Modeling multifrequency eddy current sensor interactions ... · PDF fileModeling multifrequency eddy current sensor interactions during vertical Bridgman growth of semiconductors Kumar

REVIEW OF SCIENTIFIC INSTRUMENTS VOLUME 70, NUMBER 7 JULY 1999

Modeling multifrequency eddy current sensor interactions during verticalBridgman growth of semiconductors

Kumar P. Dharmasenaa) and Haydn N. G. Wadleyb)

Intelligent Processing of Materials Laboratory, School of Engineering and Applied Science,University of Virginia, Charlottesville, Virginia 22903

~Received 13 March 1996; accepted for publication 22 February 1999!

Electromagnetic finite element modeling methods have been used to analyze the responses of two~‘‘absolute’’ and ‘‘differential’’! eddy current sensor designs for measuring liquid–solid interfacelocation and curvature during the vertical Bridgman growth of a wide variety of semiconductingmaterials. The multifrequency impedance changes due to perturbations of the interface’s locationand shape are shown to increase as the liquid/solid electrical conductivity ratio increases. Of thematerials studied, GaAs is found best suited for eddy current sensing. However, the calculationsindicate that even for CdTe with the lowest conductivity ratio studied, the impedance changes arestill sufficient to detect the interface’s position and curvature. The optimum frequency for eddycurrent sensing is found to increase as the material system’s conductivity decreases. The analysisreveals that for a given material system, high frequency measurements are more heavily weightedby the interfacial location while lower frequency data more equally sample the interface curvatureand location. This observation suggests a physical basis for potentially measuring both parametersduring vertical Bridgman growth. ©1999 American Institute of Physics.@S0034-6748~99!02306-0#

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I. INTRODUCTION

The Bridgman method has become a widely used tenique for the growth of bulk single crystals from the melt.1–5

Important semiconducting materials such as CdTe, Gaand Ge are all grown by variants of this technique.3,4,6–9 Inthe vertical variant of the Bridgman method, an axisymmric quartz, pyrolytic boron nitride (p-BN), or graphite cru-cible containing the charge material is positioned in thezone of a vertically oriented, multizone furnace with a cafully designed and controlled axial temperature gradientcrystal is produced by first melting the charge in the hot zoof the furnace and then either vertically translating the fnace~with its associated temperature profile! relative to thestationary crucible or vice versa. In either case, a solid ctal is nucleated at the bottom of the crucible and a liquisolid interface propagates along the crucible’s length ideresulting in a single crystal sample.

The yield and quality of single crystal material grownthis way is a sensitive function of the thermal fields withthe charge during the growth process. These can affect cpound semiconductor liquid stoichiometry,10,11 fluid flowpatterns in the melted part of the charge,4,12–14 the velocity~i.e., time-dependent position! and curvature~i.e., the shape!of the liquid–solid interface,15–19 the kinetics of secondphase precipitation~e.g., Te particles in Cd depleted CdT!during cooling,20 and the levels of residual stress inducdefects after cooling.21,22 Analytical models that attempt topredict the evolution of these quantities during growth ruhave emerged.5,14 However, detailed experimental validatio

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of these models has been handicapped by the lack of exmental techniques for noninvasively monitoring manythese quantities as solidification progresses throughcrucible.6,23

For instance, in order to validate predictive thermmodels,14 the temperature fieldswithin both the melt and thegrowing crystal should be continuously measured throughgrowth. However, thermocouple arrays cannot be locawithin the crucible without seriously affecting the growconditions~especially for systems like CdTe or GaAs whichave a high vapor pressure at the melting temperatu!.Readings from thermocouple arrays on the outer surfacthe ampoule are unreliable because radiative heat transfthe furnace environment can preferentially heat the thermcouple. In addition, if the thermal conductivity of the matrial is low ~e.g., as with GaAs or CdTe! and/or the ampoulediameter is large, the temperature within the charge maynificantly differ from that at the outer surface of the ampouwhere the thermocouples must be located. Several grohave attempted to image the emitted infrared~IR! radiationusing infrared cameras.24 However, unless the crucible ancharge are both transparent at the growth temperature, thmethod at best provides only an indication of surface teperature. This approach must also contend with many odifficulties associated with limited access to the furna‘‘work’’ area, the stray radiation from nearby heating elments, and uncertainties in the effective emissivity of tsample surface.

An alternative approach is to attempt the noninvasobservation of the time-varying interface position~and there-fore solidification velocity! and liquid–solid interface curvature throughout the growth process. These data would p

5 © 1999 American Institute of Physics

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3126 Rev. Sci. Instrum., Vol. 70, No. 7, July 1999 K. P. Dharmasena and H. N. G. Wadley

vide valuable information for validating/extending predictiprocess models and for improving the growth processmight also create the possibility for applying new~feedback!control approaches to crystal growth.25

The first step in developing this interface sensingproach is to identify noninvasively measurable physiproperties of the semiconductor that vary significantly btween the melt and the crystal. Ideally, the changes in thproperties accompanying solidification would be large copared to those associated with other effects~e.g., changes omelt stoichiometry, liquid temperature, and fluid flow!. Onepossibility is the use of optical techniques for visual idenfication of the liquid–solid interface. This is based upon tlarge difference between the solid and liquid optical refltion coefficients of many semiconducting materials.26 Unfor-tunately, their implementation in a vertical Bridgman furnais hindered by the high background light intensity within tfurnace, the frequently opaque nature of the crucibampoule~e.g.,p-BN or carbon coated quartz!, and the oftenpoor optical transmission of the charge material at its meltpoint. However, several other possibilities exist for semicducting materials because of sometimes large differenceelectrical conductivity~monitorable with eddy current sensing techniques!,27–37 elastic constants ~via laserultrasonics!,38 density~via x-ray radiography!,39 and specificheat~perhaps measured with a photoacoustic method!.

