Analytical model for the approximation of hysteresis loop and its application to the scanning...

8/14/2019 Analytical model for the approximation of hysteresis loop and its application to the scanning tunneling microscope

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Analytical model for the approximation of hysteresis loopand its application to the scanning tunneling microscopeRostislav V. LapshinDelta, Microelectronics and Nanotechaology Research Institute, 2 Schelkovskoye Shosse,Moscow 105122, Russia{Received 11 January 1994; accepted for publication 23 January 1995)A new model description and type classification carried out on its base of a wide variety of practicalhysteresis loops are suggested. An analysis of the loop approximating function was carried out; theparameters and characteristics of the model were defined-coersitivity, remanent polarization, valueof hysteresis, spontaneous polarization, induced piezocoefficients, value of saturation, hysteresislosses of energy per cycle. It was shown that with piezomanipulators of certain hysteresis looptypes, there is no difference in heat production. The harmonic li nearization coefficients werecalculated, and the harmonically linearized transfer function of a nonlinear hysteresis element wasdeduced. The hysteresis loop type was defined that possesses minimum phase shift. The averagerelative approximation error of the model has been evaluated as 1.5%-6% for real hysteresis loops.A procedure for definition of the model parameters by experimental data is introduced. Examples ofusing the results in a scan unit of a scanning tunneling microscope for compensation of rasterdistortion are given. 0 1995 American Institute of Physics.

I. INTRODUCTIONThe law of a closed hysteresis curve would emerge in

many physical phenomena such as dielectric hysteresis [po-larization curve P =f(E)], magnetic hysteresis [magnetiza-tion curve B =f(H)], elastic hysteresis [deformation curvee=f(F)], and some others.This phenomenon is widespread and important, howevera simple analytical equation that could approximate it with asufficient degree of precision has not existed. Therefore,rather often when analyzing various processes and systemswith a hysteresis element, the solution was being searchedfor either graphically-by using experimental data, or withthe help of straight-line approximation of the curve.12

The first method is inconvenient because of the need fortable function representation and low precision of graphics.Low precision of the second method is caused by roughnessof piecewise-linear approximation (certainly, if the numberof the segments is not too great) and thereto it impliessearching for the solutions at several intervals followed bygluing them with each other.Other methods of hysteresis loop approximation areknown in both polynomial models2 and integral operators3*4classes. But their usage is limited because of the complexityof hardware support or great time of calculations.This work pursued two objectives. First, to give a de-scription of the suggested model and its properties and char-acteristics. The part of the paper devoted to this matter is ofa wide, general scope inasmuch as the use of the model isconsidered for analyzing static nonlinearity of hysteresisloops of various types and physical nature encountered inmany scientific instruments. Second, to highlight the waysand manners of practical application of the conclusions ob-tained from the model description to particular purposes,namely, for description and compensation of nonidealities ofpiezoceramic manipulator$-2 of a scanning tunneling micro-

scope (STM)-nonlinearity and ambiguity of static charac-teristic, piezoceramic creep, thermal drift.

II. DESCRIPTION OF THE MODELA. Analytical expression for hysteresis curve family.Formation of the types and their classification

The family of hysteresis loops can be described by ageneralized transcendental equation in parametric form asfollowsx(cr)=-fa cosm atb, sin IY,

y(a)=by sin LY,where a is the split point coordinate; b,, 6, are the satura-tion point coordinates; m, y1 are integer numbers (see TableI); a is a real parameter (--~=GcK3) powers in Eq. (1) define the steepness (I,II,IIIcurvesj. The derivative types are tilted classical, doubleloop, bat [Figs. l(d)-l(f)]. They derive from main typesor from other derivative types with implementing some extraoperations.So, a curve of tilted classical derivative type [Fig. l(d)],as the tangent in inflection point of the unsplit loop makes anangle pf?rl;! with ox axis, can be built by rotating the co-ordinate system clockwise through the angle 6=~12-p.

4718 Rev. Sci. Instrum. 66 (9), September 1995 0034-6748/95/66(9)/4716/13/$6.00 Q 1995 American Institute of Physics
mailto:%[email protected]:%[email protected]


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TABLE I. Classification of hysteresis loops.

No. Qpe Vpestatus n m P Equation Figure

1 Leaf Main

2 Crescent

3 Classical

3 Tiltedclassical

5 Doubleloop

6 Bat

Main

Derivativefrom type3Derivativefrom types1/3/4Derivativefrom types

l/3/4

1 h1 3 arctan -25 bx1 T2 35 2-13 3 ir5 z

a a 37o.*.- 2a a a

a a a

n(a)=a coP ruf h, sin my,y(a)=by sin (Y

or

x(a) = b, cos a- a sin (Y,y(a)=b, cos a,occrs2rr~(n)=n(a)cos B+y(cu)sin 0,j(a)=-x(cr)sin B+y(ru)cos 8,O=(d2)-p

;(cr)=s(n)ib,,~(a)=y(a)+b,i((Y)=x(a),j?a)=ly(Nl

l(a)-11 d-IIl(a)-IIIl(b)-11 b)-IIl(b)-1111 c)-II(c)-IIl(c)-III

l(d)

W

10

aDependi ng on the initial type.

