-Z scanner for sample positioning dynamics makes it difficult to generate a waveform of pulses in the order of tens of microseconds. The approach used is to make use of the piezo dynamics to generate a testing waveform. The frequency response of the piezo was obtained using an HP control system analyzer and a model was fit to the response. The model response is compared with that obtained experimentally in FIGS. 18a and b. Curve 1800 is the model magnitude response, curve 1802 is the experimental magnitude response, curve 1804 is the model phase response and curve 1804 is the experimental phase response. It can be seen that the magnitude and phase responses of the model and experimental data substantially overlap.
Simulations were performed using the model to obtain the input signals to the piezo. FIG. 19b shows the response of the piezo to a voltage pulse shown in FIG. 19a of amplitude 0.5V, period 1000 μs and on time 500 μs with the four peaks during the on time separated by approximately 100 μs. Curve 1900 is the input voltage pulse and curve 1902 is the piezo response. The piezo dynamics results in the occurance of 4 peaks separated by approximately 100 μs during the on time. The maximum width of each peak is approximately 35 μs. The cantilever is oscillated at the first resonant frequency of 70.1 kHz. The amplitude of oscillation was approximately 80 nm and the oscillating tip was approximately 100 nm away from the sample surface. The piezo is actuated with the voltage pulse mentioned above. The oscillating cantilever interacts with the resulting peaks. The objective is to detect these peaks using the transient signal scheme. A two mode model was obtained for the cantilever with the first resonance at 70.1 kHz and the second resonance at 445 kHz. This model is used to build the Kalman filter and obtain the innovation sequence. The resulting innovation sequence is shown in FIGS. 20a, b and 21a, b. In FIGS. 20a and 20b, the cantilever deflection signal is plotted against the pulse shape generated using the piezo dynamics. Curve 2000 is the cantilever deflection, curve 2002 is the piezo position, and curve 2004 is the innovations. It is difficult to detect when the cantilever tip interacts with the peaks looking at the deflection signal 2000. The innovation process, on the other hand, bears the signature of the hits. Every time the cantilever tip interacts with the sample, the Innovation curve loses the zero mean white nature as can be seen. In other words, the innovations become non-white when there is a hit and a dynamic profile appears. This dynamic profile is detected using the techniques described above. In FIG. 21a and b, curve 2100 is the cantilever deflection, curve 2102 is the piezo position, and curve 2104 is the innovation sequence. It can be seen that the innovations show signatures of all four peaks and that the dynamic profile is clearly seen when the hits occur. The resulting likelihood ratio is shown in FIGS. 22a–c. In FIGS. 22a–c, curve 2200 is the innovation sequence, curve 2202 is the likelihood ratio, and curve 2204 is the piezo movement. Using the likelihood ratio it is possible to accurately detect the hits as predicted by the simulations proving the efficacy of the transient signal based detection scheme.
The GLRT is a computationally expensive algorithm. A simplified version of GLRT can be used where maximization over θ is replaced with {circumflex over (θ)}=n−M+1. Another simplification is SLGR where maximization over ν is replaced with a prior fixed {circumflex over (ν)}. These simplified algorithms require a wise selection of data window size M and prior fixed {circumflex over (ν)}.
From the foregoing, it can be seen that a new approach to determine cantilever movement has been described. An observer based state estimation and statistical signal detection and estimation techniques have been applied to Atomic Force Microscopes. A first mode approximation model of the cantilever is considered and a Kalman filter is designed to estimate the dynamic states. The tip-sample interaction is modeled as an impulsive force applied to the cantilever in order to detect the presence of sample. Generalized likelihood ratio test is performed to obtain the decision rule and the maximum likelihood estimation of the unknown arrival time of the sample profile and unknown magnitude of it. Simulations on a realistic model indicate significant gains in the speed of detection using these techniques.
All references, including publications, patent applications, and patents, cited herein are hereby incorporated by reference to the same extent as if each reference were individually and specifically indicated to be incorporated by reference and were set forth in its entirety herein.
The use of the terms “a” and “an” and “the” and similar referents in the context of describing the invention (especially in the context of the following claims) are to be construed to cover both the singular and the plural, unless otherwise indicated herein or clearly contradicted by context. The terms “comprising,” “having,” “including,” and “containing” are to be construed as open-ended terms (i.e., meaning “including, but not limited to,”) unless otherwise noted. Recitation of ranges of values herein are merely intended to serve as a shorthand method of referring individually to each separate value falling within the range, unless otherwise indicated herein, and each separate value is incorporated into the specification as if it were individually recited herein. All methods described herein can be performed in any suitable order unless otherwise indicated herein or otherwise clearly contradicted by context. The use of any and all examples, or exemplary language (e.g., “such as”) provided herein, is intended merely to better illuminate the invention and does not pose a limitation on the scope of the invention unless otherwise claimed. No language in the specification should be construed as indicating any non-claimed element as essential to the practice of the invention.
Preferred embodiments of this invention are described herein, including the best mode known to the inventors for carrying out the invention. Variations of those preferred embodiments may become apparent to those of ordinary skill in the art upon reading the foregoing description. The inventors expect skilled artisans to employ such variations as appropriate, and the inventors intend for the invention to be practiced otherwise than as specifically described herein. Accordingly, this invention includes all modifications and equivalents of the subject matter recited in the claims appended hereto as permitted by applicable law. Moreover, any combination of the above-described elements in all possible variations thereof is encompassed by the invention unless otherwise indicated herein or otherwise clearly contradicted by context.