In this, and several related articles,40–42 the use of eddycurrent sensors for monitoring the vertical Bridgman growof semiconducting materials is explored. The eddy currtechnique exploits the sometimes very large electrical cductivity differences between the solid (ss) and liquid (s l)phases of many semiconducting systems.43,44 Since both theabsolute conductivity and the conductivity difference alikely to affect the performance of this sensing approach,study explores the application of eddy current methodsvariety of semiconductors, Table I. Silicon, though not comercially produced by a Bridgman method, is included inmaterials analyzed to span a broader conductivity range,to establish the sensor performance–test material conduity relationship.

Table I shows that for most semiconductors, the eleccal conductivity of the liquid is many times that of the solat the melting point. The principle underlying the applicatiof an eddy current sensor approach to crystal growth is baon the observation that the eddy current density inducedpoint within a test sample by the electromagnetic field ofalternating current~ac! excited coil is proportional to thesample’s electrical conductivity at that point. Since the eltrical conductivity of liquid semiconductors exceeds thatthe solid, higher eddy current densities are expected to ewithin liquid regions of a solidifying charge. Sensors basupon this principle have been previously proposed for msuring solidification conditions and temperature profiles ding the Czochralski growth of GaAs and Silicon.35,36 Theyare widely used in other types of high temperature materprocessing, e.g., for determining internal temperatures wialuminum alloy extrusions27 and for the measurement of dmensional changes during hot isostatic processing.45

The response of an eddy current sensor is a complic


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function of the eddy current distribution induced within thsample by the fluctuating electromagnetic field of an exctional coil. This will be affected by the geometry of the eciting coil ~which governs the electromagnetic field’s distbution within the test material!, the coil’s excitation currentfrequency, the fraction of material solidified in the interrgated volume, the shape of the boundary between the sand liquid regions, and the respective conductivities ofsolid and liquid. In the eddy current technique, the distribtion of eddy currents induced in the sample is sensed frtheir effect on the impedance of either the exciting coil oseparate ‘‘pickup’’ coil. It will be a sensitive function of thsensor’s design and test frequency as well as all the matand growth parameters listed above. Experimental methmight be used to perfect the sensor design, optimize thefrequency, and develop data analysis protocols. Howeveis costly and time consuming to equip a crystal grower wa variety of eddy current sensors, and experimentally desa sensor approach. Furthermore, a definitive validation ofresponse is almost impossible because of the lack of inpendent observations of the solidification front.

An alternative approach to sensor design has beensued here. The responses of several sensors have beenlated ~using electromagnetic finite element techniques! for a

FIG. 1. Schematic diagram of a dual coil eddy current sensing arrangemA seven-turn driver coil is used to excite an electromagnetic field. Eithesingle coil or a pair of opposingly wound coils are used to ‘‘pick up’’ thperturbed flux.

TABLE I. The electrical conductivities of selected solid (ss) and liquid(s l) semiconductors close to their melting points.

Electricalconductivity

Semiconductor material

CdTea GaAsb Sib Geb

ss ~S/m! 1200 3.03104 5.83104 1.253105

s l ~S/m! 6600 7.93105 1.23106 1.43106

s l /ss 5.5 26.3 20.7 11.2

aReference 43.bReference 44.


3127Rev. Sci. Instrum., Vol. 70, No. 7, July 1999 Eddy current sensor

FIG. 2. ~a! Finite element model geometry;~b! finite element mesh in interface region.

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variety of liquid–solid interface locations/curvatures, andseveral different materials systems. This allows quantitarelationships to be obtained between growth parametersas the liquid–solid interface location/shape and measurquantities of an eddy current sensor’s response~e.g., thefrequency-dependent test coil impedance!. This approachalso has the advantage of allowing anomaly free protocolbe designed for deducing the growth parameters from msured experimental data, and provides guidelines for evating their potential for other material systems.

Here, the simulated responses for two candidate ecurrent sensor designs are obtained for each of the fourterial systems given in Table I. The results are used to invtigate the effects of changing either the interface positionits shape on the sensor’s complex impedance in the mexperimentally accessible 200 Hz–2 MHz frequency ranIt is shown that the impedance change due to a perturbaof the interface’s position or shape is greatest for GaAs~thematerial with the highest liquid/solid electrical conductiviratio! and is least for CdTe~with the lowest conductivityratio!. The sensitivity to both location and shape has befound to depend strongly upon frequency. The frequencymaximum sensitivity to interface shape change increasethe test material’s conductivity decreases. Thus, the besterating frequency range is unique to each material. At htest frequencies, the sensor’s response is shown to be dnated by the interface location and is almost independen


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interface shape. At lower frequencies, both shape and lotion contribute to the predicted impedance. This providesphysical basis for the possible discrimination of the locatand shape contributions to eddy current sensor responsethus their independent monitoring during vertical Bridgmgrowth.