Thus, using the foregone formula for transformation of Car-tesian coordinates with the axes rotated,14 the following ,qa)=x(a)fxg,(5)equation is obtained: Y(aj=Y(aj+Yo,

i(a) =x( ajcos B+y(a)sin 8,(2)y((~)= -vx(ajsin ~+.Y(CX)COS,

where the equations for x(a), y(a) are determined by thecomposition base type.

where X(a), y(a) are the coordinates of the rotated system.When rotating, the split point +a and the saturationpoint 51b change their positions relative to the origin system.Therefore a preliminary distortion of their coordinates isneeded to get these points to coincide with the originals afterrotation. Here, the following transformation formulas14 willbe used:

To form a model of bat type [Fig. l(f)], it is sufficient totakti the moduIe of y(a).

cix=a cos 19 (31and

b,=b, cos 0-b, sin 0,by=bS sin O+b, cos 8. (4)

The direction of passing along a hysteresis loop is sup-posed to be determined in the following way.. For a coor-dinate lag system, the movement along the lower half of thecurve is associated with the inequality dxldt>O, and alongthe upper half .with dxldt


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Cc)

(d)

II-.-L.---l 4 I I I I(e)

FIG. 1. Hysteresislooptypessupportedbythemodel: (a)JLeaf(u=O.2;b,=O.6;6,=0.8;m=l,3,5;n=l; 8=O);(b)crescent(n=0.2;b,=0.6;by=0.8;m=l,3,5; n=2; 0~0); (c) classical (~~0.2; b,-0.6; b,=0.8; m= 1,3,5; n=3; o=O); (cl) tilted classics (~=0.2; b,=O.fj; b,=O.g; m=3; a=3;0=15); (e) double loop (u=O.l; b,=O.4; b,=0.4; m=3; n=3; 0=15); (f) bat (u=O.2; b,=O.6; b;=O.fj; m=3; a=3; +l5o).

case, instead of Eq. (6). any suitable function that passes derivative of the input signal is changing (this moment cor-through the points + b and the origin of coordinates can be responds to a saturation point of a particular cycle). Next, asused. the value of the CY arameter is known at that moment, theTo trace the history of motion, it should be done in thefollowing way. First, the moment is defined when the sign of parameters t bz = x( cr), 2 b; = y(a) can be calculated.Then, as the law of disposal of particular cycles inside Limit4720 Rev. Sci. Instrum., Vol. 66, No. 9, September 1995 Analytical model for the approximation of hysteresis loop


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FIG. 2. Embedde d loops. Determination of spontaneous polarization value(a=0.2; b,=O.6; b,=O.S;m= 1; n=3; b: = 0.3).

one is known, it becomes possible to define a and f? pa-rameters. Finally, the a parameter is assigned the value ofrrkl2;. where k=1,3,5,... Here are all the parametersneeded to proceed with the motion along the new cycle.Beginning with m =3 and IZ= 1, function (1) cannot beresolved in explicit form y=f(x) because it would result inan equation of more than fourth degree. However, an explicitrecord can be obtained for me function, inverse to Eq. (l),which is yielded by simply swapping x(a) and y(a)

x(a) = b, sin a,y(a)=a COP a+b, sin a. (7)

That inverse function will serve as the base for building ahysteresis compensation system. An explicit record for Eq. (7) can be given byy=~x~~,~~, -b,sxG+b,. (8)Y YTaking into consideration, on one hand, that the opera-tions of division and square root are not facile in hardwareimplementation, require much time of calculation, and arenot typical in digital signal processor (DSP) algorithms and,on the other hand, the convenience of definition of the in-verse function and the contour pass direction, the parametricform (1) must be admitted to be the most suitable to be used

further.

6. Analysis of the curve: Coersitivity, remanentpolarization, dielectric spontaneous polarization,induced piezocoefficients, value of saturationFunction (l), which was taken as the model of hyster-esis, is a continuous, nonlinear, ambiguous, ~limited one de-fined on the r-b, ,b,] segment. Besides, this function isperiodical with the period 271; since the following equalitiesare true:x(a)=x(a+2rk),yW=yb+2~W, (9)

where k= 1,2,3,... Owning to the property of periodicity offunction (l), use of the model will prove to be most effectivewith cycle processes, e.g., in scann ing systems (see Sec.III A).Let us find the zeros of function (1). Here, the value ofthe (Y parameter should be defined such that j(a) becomesequal to 0, i.e., b, sin cr=O, whence (Y= nk. Substituting thisvalue of (Y n x(a), it will yield x = t a. Note that the physi-cal sense of a zero of function (1 is coerditivity.Now, the coordinates of the +c point are to be definedas the intersection points of the loop and oy axis. The pointtc defines the value of remanent polarization. Writing

a COP a+b, sin LY=O,b, sin LY= kc. (10)