## Claims

1. A method to detect transient cantilever movement in an atomic force microscope, the cantilever having a tip, the method comprising the steps of:
receiving an input signal and tip deflection data corresponding to the input signal;
generating an estimated dynamic state of the cantilever movement using the input signal and tip deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude with additive white Gaussian noise; and
computing maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test.

2. The method of claim 1 wherein the step of computing maximum likelihood estimates includes the step of selecting a threshold ε such that a false alarm rate is below one percent.

3. The method of claim 1 further comprising the step of updating the estimated dynamic state with the maximum likelihood estimates of the arrival time and the magnitude.

4. The method of claim 1 wherein the dynamic profile is defined by the equation


Yi;θ=H[Φ(i,θ)−FX(i−1;θ)]


where


X(i;θ)=KiYi;θ+FX(i−1;θ)


and where Φ(i,θ)=Πj=θiF, X(i;i)=KiH, Ki is a Kalman filter gain, F is a function of the first resonance frequency of the cantilever and the damping factor in free medium, θ is an arrival time, and H is a function of cantilever tip position and cantilever tip velocity.

5. The method of claim 4 wherein step of generating an estimated dynamic state includes the step of generating a maximum likelihood estimate of arrival time, the maximum likelihood estimate of arrival time being given by a value θ≦n such that {circumflex over (θ)}n=arg max1≦θ≦nl(n;θ) where l(n;θ)=dT(i;θ)C−1(n;θ)d(n;θ),




  
    
      C
      ⁡
      
        (
        
          n
          ;
          θ
        
        )
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          
            i
            ;
            θ
          
        
      
    
  
  ,
  
    
      d
      ⁡
      
        (
        
          n
          ;
          θ
        
        )
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          i
        
      
    
  
  ,
  
    V
    =
    
      
        HPH
        T
      
      +
      R
    
  
  ,




H is a function of cantilever tip position and cantilever tip velocity, P is a state estimation error covariance matrix and R is measurement noise power.

6. The method of claim 5 wherein the step of generating the maximum likelihood estimate of arrival time includes the step of generating the maximum likelihood estimate of arrival time in a data window of finite length.

7. The method of claim 4 wherein the maximum likelihood estimate of magnitude is given by a value {circumflex over (ν)}n(θ)=C−1(n;θ)d(n;θ) where




  
    
      C
      ⁡
      
        (
        
          n
          ;
          θ
        
        )
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          
            i
            ;
            θ
          
        
      
    
  
  ,
  
    
      d
      ⁡
      
        (
        
          n
          ;
          θ
        
        )
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          i
        
      
    
  
  ,




V=HPHT+R, H is a function of cantilever tip position and cantilever tip velocity, P is a state estimation error covariance matrix and R is measurement noise power.