II. EDDY CURRENT SENSOR DESIGN CONCEPTS

The physical basis of all eddy current sensor approacis electromagnetic induction. A coil carrying an alternaticurrent is first used to create a fluctuating electromagnfield.46 Eddy currents are then induced in any conduct

TABLE II. Electromagnetic skin depthsa in several solid (ds) and liquid(d l) semiconductors.

Frequency~kHz!

Skin depth~mm! CdTe GaAs Si Ge

10 ds 205.5 29 20.9 14.210 d l 65 5.7 4.6 4.3

500 ds 29 4.1 3.0 2.0500 d l 9.2 0.8 0.65 0.6

2000 ds 14 2.1 1.48 1.02000 d l 4.6 0.4 0.32 0.3

aFor cylindrical samples, Eq.~1! can be used only if the skin depth isignificantly smaller than the radius of the cylinder. Significant errorscalculated eddy current densities arise if the skin depth is larger than,of the same order of magnitude as the sample radius~Ref. 48!.



FIG. 3. Calculated absolute sensor impedance curves for the liquid and solid states of~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

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medium placed within this field. These currents create a sondary electromagnetic field which perturbs the primary fiand changes the inductance of the coil. If the impedanceother circuit elements are small compared to the coil indtance, the impedance of a test circuit containing the coil wbe directly related to the coil’s inductance and those propties of the test sample that control the eddy current distrition within it ~e.g., its electrical conductivity or magnetpermeability!. By analyzing the coil’s impedance, it is posible to infer the electrical conductivity/magnetic permeabity ~and even its spatial distribution! within the region of thetest material sampled by the electromagnetic field.27–36 Thisfield depends upon the coil’s geometry~its number of turns,diameter, axial length, etc.! and the extent of penetration othe primary field into the sample. The latter is governedthe test material’s electrical conductivity~s!, the magnetic


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permeability ~m!, and by the angular frequency~v52p f ,wheref is the frequency in hertz! of the excitation current.

The depth at which the magnetic field intensity~or in-duced eddy current! falls to a value 1/e(50.368) from that atthe sample surface, is defined as the skin depth,d, givenby47,48

d5A 2

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Values of this skin depth at three readily accessiblefrequencies~10 kHz, 500 kHz, and 2 MHz! are given inTable II for the four materials listed in Table I. For eacmaterial, the magnetic permeability has been taken to beof free space (m54p31027 H/m). Large skin depths~greater penetration of the field into the sample! are obtained



FIG. 4. Imaginary component of impedance vs frequency for the liquid and solid states of~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

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when the conductivity and/or the frequency are low. Feach of the four materials, it is possible to probe to differdepths below the sample surface by varying the sensor’scitation frequency. If the electrical properties were to vawith radial position in a cylinder~e.g., if the liquid–solidinterface were curved!, the sensor’s frequency-dependentsponse will be perturbed from that of a sample with no radgradient in properties~i.e., one with a flat interface! and in-sight might be gained about the interface shape. Tablsuggests that the best range of frequencies to revealeffects are test material dependent.

Several experimental approaches have been develfor eddy current sensing. In the simplest,33 a single coilsenses small sample induced perturbations to its own fiWhen this approach is applied to the crystal growth envirment, anomalous changes in coil impedance can accomtemperature changes of the coil. These arise from the cha


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in the coil’s ac resistance due to the winding’s temperatudependent resistivity. They can be reduced by using coil mterials with a low thermal coefficient of conductivity, buthey cannot be eliminated. This drawback can to some exbe overcome with a dual coil system. In this approach, ocoil is excited with an alternating current to create the pmary electromagnetic field, and a second coil is then usedetect sample induced perturbations to the field.

Figure 1 shows a schematic diagram of a two-coil stem embodiment. The transfer impedance of such a senZ, is given by

Z5Vs / I p , ~2!

where I p is the phasor excitation current in the primary~ordriving! coil andVs is the phasor voltage induced across tterminals of the secondary coil. The transfer impedance



FIG. 5. Magnetic vector potential contours for liquid CdTe at frequencies of~a! 10 kHz, ~b! 500 kHz, and~c! 2 MHz.

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such a system is relatively insensitive to the resistance ofcoils if, ~a! the induced voltage,Vs , is measured with a highimpedance instrument and~b! the current,I p , is continu-ously monitored~e.g., by observing the voltage across a pcision resistor placed in series with the primary coil!. Thisenables operation of the sensor at high temperatures witthe need for either correcting for coil temperature or~poten-tially invasive! coil cooling.46 With two-coil systems likethese, the primary coil’s axial length can also be made mlonger than the secondary coil, so producing a relatively uform field in the sensing region.

For a two-coil system, the ratio of the induced voltagethe secondary coil to the current in the primary coil canconveniently obtained from its multifrequency gain-phasesponse measured with a network analyzer.46 To simplify in-terpretation of this type of data, the resulting complex ipedance components are usually normalized with respethe empty coil’s impedance measured in the sample’ssence. The results can be presented in the form of impedplane diagrams which are plots of the real and imagincomponents of the impedance~the abscissa and ordinate, rspectively! as a function of frequency. The resulting impeance curves are also functions of the sample and sensoometries and the electrical/magnetic properties of thematerial.