In the case of m = n, this will result in the following solution:+c=

?b,,dl +(b,/a)2 (11)

The value of hysteresis H, is defined as clb,( 100%);by using expression (1 ), it can be written asHy= 100%41 +(b,la)2m (12)

Note, that the ratio b,la in expression (12) is none other than100%/H,, where H, is hysteresis along ox axis. Thus, acorrelation is obtained that links the values H, and HI witheach other (m = n)(fy (%)I- 1. (13)Let us find the coordinate of the c point (see Fig. 2). Itis the intersection point of the oy axis and bd tangent drawnthrough the saturation point of the unsplit curve (6). Notethat the point c defines the value of dielectric spontaneouspolarization. The equation for the bd straight line is given byy = kx + c , where k = tan y. The derivative k = dyldx of theunsplit curve (6) in the point x= b,r is defined by b,l(nb,).From the equation for the bd line with x = b, , y = b, , takinginto account the found k value, the value of C is defined asc=by 1-i.( i (14)It will be shown (not rigorously) that function (1) is odd(except for crescent type). By definition, a function y =f(x>would be odd provided f( -x) = -f(x). .In regard to func-tion (1) p-x(h)=-(a COP a+b, sin a),-y(a)= -by sin (Y. (15)

Using the identities -sin cr=sin(-cr), cos cr=cos(-a), mak-ing a formal replacement of the -a parameter with l, andconsidering the oddness of m and n as well as the fact thatthe sign of the split parameter a in formula (1) defines onlythe beginning point and the curve pass ,direction and, there-fore, can be chosen as negative here, the following state-ments can be formulated:

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-x(b) = a COP l+ b, sinn 5,-y(l)=b, sin 5.

Thus, the right-hand parts of the equations in (16) have nochanges beside those i n Eq. (1); consequently the function isodd. As the graph of an odd function is symmetrical relativeto the origin of coordinates, the information of a half is suf-ficient to have the whole function set. The property of sy m-metry of function (1) gives the opportunity, for example atestimating the approximation error (see Sec. II E), to carryout the calculations at a half of the domain.Let us find first and second derivatives of function (l),which i n case of a parametrically set function are given by14

dY 3(a)-=-dx X(a) (17)and

dy i( a)j(a) -i( ct)j( n)z= i( ff)3 , (18)

where i((~)=dx/da#O, .;C.(~~)=d~xldc? , j(n)=dylda,j;(a) = dyldcu are first and second derivatives of the func-tions X( a)) and y ( cuj by cr parameter, respectively. For thefunction being consideredi( CY) -am cosm CYan CY+ ,n sin a cot cr, (19),?(~~)=am COP o$(m- 1)tan CY- l]+b,n

Xsin Lt[(n- 1 C0t2 a- 11, CQj(a)=b, cos LY,j(a)= -I?, sin a.

In accordance with Eq.

iwG3

(17), the first derivative is defined asdy b, cos .CYdx= -am sin LY osm-t ff -I-h,n cos ff sir?- (Y (23)and, in accordance with Eq. (18) after simple transforma-tions, the second derivative is defined as

Id2yz= abym(2-m)sin2 (Y ~0s.- a+abym cos+t cu-b,b,n(n- 1 sin+ CY ot3 CY(-am sin (L cos-t cr+b,n cos a sinn-* a)3 (24)

Expression (23) allows for calculating the induced pi-ezocoefficients of a hysteresis curve in ,the coordinates ofdisplacement versus electric field strength as well as the dif-ferential magnetic permeance at any point of the curve.B=f(H). By analyzing expressions (23) and (24), it can be shownthat for m = 1 function (1) reaches its maximum value in thepoint ( + b, , + b,) and minimum-in the point ( - b, , - br).For ail m# 1, the lower and the upper parts of hysteresisloop (1) are monotonously increasing functions, therefore atthe edges of the domain, i.e., in the points I!Z , , function (1)has the greatest + b, and the least -b, values. Thus, the( %b, , + by) points of curve (1) are the saturation points of ahysteresis loop.