8. The method of claim 7 wherein the step of generating the maximum likelihood estimate of magnitude includes the step of generating the maximum likelihood estimate of magnitude in a data window of finite length.

9. A method of estimating the transient deflection of a cantilever operating in dynamic mode in an atomic force microscope system, the method comprising the steps of:
receiving, by a Kalman observer, an input signal and tip deflection data corresponding to the input signal;
generating, by the Kalman observer, an estimated dynamic state of the cantilever movement using the input signal and tip deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude;
computing, by the Kalman observer, maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test; and
estimating the transient deflection of the cantilever using the maximum likelihood estimates of the arrival time and the magnitude.

10. The method of claim 9 wherein the step of generating an estimated dynamic state includes the step of generating a measurement residual, the measurement residual equal to the dynamic profile plus an offset, the offset being a non-jump measurement residual.

11. The method of claim 10 wherein the non-jump measurement residual comprises white Gaussian noise.

12. The method of claim 9 wherein the dynamic profile is defined by the equation


Yi;θ=H[Φ(i,θ)−FX(i−1;θ)]


where


X(i;θ)=KiYi,θ+FX(i−1;θ)


and where Φ(i,θ)=Πj=θiF, X(i;i)=KiH, Ki is a Kalman filter gain, F is a function of the first resonance frequency of the cantilever and the damping factor in free medium, θ is an arrival time, and H is a function of cantilever tip position and cantilever tip velocity.

13. The method of claim 12 wherein the maximum likelihood estimate of arrival time is given by a value θ≦n such that {circumflex over (θ)}n=arg max1≦θ≦nl(n;θ) where l(n;θ)=dT(i;θ)C−1(n;θ)d(n;θ),




  
    
      C
      ⁡
      
        (
        
          n
          ;
          θ
        
        )
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          
            i
            ;
            θ
          
        
      
    
  
  ,
  
    
      d
      ⁡
      
        (
        
          n
          ;
          θ
        
        )
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          i
        
      
    
  
  ,




V=HPHT+R , H is a function of cantilever tip position and cantilever tip velocity, P is a state estimation error covariance matrix and R is measurement noise power.

14. The method of claim 12 wherein the maximum likelihood estimate of arrival time is given by a value {circumflex over (ν)}n(θ)=C−1(n;θ)d(n;θ) where




  
    
      C
      ⁡
      
        (
        
          n
          ;
          θ
        
        )
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          
            i
            ;
            θ
          
        
      
    
  
  ,
  
    
      d
      ⁡
      
        (
        
          n
          ;
          θ
        
        )
      
    
    =
    
      
        ∑
        
          i
          =
          θ
        
        n
      
      ⁢
      
        
          Y
          
            i
            ;
            θ
          
          T
        
        ⁢
        
          V
          
            -
            1
          
        
        ⁢
        
          Y
          i
        
      
    
  
  ,
  
    V
    =
    
      
        HPH
        T
      
      +
      R
    
  
  ,




H is a function of cantilever tip position and cantilever tip velocity, P is a state estimation error covariance matrix and R is measurement noise power.

15. A control system adapted to perform the method of claim 9 comprising:
a first mode approximation model of the cantilever;
a tip-sample interaction model in communication with the first mode approximation model; and
a Kalman observer designed to estimate the dynamic state of the cantilever movement, the Kalman observer in communication with the first mode approximation model of the cantilever.

16. The control system of claim 15 wherein the tip-sample interaction model is modeled as an impulsive force applied to the cantilever.

17. The control system of claim 16 wherein the impulsive force comprises a repulsive part of the tip-sample interaction.

18. The control system of claim 15 wherein the first mode approximation model of the cantilever and the Kalman observer are adapted to receive an input signal of the cantilever.

19. The control system of claim 15 wherein the Kalman observer is adapted to:
receive an input signal and tip deflection data corresponding to the input signal;
generate an estimated dynamic state of the cantilever movement using the input signal and tip deflection data, the estimated dynamic state having a dynamic profile of unknown arrival time and unknown magnitude;
compute maximum likelihood estimates of the arrival time and magnitude using a generalized likelihood ratio test; and
estimate a transient deflection of the cantilever using the maximum likelihood estimates of the arrival time and the magnitude.