Figure 1 shows two encircling eddy current sensorsigns selected for detailed study. Both have the same seturn primary coil for excitation. One uses a single-tupickup coil located at the primary coil’s midpoint, while thother uses two opposingly wound pickup coils located nthe ends of the primary coil. The single pickup coil sensarrangement is called an ‘‘absolute’’ sensor, while the tpickup coil design sensor is referred to as a ‘‘differentiasensor.

To envision the way such sensors might be used


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monitor solidification during vertical Bridgman growth, consider a control volume element in the vicinity of either sesor ~as indicated by the dashed area in Fig. 1!. The length ofthis control volume is determined by the ‘‘effective’’ axiarange of the electromagnetic flux created by the primary cIts penetration into the sample is controlled by the skin dein the material under investigation. The induced eddy crents will be influenced by the volume fractions of the soand liquid within this control volume. This changes if eiththe sample~containing a liquid–solid interface! remains sta-tionary and the sensor is translated, or if the sensor/samare both stationary and the growth furnace is transla~causing the solidification front to propagate along tsample!. For this latter scenario, the region initially samplewould be liquid whereas at the end of growth~after the in-terface has moved through the control volume!, the sensedregion would be fully solid. It is shown that the sensorresponse will always be bounded by its response to thesestates, and all measured responses during growth musbetween these two extremes.

A single secondary coil sensor design will be mostsponsive to the eddy current density closest to the seconcoil’s location. It is likely to be relatively unaffected by eddcurrents excited far from the coil. The sensitivity of a sensto the presence of an interface could potentially be improby using a pair of opposing wound secondary coils locaabove and below an interface. The response of such a diential sensor will be dependent on the difference in samcreated field perturbation at the two coil locations; commcontributions to the two coils’ induction will be canceled oin this configuration. The sensitivity of a differential sensmeasurement is likely to vary with the spacing betweentwo pickup coils providing an additional degree of freedofor sensor design.


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III. FINITE ELEMENT SIMULATION MODEL

The differential equation governing eddy current genetion in conducting materials is derived from Maxwell’s equtions and can be expressed in terms of an unknown magnvector potential~A!. For a sinusoidal current of angular frequency~v!, the governing equation is

,2A1 j vmsA52mJs , ~3!

where,Js is the source current density. For inhomogeneoproblems, the boundary conditions require thatA and its nor-mal derivative be continuous across all material interface

If the spacing of the primary coil turns is small compared to their diameter, it can be assumed that the cocircular symmetric~i.e., the helical effect of the coil can bignored!. The primary coil can then be modeled as a seriecircular loops of a known radius spaced a distance aequal to the pitch of the primary coil winding. Since thsample encircled by the eddy current sensor is containedcylindrical crucible, the entire geometry allows axisymmetcalculations rather than full three-dimensional simulatioConsiderations of the cylindrical geometry~Fig. 1! show thatonly one half of an axisymmetric plane must be analyzFig. 2~a!. In addition, no electromagnetic flux can cross taxis of symmetry, and hence a zero vector potential bouary condition can be specified on the axis.

Closed form theoretical solutions to electromagneticduction problems are in general limited to simple geometand are based on simplifying assumptions for the geom~for example, the sample is assumed to be infinitely locylindrical etc.!, and its electrical properties are assumed uform throughout the interrogated volume.49 Electromagneticfinite element modeling provides a convenient tool to evaate eddy current sensor responses for the sensor and sageometries encountered here provided the electromag

FIG. 6. Variation of the imaginary component of impedance with the liqusolid conductivity ratio for the absolute sensor.


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properties~i.e., the electrical conductivity and the magnepermeability! of the sample material are known along withe sensor’s excitation frequency.50–55

The problem modeled consisted of a cylindrical 76 mdiam sample containing one of five interface locations afive interface curvatures. Two of the interfaces had convshapes ~defined by a convexity parameteru5z/D510.167,10.333 wherez is the interface curvature heighon the axis andD the sample diameter!, one interface wasflat (u50.0), and the remaining two were conca(u520.167,20.333). The nonplanar interfaces were hemspherical surfaces of differing radii of curvature. In orderminimize the effects of mesh size on the solution, all fiinterface shapes were incorporated into one finite elemmodel and the same finite element mesh@see Fig. 2~b!# wasused for all the calculations. The different models corsponding to each interface shape/position were built frthis mesh by changing the assigned material properties~i.e.,the electrical conductivity! of the elements in the mesh tcreate regions of solid, liquid, or air.

In order to account for the skin effect at high frequecies, the finite element mesh was refined in the interfregion with an increased number of elements concentratowards the edge of the charge. As a result, the elementsthe smallest depth~0.038 mm! were placed along the outesurface of the sample. These element sizes were smallerthe skin depth at the highest frequency analyzed~2 MHz! forthe most conductive sample condition~liquid Ge!. Themodel had a total of 913 grid points and 1007~triangular andquadrilateral! elements. Additional mesh refinement wconstrained by the limitations of the commercial electromnetic analysis package56 used for the creation of the axisymmetric finite element model. However, this step was not csidered to be important since calculations were performwith and without the sample using the same element mesobtain normalized impedance values. The output ofmodel allowed calculation of the inductive reactance ofcoil. The model did not incorporate the capacitive reactaor the ac resistance of the coils, nor the impedance contrtions of other test circuit elements.