C. Square of a loop.)

Physically, the square of a hysteresis loop characterizesthe heat losses that cause heating of the material and, there-fore, define its efficiency coeffic ient. To find the square S ofa bop, the following integralI should be calculated:

x(a) g-y(a) $ dq (25)then taking into account expressions (l), (19), and (21), writ-ings=; J-

2.nn [( a cosm a+bx sin cu)b, cos a-b, sin cr

X(-am sin a cosmml a+b,n cos a sin-*.cw)]da.CW

4722 Rev. Sci. Instrum., Vol. 66, No. 9, September 1995

Opening the parenthesis, grouping up the terms, then usingDe Moivres expansion4 of cosm C Yand CO?- a, andsolving the integral for odd n, the hysteresis loop square (26)will be given by

(27)where Cf=I!l[k!(l-- k) !] are binomial coefficients. Thus,the square of leaf and classical hysteresis loops would notapparently depend on the saturation value by x coordinateand is only determined by the split constant and the satura-tion value by y coordinate.Formula (27) is applicable to odd n, but the coefficient PZitself does not participate in it explicitly. Therefore, the fol-lowing theorem can be formulated: the quantity of the heatproduced by the piezomanipulator for a cycle is the same forhysteresis loops of both leaf and classical types, provided thevalues of m, a, b, involved are the same.Since the square of a geometrical figure wou ld be invari-ant to rotation, by substituting a and b, in formula (27) withtheir corrected values Gb and b,, [see Eqs. (3) and (4j], aformula can be written to define the square of tilted classicalhysteresis loop

s= ; clnm,:u2.+m( c-m-l WA cy12)[ 1~.p+lXrra[b, sin 28+b,(cos 28+1)]. CW

As the terms containing the m coefficient are consideredat various values of m in formulas (27), (28), it can be con-Analytical model for the approximation of hysteresis loop


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eluded that with fixed values of the rest model parametersthe inequality Snr=5CSm=32mfnC~~~1j/2b,b,,-j22nfmC~~,)aby

= 2211( ~++~1)/2j2a2+22m( C~,+,/2)2b,2 (36)

which only depends upon 6, amplitude and does not upon wfrequency. Since W(b,) =q(b,) +jG(b,), from Eq. (36)comes the next22m+nCll;+11)/2bb

4ibJ= 22rztC(m+~ r2 2 2 5 ym+l ) ) a +22m(C$+l*)2)2be;22n+mcc$++,i1)r2uby (37)

4(bJ=- 22ncC(m+1v2 2 22which are the required harmonic linearization coefficients offunction (1). For the case of m = n = 3, coefficients (37) aregiven by q(b,)=4b,b,l[3(a2+b~)] and {(b ,)=-4abyl[3(a2+bf)]; their graphs are shown in Fig. 3.

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The coefficient q(b,) determines the steepness.of theaveraging line. From formula (37), it is easy to see that withthe amplitude of input harmonic signal b, increasing, q(b,)approaches zero since in that case nonlinearity (1) gets satu-rated.The amplitude of first harmonic A(b,) = [ W(b,) 1 can bedefined by the transfer function (36) as

A(.b,) h&-4 + Li2!b,>3minc by= J22n(C~~f1)/2j2u2+22m(C~~,1)/2)2bx2 (38)and its phase cp(b,) =arg W(b,)-as

ci(bx) yp+ 1)1Zucp( ,) = arctan q0 = - arctan m+12CI(,+,)2), . (39)IFor example, the phase shift of first harmonic at the output ofclassical nonlinear element shown in Fig. l(c), curve II willmake - 18.4.The minus sign in expression (37) for G(b,) [this coef-ficient is placed by the derivative in formula (29)] as well asthe minus sign in Eq. (39) mean that the presence of a hys-teresis element results in a phase lag of the output signalbeside the input one. As is derived from Eq. (39), the widerthe hysteresis loop is (the greater the split parameter a), thegreater the phase shift becomes.By analyzing expression (39), it can be shown that thefollowing theorem would take place: with the same values ofm, a, and b, , minimum phase shift wou ld occur at the leaf-type loops, and at that IP~=~I


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(4

i j -mri i Ib)

HCE element. The inverse function argumentX is a controls ignal, for exampl e X In scan signal, which und ergoe s re-l iminary distort ion in the HCE and then is appl ied, hrough ahigh voltage ampli fier (HVA), to the piezomanipulator. hus,the result ing displacement of the manipul ator wi l l corre-spo nd o the input function of the scan.

I f l4M801 20Y2Do 280 I

I I I I I I I I I II I I I I I I I I(d)

(4

FIG. 4. Accuracy of approximation: -model data: I I-experimental data.(a ) Displacementof STM PZT X, Y p iezomanipulators s appl ied vol tage(leaf; a=32 .6; b,=300; b,=955; m=3; n= I: p=72.6 ; H,-11%; (6)=1.5%); (b ) ceramic polar izat ion vs electr ic f ield strength (Ref. 15) ( t i l tedclassical,a=S4; b,= 130; b,=36.4; t-n= 1; n=3; 8=27.7 ; (4=6%); (c )cerami c polari zation vs electric field strength (Ref. 16); (tilted classical;u= VO ; b,=340; b,=39.6; m=5; a=3; 19=3 0; 6)=4.1%); (d ) ceramicpolar izat ion vs electr ic f ield strength (Ref. 16) (l eaf; u=180; b,=195;b,=63.8; rn= 1; n= 1; p=18.1; ( s)=3.9%); (e ) ceramic polar izat ion vselectric field strength (Ref. 16) (tilted classical ; cl= 122.5; b,=341;b,=23.2; m= 1; n=3; .9=14 ; (5)=2.7%).