The finite element code solved Eq.~3! for the magneticvector potential~A(r ,z), wherer is the radial andz the axialposition! subject to a prescribed source current~applied load!distribution and boundary conditions. The applied loadthis problem was the driving current in the multiple tuprimary coil. This was specified as a point current at eachthe seven grid points corresponding to the location of eacthe seven turns on the primary coil. Since each calculawas normalized with respect to the empty coil condition, tactual value of current in the primary coil was not importaand for convenience was taken to be unity.

The magnetic vector potential obtained from the finelement calculations can be directly used to obtain the ssor’s transfer impedance~see Ref. 57 for details!. For anabsolute sensor, it can be shown that

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FIG. 7. Normalized impedance curves for three positions of a flat interface for the absolute sensor~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

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where r s is the secondary coil radius,Ns is the number ofturns in the absolute secondary coil,f is the test frequencyandAave is the average vector potential over the cross secof the secondary coil wire.

For an axially separated differential sensor

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whereNs1 andNs2 are the number of turns at the two loctions of the differential secondary. All of the calculated cimpedances were normalized with respect to the coil impance at the calculation frequency. This was obtained byplacing the relevant electromagnetic properties~m ands! of


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the ‘‘solid’’ and ‘‘liquid’’ region elements of the charge bythose of the ‘‘air’’ elements, and repeating the finite elemeanalysis.

IV. ABSOLUTE SENSOR RESPONSE

A. Homogeneous liquid or solid states

The simplest problems to analyze are the initial and fistates of a growth run when the sensor observes eitherthe melt ~prior to solidification! or only the solid ~aftercompletion of growth!. In this case, the test material waassumed to have a uniform conductivity as defined in TaI. Figure 3 shows the absolute sensor’s calculated normalimpedance response for the liquid~circles! and solid~squares! states of CdTe, GaAs, Si, and Ge at 13 frequenc



FIG. 8. Magnetic vector potential contours for CdTe at a frequency of 500 kHz for a flat interface position at~a! h5212.7 mm,~b! h50 mm, and~c! h512.7 mm.

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between 200 Hz and 2 MHz. The impedance data for bthe liquid and solid states of all four materials fall on tsame characteristic ‘‘comma-shaped’’ curve. The shapalmost identical to that expected for an infinite conductcylinder contained in a long solenoid.47,49The ‘‘size’’ of thiscurve can be characterized by its high frequency intercep~I!with the normalized imaginary impedance component aThis is a function of the sample and pickup coil diameteand is independent of the test material conductivity.27 For aninfinitely long cylinder contained in a long solenoid, fringfield effects are insignificant andI 512(ds /dc)

2512fillfactor, whereds is the sample diameter anddc the diameterof the secondary coil. Both the liquid and solid forms offour test materials intercept the imaginary axis at the sapoint because the intercept is independent of the test maal’s conductivity in the high frequency limit.~In this limit allconductors totally exclude the penetration of flux into tsample.! This well understood phenomenon is the basiseddy current dimensional sensing27 and could be exploited invertical Bridgman growth~e.g., to detect debonding of thsolid from its ampoule during cooling!.

The only difference between the sensor’s responseeither a solid or liquid test material is a shifting of frequenpoints along the impedance curve. A decrease in conducity, associated for example with solidification, causes thepedance at a fixed frequency to move counter clockwaround the curve because the sample becomes less induThis also explains why the length of the impedance cucalculated up to 2 MHz decreases as the test materialductivity decrease. In the limit, as the conductivity of the tmaterial approaches zero, the normalized impedance atthe highest frequencies would be located at (01 j ), i.e., atthe upper left corner of the impedance plane, which issame as the ‘‘no sample’’~or ‘‘coil in air’’ ! situation.

The imaginary component of impedance~i.e., the nor-


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malized inductive reactance! is plotted as a function of fre-quency for the liquid and solid states of the four test mateals in Fig. 4. At low frequencies~e.g., below 10 kHz forCdTe!, each material system gives a null response. Tarises because the rate of change of the electromagneticwithin the test material is insufficient to induce detectabeddy currents. The sample is effectively transparent tofield, and the sensor’s response is similar to that whensample is present. Figure 5 shows vector potential~magneticflux! contours for liquid CdTe for three test frequencieNote that at a frequency of 10 kHz, Fig. 5~a!, the vectorpotential contours are indistinguishable from those ofempty coil.

Figure 4 shows that beyond this threshold frequencyclear separation of the liquid and solid impedance curveseen, and a measurement of the imaginary impedance cponent in this region could be used to distinguish betwethe solid and liquid states. Beyond the threshold frequenthe separation of the curves is seen at first to increase, ra maximum, and finally decrease as the frequency iscreased. For the highest conductivity material~Ge!, theliquid/solid impedance separation decreases more rapidlthe test frequency increases because the skin effect meffectively expels flux in higher conductivity materials. Thflux expulsion can be clearly seen in the vector poten~magnetic flux! contour plots of Figs. 5~b! and 5~c!. Theimpedance of the sensor in the intermediate range ofquencies~where detectable eddy currents are excited insample but flux expulsion is not complete! depends bothonthe test sample’s diameter and its conductivity. Data clected at these frequencies~where skin depths are around 0times the sample radius! is widely used to measure the conductivity of test materials of known diameter~obtained fromhigh frequency data! and to infer sample conditions that afect it ~e.g., temperature!.27



FIG. 9. Variation of the imaginary component of impedance with frequency for an absolute sensor for five positions of the flat interface~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

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The solid and liquid impedance curves represent the ssor’s response to two extremes of a growth process; theythe ‘‘upper’’ and ‘‘lower’’ bounds of all sensor responsethat could be encountered during growth. All the impedancurves observed, no matter what interface shape or posimust lie between these bounds. The relative sensitivity oeddy current solidification sensor to interfacial shaposition will depend upon the magnitude of separation of

TABLE III. The maximum difference between the imaginary impedancomponents (ImDZmax) of an absolute sensor for the homogeneous liqand solid states and the frequency at which it occurs.