A. Compensat io n of raster distort ion in the STM scanunitThe flow chart of the hysteresis compen sati on unit isshown in Fig. 6(a). The scheme s bui l t on an alog elements

and consists of the fol lowing units: GE N sinusoidalgenera-tor; a chan nel of MU L multipliers that carries out the oper a-Rev. Sci. Instrum., Vol. 66, No. 9, Septemb er 1995 Analyt ical model for the approximation of hysteresis loop 4725


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TABLE II. Results of calculation of the approximation error.Source ofexperimentaldata and figure

Fig. 4(a)

Type of model and itsparameters:a;b, ;b, ;tn;n;O or pLeaf32.6;300;955;3;1;72.6

A m49.5 5.2 I .5 20.5

Ref. 15, Fig. 4(b) TiIted classical 5.4 14.8 6.0 2.854:130;36.4;1;3:27.7Ref. 16, Fig. 4(c) Tilted classical 3.3 8.3 4.1 I.990;340;39.6;5;3;30Ref. 16, Fig. 4(d) Leaf 9.0 14.1 3.9 3.2180;195;63.8;1;1;18.1Ref. 16, Fig. 4(e) Tilted classical 1.7 8.1 27 0.9L22.5;341;23.2;1;3;14

tion of raising the input oscillation sin( wt) to m and n pow-ers; PS phase-shifting element that converts the sin(wt)oscillation into co?(wt) by shifting it by a quarter of theperiod; AMP2, AMP3 operational amplifiers with the gainsK=a and K=b,, respectively; SUM summing amplifier;AMP1 amplifier wi th the gain K =b, ; COMP comparator,and AS analog storage unit. The last three components of thescheme are destined to set the accordance between theXIn input scan signal and the parameter a = wt by succes-sive approximation [such embodiment is simpler than an ex-plicit calculation of the function arcsin(X(t)lb,)].In fact, the scheme implements the calculation of the

will look like in case of a circular scan microscope. Note thata circular scan is more preferable beside a raster scan for thefollowing reasons: the design of X and Y manipulatorsproves to be symmetrical, an even employment of the ma-nipulators occurs during the operation, the scan growssmoother and to those the possibility appears of getting theadditional features of the microscope which have been de-scribed in paper.t7 As a disadvantage of a circular scan, therecan be admitted the necessity of converting the points of thecircle into a conventional rectangular display window whenvisualizing.

inverse function (7). Its operation principle is clear with noextra elucidation, to only point out that, to ensure the condi-tion of in phaseness, the SCAN unit forming the scan worksout an XStart signal at the beginning of each half-periodof the scan, which starts the waiting generator GEN so that itwould generate the proper half of the sine function accord-ingly to the sign of the scan signal derivative [the probingfrequency of GEN is much greater than the frequency of thescan signal X(t)]. Pay attention to the triangle (not ramp)impulses of the scan signal shown in Fig. 6(a), the circum-stance pointing .out to operation with no idle stroke.Let us show now what the scheme of the HCE element

To have a circular scan, it is necessary to apply a sin(wt)voltage to the X manipula tor and cos(wt) to Y. Let us, in-stead of the triangle voltage [see Fig. 6(a)], mentally applythe X(t) = 6, sin a voltage, where LY=wt. It can be easy tosee that the chain of units-AMP1 amplifier, COMP com-parator, and AS analog storage unit-becomes out of useand, therefore, can be excluded from the scheme. Now, in-stead of the triangle voltage. Y(t), let the voltageY(t)= b, cos a-be applied (structurally, X and Y channelsare identical). Since a cosine function would outstrip a sineat ?Tl2, formula (7) can be given by y(a)= a cosm( a + 74 2) + b, sin( CL 7r/2), whence the inversefunction can be written in another recordx(aj=by cos a,y( a)=bx CO? a-a sin CY,

which can be used abreast with Eq. (7). From the last rea-sonings, it can be concluded tha t the circular scan generationscheme with simultaneous hysteresis compensation will takethe outline presented in Fig. 6(b), where SUB is an opera-tional amplifier connected in a differential mode. If compar-ing Figs. 6(a) and 6(b), the decrease of hardware expendi-tures can be revealed with the circular scan implementation;note that the parameter b,, becomes unnecessary. The circularscan scheme described here would serve as the base forbuilding cycloid and spiral scans.

FIG. 5. HCE hysteresis compensation elements included in S TM controlsystem.

The units of hysteresis compensation described aboveare the simplest ones and, generally speaking, do not permitus to work with loops of derivative types or various imagesizes either to shift the image window along the scanning

4726 Rev. Sci. Instrum., Vol. 66, No. 9, September 1995 Analytical model for the approximation of hysteresis loop


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HCE _ . . . _ . - - . . - . 1 - - . -)X Channel ---I__aWos(ut)

1 bxEsin(ut)

siram

nrir1I I aWosm(ut) I

sinqut)

field or to implement vector access to a surface point. Ahardware realization of the features mentioned above wouldresult in a substantial complication of the equipment. Theway out could be to build a digital system or to compute themodel by a program.4 digital hysteresis compensation system will be de-signed by using the structures and the conclusions yieldedwhile synthesizing the analog system. So, for the schemespresented in Figs. 6(a) and 6(b), the analog multipliers MULand the operational amplifiers AMP1/2 /3/4 should be re-placed with digital multipliers: the summing operationa l am-plifier SUM and the differentiating amplifier SUB-with anarithmetic-logic unit; the analog storage unit AS-with astrobed register with a digital-to-analog converter (DAC)connected up to its output; the sinusoidal generator GEN-with a read-only memory (ROM) scheme with a sine tablewritten down in it; the analog comparator COMP-with adigital one. A ring counter shou ld be connected to the ROMaddress inputs. The function of the phase-shifting element PSconsists of shifting the value of the address worked out bythe ring counter so as to skip exactly a quarter of the periodof sine in the ROM table.

At program realization of the model, the functionscosm a, sin a are stored in the computer data memory assome table structures. The scan voltage can be transformedinto the correspondi ng CY alue by either successive approxi-mation (similar to the operation of the unit chain AMPl-COMP-AS) or immediately by an arcsin table also kept inthe data memory, or by directly calculating the arcsin by theforegone identityIfit)arcsin - = t @it)by I oJijqq

where .f(t) is a scan function ([f(t) 1


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I Channel 1

FIG. 7. Fl ow chart of digital device for hysteresis compensation (soft-hardware realization).

solved at a DSP because the operation AB,+C (whereA,B, C are real numbers) is encountered in the calculationswhich is typical of DSP algorithms. Here, A=a,Bi=COSm sir C=DEjfF, and D=b,, Ei=sinn ~j, F=O.The indexes i and j refer to the addresses of the memorycells where the tabu lated values of cos CY~ nd sin Y~ arecontained, respectively.The flow chart of a hysteresis compensation digital unitis shown in Fig. 7. This unit is realized on a soft-hardwarebase and ensures working with all the hysteresis types andperforming a scan along an arbitrary trajectory so as to sup-port any scan type, to carry out rotation of the scan windowaround oz axis (in order, e.g., to reduce moire distortionj aswell as to vary the window size and shift it within the scanfield.A ring programmable counter PCT2 is involved in thescheme which, in the rate defined by the Clock signal andthe division coefficient DC, generates the addressAddr2 and the read signal Read2 for the Dual-PortRAM (random-access memory) scheme with the model datawritten in it. From the RAM output, the MData2 code,which corresponds to the preliminary distorted current valueof raster voltage, comes into the DAC from where the signal,after having passed through a high voltage amplifier, is ap-plied to the manipulator. Another RAM port is intended forwriting the model data MDatal calculated by the micro-computer processor. The use of a dual-port RAM permits usto get the processes of calculating the model and generatingthe control signal for the manipulator coincided in time and,therefore, to increase the unit fast acting.Built on a dual-port RAM base, the scan subsystem iscapable of being transformed from synchronous into asyn-chronous. To do that, it is sufficient, instead of the Clocksignal, to apply a signal pointing out to readiness of the datain the tunnel junction stabilization system.* Besides, if thatsystem also uses a dual-port RAM as a Z-Buffer, then for thenext memory cell sampling, the Addr2 signal generated bythe PCT2 counter can be used. Thus, there can be seen agood mutual coordination in work between the tunnel junc-4728 Rev. Sci. Instrum., Vol. 66, No. 9, September 1995

tion stabilization system and the scan signal generation sys-tem.It is appropriate to note that when using the scan modesthat require frequent changing the model parameters a, b,,b, (e.g., in order to change the image size), in a certainoperation time, some error will have been accumulated in themodel. At the moment when it reaches a certain thresholddefined by the admissible approximation error, a compulsorycorrection must be done: for the model it is assigning 5-12 othe Q! parameter and for the piezoelement-applying thesaturation voltage to it so as to ensure it set at a fixed point o fthe limit characteristic-the saturation point.IV. EXPERIMENTAL RESULTS