Material CdTe GaAs Si Ge

Im DZmax 0.2417 0.3817 0.3625 0.3167Frequency~Hz! 2.53105 7.53103 4.53103 2.23103


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liquid and solid impedance curves (ImDZ) which is a func-tion of the excitation frequency. Examination of Fig. 4 rveals that there exists a characteristic frequency whermaximum impedance separation~and thus, sensor sensitivity! occurs. Values for ImDZmax and the frequencies at whicthey occur are tabulated for each of the test materialsTable III.

Examination of Tables I and III reveals that ImDZmax

monotonically increases with the liquid to solid conductiviratio, Fig. 6. The frequency at which the maximum diffeence occurs varies inversely with the melt~or solid! conduc-tivity. Germanium~which has the highest liquid and soliconductivities! has the lowest frequency where the maximuimaginary impedance change occurs. The highest frequeoccurs in CdTe which has the lowest liquid and solid coductivities. Clearly, the best frequency for operation of



FIG. 10. Normalized impedance curves for three interface shapes for the absolute sensor~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

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eddy current solidification sensor is a material dependentrameter, and varies from one system to another. All offour systems analyzed here have a sufficiently high condtivity that the frequency of ‘‘maximum response’’ is wewithin the range of frequencies experimentally accesswith conventional eddy current sensing instrumentationhave a sufficiently large ImDZmax value for reliable eddycurrent sensing.

B. Interface position effects

To assess the response of an absolute sensor to thsition of an interface, a series of calculations were performfor five locations of a flat interface. Figure 7 shows calclated normalized impedance curves for three of these ptions. The impedances of all four materials are seen to cverge at high frequency and again approach a commintercept with the imaginary axis~because in the modele


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problem all four materials have the same diameter and thfore identical fill factors!. In the limit, as the test frequencapproaches infinity, the sensor’s response depends only uthis fill factor and is independent of the interface’s positiwithin it, even when a liquid–solid interface exists within thinterrogated volume.

The impedance curves are seen to be a strong functiointerface position at lower frequencies, Fig. 7. Recall thathe completely liquid or solid cases~Fig. 3!, the sample actedlike an infinite cylinder of uniform conductivity encircled ba long solenoid, and changes of conductivity only shiftedimpedances at specific frequencies around a common cuHowever, when an interface between dissimilar conductivmaterials exists within the field of an eddy current sensor,observed response can be viewed as the net effect of sitaneous interactions with two finite length cylinders of dferent conductivities. Fringing of the field at the interfa



FIG. 11. Variation of imaginary component of impedance with frequency for five interface shapes for the absolute sensor~a! CdTe,~b! GaAs,~c! Si, and~d!Ge.

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allows the impedance for a given frequency to move anonzero angle to the characteristic impedance curve, in eshrinking its size. The contribution of this effect must dpend upon where the interface is located with respect tosensor. Thus, when the interface is below the center pointhe sensor~i.e., h5212.7 mm, Fig. 7!, more of the higherconductivity liquid cylinder is sampled by the encirclineddy current sensor. The solid cylinder is still in the fieldview of the sensor, but contributes less to the sensor’ssponse. Later in a growth process when the interfacegrown upward beyond the center of the~stationary! sensor,say to a positionh512.7 mm above the sensor’s centemore of the lower conductivity solid cylinder is encircled bthe sensor and a lesser contribution is made by the liqregion.

The consequence of this phenomenon can be cle


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seen from the magnetic vector equipotential contours~fluxlines!. Figure 8 shows the 500 kHz vector potential field fthe CdTe material system for interface heights of212.7, 0and112.7 mm. Because of the bigger skin depth in the sophase, the depth of penetration into the solid is always mgreater than that of the liquid. Since the vector potentiacontinuous across the liquid/solid interface, the field nearinterface is perturbed from that expected for a homogenecylinder of either conductivity. The extent of this perturbtion depends on the frequency of excitation~through the skineffect! and the relative position of the interface within thsensing coil. Since the fields are no longer the same as tof an infinite uniform cylinder, the sensor’s response depafrom that of an ‘‘ideal’’ uniform cylinder~Fig. 3! and pro-vides the potential for a method of sensing position.

Since this behavior again originates from the skin effe



FIG. 12. Magnetic vector potential contours for CdTe at a frequency of 500 kHz for three interface shapes~a! u510.333,~b! u50, and~c! u520.333.