The surface image of a test pattern is shown in Figs. 8(a)and 8(c), which was taken by a STM when scanning withidle stroke and without, accordingly. As the test pattern, adiffraction grating with 0.3 ,ccmperiod coated with gold wasused. Since the grating structure changes along one directiononly and scan piezomanipulators, because of hysteresis, in-troduce distortions along two directions, then for reflectingthese distortions on the image taken, the test object was fittedup so that its stripes would make an angle of some 45 withthe manipulator X axis.On the images, the distortions of the test pattern are seenwell, which consist in curving the stripes and changing theirwidth from one stripe to another [Fig. 8(a), cf. Ref. 111,splitting the stripes and formation of a double-sided comb-like structure [Fig. 8(c)], as well as parallel stripes lookinglike divergent ones [Fig. 8(e), cf. Ref. 121.Figures 8(b) and 8(d) present the corrected images of thesame area that were taken using the hysteresis compensationsystem shown in Fig. 7 [the model parameters were extractedfrom the data presented in Fig. 4(a)]. A visual comparison ofFigs. S(a) and 8(b) [8(c) and 8(d)] will show that nonlineardistortions caused by hysteresis of manipulator piezoceramicare practically removed.Hysteresis loops with parameter m equahng to unitwould have a section of negative derivative [see Fig. 4(d)].The behavior of piezomanipulator at that part of the curve issimilar to creep: on achieving e point the voltage begins todecrease, the displacement still increasing for a certain timeuntil the point b is reached. [The parameter (Y that corresponds to e point can be found by setting the denominator inEq. (23) equal to zero.] From works,7p8 t is known that creepwould cause distortions at the image boundaries where thetip motion is reversed (e.g., straight lines would get hooks atthe ends). Distortions of the same kind would appear when apiezomanipulator with the hysteresis loop said above is ap-plied. Thus, the resulting picture wilI contain distortions oftrue creep and hysteresis mixed up together.V. DISCUSSION

The approximating model suggested belongs to the realtime model class. It allows for compensation of the STMpiezomanipulator nonidealities such as nonlinearity and am-biguity, which makes it possible to get rid of the image dis-tortions (see Fig. 8 and Refs. lo-12), which is especially

Analytical model for the approximation of hysteresis loop


12/13

ij

...:/z :.::j;.::..::.:.:::..::...

: :.

(a)

important with great scans, to refuse the idle stroke in the As was shown in Sec. II C, the Q quantity of the heatraster scan, and consequently to reduce the scanning time; in produced by the piezomanipulator for a hysteresis cycle taddition, the model supports the mode of direct access to a determined by the system timer or by the scan frequency candesired point. Therein, the reconstruction of the true image is be easily calculated, for instance, by formula (28). The val-available for either preprocessing (preliminary distortion of ues Q and t might serve as the input data for some modelthe scan trajectory) or post-processing of the distorted image considering the heat exchange proceeding among the STMtaken. construction elements. After having defined the difference in

... ...._:., ,i >. ;.*.> .., I.i.: ..l.. . ..: .._ ...... i.. .......

(4 .

z . . . . . . . .. . .; . J 7 . .. .. .. . ..

FIG. 8. 128X 128 STM scan of the same 1X1 pm area of test patternsurface (diffraction grating with 0.3 pm peri od coated with gold) (a),(c)Surface image data obtained by scanni ng with and without idle stroke, ac-cordingly (the test object has been rotated by 45 counterclockwise) , thehysteresis compensation system turned off. (b),(d) Surface image data cor-responding to (a),(c) obtained with the hysteresis compensation systemturned on. (e) Surface image data obtained by scanning with idle stroke (thetest object has been rotated by 4S clockwise) . After compensation of hys-teresis, the image will look like the one shown in (b).

Rev. Sci. Instrum., Vol. 66, No. 9, September 1995 Analytical model for the approximation of hysteresis loop 4729


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the manipulator temperature AT by using such model, thedifference in the manipulator length A1 can be found by theformulaAl=@AT, (46)