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the frequency at which it occurs will be conductivity~andthus test material! dependent. This can be seen more cleaby plotting the normalized imaginary impedance componagainst excitation frequency for each interface position, F9. Again, there exist material dependent low and high fquency thresholds below/above which the sensor’s respis independent of frequency. However, for each matethere also exists an intermediate range of frequencies wsensitivity to interfacial position is a maximum. The sentivity ~i.e., the difference in impedance forh5612.7 mm!and optimal frequency are listed in Table IV for each tmaterial. For the highest conductivity materials~like germa-nium or silicon!, the curves converge at significantly lowfrequencies than for lower conductivity materials suchCdTe. This is because the penetration depth of the elecmagnetic field becomes infinitesimal~i.e., approaches the infinite frequency limit! at lower frequencies in higher condutivity materials. From Tables I and IV it is also observed ththe maximum separation due to interfacial positi(Im DZmax), increases/decreases as thes liquid /ssolid ratioincreases/decreases. The frequency at which this maximposition effect occurs varies inversely with either the liquor solid electrical conductivity.

Figure 9 shows that in the intermediate range of frequ

TABLE IV. Imaginary impedance component values at the frequencymaximum sensitivity to interface position for the absolute sensor.

Relative interfaceposition,h ~mm! CdTe GaAs Si Ge

212.7 0.610 0.567 0.570 0.59826.4 0.640 0.600 0.602 0.628

0 0.682 0.654 0.654 0.67416.4 0.734 0.722 0.717 0.728

112.7 0.778 0.782 0.774 0.773Im DZmax 0.168 0.215 0.204 0.175Frequency~Hz! 43105 13104 63103 3.43103


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cies, the sensor’s imaginary impedance component imonotonic function of interface position. If such a senswere used to monitor an interface that moved throughsensor, the ‘‘sampled’’ fractions of liquid and solid woukeep changing within the volume interrogated by the senWhen the interface was well below the sensor~i.e., h5212.7 mm!, a larger fraction of the high conductivity liquid would be sampled~in the limit of h tending to minusinfinity, a uniform liquid response like that of Fig. 4 woulbe obtained!. As the interface approached the sensor, a pgressively increasing fraction of solid would be ‘‘sensedand as the interface moved past the sensor, its respwould approach that for a uniform solid, Fig. 4. Since tliquid conductivity is always much greater than the solidthe net effect would always be an increase in the imaginimpedance as each of the liquids studied gradually turinto solid during the growth process.

C. Interface shape effects

During Bridgman growth, the liquid–solid interfacshape can be concave, flat, or convex and can change cture as the governing heat and fluid flow conditions evoduring growth.14 In order to assess the response of the ab

fTABLE V. Imaginary impedance component values where maximum inface shape effect occurs for the absolute sensor.

Interface convexity,u CdTe GaAs Si Ge

10.333 0.818 0.808 0.818 0.82410.167 0.786 0.769 0.780 0.790

0 0.762 0.740 0.752 0.76520.167 0.738 0.714 0.726 0.74020.333 0.711 0.684 0.696 0.712

Im DZmax 0.107 0.124 0.122 0.112Frequency~Hz! 23105 3.73103 2.13103 1.53103



FIG. 13. Normalized impedance curves for three positions of a flat interface for the differential sensor~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

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lute sensor to this curvature, a series of calculations hbeen conducted where the interface shape has been allto have one of five shapes specified by a convexity pareter,u5z/D, wherez is the difference in axial intersectioof the interface central axis and periphery of the sample~i.e.,the interface height!, andD is the diameter of the sample~seeFig. 2!. For each calculation, the point where the interfaintersected the outer boundary of the test material was fiat the axial location of the secondary coil~i.e., h50!.

Figure 10 shows examples of the normalized impedacurves for a convex (u510.333), a flat (u50.0), and aconcave (u520.333) interface. The shape of the interfaceseen to have a small but significant effect upon the strucof the impedance plane curve. The dependence upon intcial curvature disappears at low and high frequenciesexhibits a similar frequency dependence to the interfacesition effect. This can be more clearly seen when the ima


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nary impedance component is plotted as a function ofquency for the five interface shapes, Fig. 11.

The dependence of the intermediate frequency impance upon interface curvature again results from the elecmagnetic flux interaction with each interface, Fig. 12. In thigh frequency limit, the flux is confined closer to the sampsurface, and the sensor’s response is insensitive to the inal interface shape. This limit is most nearly approachedthe higher conductivity materials~Ge, Si, GaAs! for frequen-cies beyond 1 MHz. For lower conductivity materials suchCdTe, it would be necessary to increase the frequencyward 10 MHz in order to obtain an impedance that is almindependent of interface shape.

At lower frequencies, the sensor’s imaginary impedanshows a significant dependence upon interfacial curvatThe frequencies at which the interface shape effect is a mmum and the magnitude of the impedance changes are



FIG. 14. Variation of the imaginary component of impedance with frequency for a differential sensor for five positions of a flat interface~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

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given in Table V for each of the test materials.Although the effect of curvature upon the imagina

impedance–frequency relationship was similar to that sfor interface location, the maximum interface shape effeoccurred at a lower frequency than the interface positionfect for all four materials. To understand why this occurecall that all of the interfaces meet at the same point ontest sample’s outer boundary. It is only when the magnflux penetrates sufficiently deep into the material thatsamples the interior solid–liquid boundary that each intface will differently perturb the flux at the secondary colocation. In CdTe, this is seen to occur at;500 kHz, Fig. 12.In contrast, the interface position still affects the responsethe sensor even when the flux is concentrated very closthe edge of the crystal, i.e., when operating at higherquencies. It is only when the infinite frequency limit is a


ntsf-,e

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proached that one loses sensitivity to the position.This analysis of an absolute sensor’s response has sh

it to be sensitive to both the position and shape of the inface. The sensitivity to both phenomena is frequency depdent and a maximum sensitivity exists at intermediate fquencies. The analysis has shown that both locationposition effects are coupled in an impedance measuremethe intermediate frequency range. However, careful measments over a range of frequencies may be able to separresolve the two growth parameters because of their diffefrequency dependencies.

V. DIFFERENTIAL SENSOR

The essential idea of an axially displaced differentsensor is to sample the difference in field perturbation at



FIG. 15. Normalized impedance curves for three interface shapes for the differential sensor~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

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positions along the axis of a sample. By placing two oppingly wound secondary coils at these locations and ensuthat they are symmetrically located within the primary coequal magnitude, but opposite sign voltages are inducethe coils when a homogeneous sample is present. The induction of an inhomogeneous sample with different condtivities near the two pickup coils will perturb the electromanetic flux at one coil more than the other, and a nonzresultant voltage will be observed. Thus, such a sensorbe incapable of distinguishing between an entirely liquidsolid sample~because of equal but opposite induced voltaat the two coil locations!, but might exhibit enhanced senstivity to the location and curvature of an interface separatmaterials of different electrical conductivity.

A. Interface position effects

Figure 13 shows the effect upon the normalized impance curve of moving a flat interface through a differen


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sensor. For these calculations, the two secondary coils wplaced close to either end of the primary coil~they were 34mm apart!. It can be seen that the imaginary componentimpedance at first increased with frequency, reached a mmum, and then decreased for each location. Figure 13 icates that the frequency corresponding to the maximimaginary component was interface position dependent~thefrequency increased as the interface passed upwards thrthe sensor!. This can be seen more clearly in Fig. 14 whethe imaginary impedance component is plotted as a funcof frequency for eachh value. At or above the frequency omaximum response, the impedance reached its maximvalue well after the interface had passed through the ceof the primary coil. The exact location at which this occurrwas determined by the relative magnetic vector potentialeach secondary coil location. This depends on the testquency and the electrical conductivities of the solid and l



FIG. 16. Variation of imaginary component of impedance with frequency for five interface shapes for the differential sensor~a! CdTe,~b! GaAs,~c! Si, and~d! Ge.

aad

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uid. As the frequency is further increased beyond this pethe curves for each interface location continue to remseparated until very high frequencies are reached, againto differences in the fringe field at the two coil locations.

B. Interface shape effects

Interface shape effects were investigated by changthe interface shape while maintaining the outer edge ofinterface midway along the primary coil. The normalizimpedance curves for concave, flat and convex interfacesshown in Fig. 15. The size of the impedance curve wasserved to increase as the interface curvature changedconcave to convex. The sensitivity to interface shape atincreased with frequency, went through a maximum at a mterial dependent frequency, and then decreased again atfrequency~beyond 2 MHz! before the curves eventually con


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verged, Fig. 15~a!. This can be seen more clearly in Fig. 1which shows the imaginary impedance component’s fquency dependence. These calculations reveal the existof a relatively narrow, material specific, intermediate ranof frequencies where a strong sensitivity to the interfaccurvature exists. In this region, the imaginary impedancomponent monotonically increases as the interface’s shchanges from concave to convex. Above and below thisgion the sensor has little or no sensitivity to curvature.

If Figs. 14 and 16 are compared, it is again apparentthe calculated impedance above 105 Hz ~107 Hz for CdTe! isdominated by the interface’s location while lower frequendata are sensitive to both the interface curvature and thesition. Therefore, data collected over a range of frequenmay be sufficient to separately discriminate interface lotion and shape. The range of frequencies where the se


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significantly responds to interfacial curvature is seen toreduced with the differential sensor arrangement becausfringe field effects at the ends of the primary coil. The manitude of the imaginary impedance component’s changesociated either with movement of the interface or a changits curvature are also significantly enhanced with the diffential sensor design~compare the ordinate scales of Figs.and 16!. Thus, from a practical point of view, higher qualitinformation about interfacial curvature and location mightobtained with a differential sensor. However, this methwould preclude the measurement of conductivity and thetors that affect it~e.g., melt composition or temperature!.

VI. DISCUSSION

Because inductive contributions to an eddy current ssor’s test circuit eventually become overwhelmed by paraics and other circuit component impedances at high frequcies, the eddy current sensing method appears to bepromising for high liquid conductivity materials like Ge, Sand GaAs. Lower conductivity systems such as CdTe worequire careful test circuit design to enable observationthe high frequencies predicted to be needed for locationtermination. Axially separated differential coils are mosensitive to changes in the position and curvature ofliquid/solid than the absolute sensor design and providehanced discrimination of these two contributions to tsensed response.

ACKNOWLEDGMENTS

This work was performed as a part of the research ofInfrared Materials Producibility Program conducted byconsortium including Johnson Matthey Electronics, TexInstruments, II–VI Inc., Loral, the University of Minnesotaand the University of Virginia. The authors are grateful fthe many helpful discussions with their colleagues in thorganizations. The consortium work was supported byARPA/CMO contract monitored by Raymond Balcerak.

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