where p is the heat expansion coefficient ( 1 K) ; 1 is theinitial manipulator length (m) . Calculated in such a way, thethermal drift AZ can be compensated in the scan unit byproperly shifting the scan window. Thermal drift compensa-tion is especially important in a scan device since there is notany active servo system mounted in it. Note that the STMheat processes, depending much upon construction featuresof the microscope and the materials used, are of a rathercomplicated nature and deserve a special investigation whichcomes out of the limits of this paper.The main advantage of the model, its simplicity, is calledforth by no need of calculating the parameters-they aretaken right from the experimental curve, neither the inversefunction is to be calculated. To that, the model parametersmake clear physical sense and have a simple geometricalinterpretation. A distinctive feature of the model is the pos-sibility of comparing different hysteresis loop types witheach other (see theorems in Sets. II C and II D).As a drawback of the model, there could be mentioned arestriction on the approximation accuracy to be achieved,i.e., its dependence upon the particular shape of the hyster-esis curve. Though, the reverse task could be apparently re-solved, the piezoceramic hysteresis curve being adopted tothe desired model by changing the technological parameters,namely, the chemical composition, the baking and coolingconditions, the mechanical influences, etc.Owing to application of the model, it becomes possibleto linearize the STM tunnel junction stabilization systemcontour by methods similar to those described in Sec. III A,which allows us to prevent the appearance of auto-oscillations and to get rid of distortions in 2 direction,* al-though the method that was used in Ref. 18 must be admittedas the most acceptable solution here.When measuring the frequency characteristic of Z ma-nipulators, an auxiliary piezoelement is often employed,which is supposed to modulate the tunnel junction at a har-monic law. With this method, if a sine voltage is applied tothe auxiliary piezoelement, the law of its mechanical dis-placement would not be exactly the sine because of the pres-ence of hysteresis. The model for approximation of hyster-esis loops would allow to increase the precision of thismethod. To do that, the voltage applied to the auxiliary pi-ezoelement must match x(a) by formula (1). In that case, thedisplacement y(a) will be a harmonica1 function.The model described may prove useful with the tasks ofimitation modeling as well as in engineer calculations ofnonlinear control systems containing hysteresis elements.

ACKNOWLEDGMENTS York, 1968).I want to thank Oleg E. Lyapin, Valery V. Efremov, OlegD. Cnab, Vladimir N. Yakovlev, and Oleg V. Obyedkov fortheir helpful advice and discussions.

15S Hirano, T. Yogo, K. Kikuta, Y. Araki, M. Saitoh, and S. Ogasahara, J.Ak. Ceram. Sot. 75, 2785 (1992).r6G D. E. Lakeman and D.A. Payne, J. Am. Ceram. Sot. 75, 3091 (1992).D. W. Pohl and R. MoIIer, Rev. Sci. Instrum. 59, 840 (1988).R. V. Lapshin and 0. V. Obyedkov, Rev. Sci. Instrum. 64, 2883 (1993).

APPENDIXBeside smooth l oops, the model suggested can be imple-mented for description of piecewise-linear loops as well (seeRefs. 1 and 2). To obtain a set of piecewise-linear hysteresisloop primitives, in formula (1) there must be used somepiecewise-linear functions instead of sin c~and cos IX. Theycould be, for instance, trapeziumlike pulses with unit ampli-tude

trhCaj=iA (a-iZJC->rectltwY+(-I) rectz(a,i) ,i (Al)trp,(a)=trp,i+;

where the subscripts s and c refer to sine and cosine, respec-tively; d and D are the upper and lower bases of the trape-zium, respectively; T= d + D is the period of pulses;

rectt(cu,i)=l(cu+(D-d)/4-iT/2)-l(a-(D-d)/4-iT/2)

rect,(a,i)=l(a-(D-dj/4-iT/2)-l(c~-(D-d)/4-d-iT/2)

are the ith rectangular pulses; 1 a?i) is the ith unit stepfunction. Triangle functions [tri,( a) =lim,,, trp,( n) andtri,( a) =tri,( (Y+ T/4)] or rectangle functions [rect,( a)=lim,,, &p,(a) and rectJaj=rect#(a+ T/4)] could bealso used to that purpose.

E. l? Popov, The Theory of Nonlinear Automation Control Systems(Nauka, Moscow, 1988) (in Russian).E. P. Popov and I. P. Paltov, Approximation Methods for Investigation ofNonlinear Automatic Systems Pizmatgiz, Moscow, 1960) (in Russian).3B. Jankovic, Proceedings of the 5th International Conference on NonlinearOscillations, Kiev, 1969 Vol. 4, p. 503.4R. Bout in Ref. 3, p. 100.L. E. C. van de Leemput, P. H. H. Rongen, B. H. Timmerman, and H. vanKempen, Rev. Sci. Instrum. 62, 989 (1991).S. Vieira, IBM J. Res. Dev. 30, 553 (1986).E. P . Stall, Ultramicroscop y 42-44, 1585 (1992).*E. P Stall, Rev. Sci. Instrum. 65, 2864 (1994).D. W. Pohl, IBM J. Res. Dev. 30, 417 (1986).*O. Nishikawa, M. Tomitori, and A. Minakuchi, Surf. Sci. 181,210 (1987).R. C. Barrett and C. E Quate, Rev. Sci. Ins&urn. 62, 13 93 (1991).*L Libioulle A. Ronda, M. Taborelli, and J. M. Gilles, 3. Vat. Sci. Tech-ni B 9, 65; (1991).13S.-Y. Liu and I-W. Chen, J. Am. Ceram. Sot. 75, 1191 (1992).r4G. A. Kom and T. M. Korn, Mathematical Handbook (McGraw-Hill, New
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