PHYSICAL REVIEW VOLUME 117, NUMBER 6 MAR C H 15, 1960

Determination of the Surface Tension and Surface Migration Constants for Tungsten*

J. P. BARBoUR, F. M. CHARBQNNIER) W. W. DoLAN) %. P. DYKE, E. E. MARTIN) AND J. K. TRQLANIsgfield Research Ilstetgte, Mc3fieeeslle, Oregors

(Received October 1, 1959)

A determination of the surface tension and surface migration constants of metals in the solid phase, basedon the use of pulsed Geld emission microscopy, is discussed. The experimental technique is described. Anapplication to 6eld emission cathodes of Herring's theory of transport phenomena in solids is given, whichyields the necessary relations required for the determination of the basic physical constants from the experi-mental data. Results are presented and discussed for the case of tungsten. The method is applicable to anumber of other metals, several of which are currently under investigation.

I. INTRODUCTION

~ 'RANSPORT phenomena are known to occur inheated solids. Several physical processes may be

involved, such as volume diGusion, evaporation andcondensation, or plastic Row; the dominant process isdetermined by the experimental conditions, particularlythe temperature and the size of the object under study.The experimental determination of the physical con-stants associated with transport phenomena provides

, basic information concerning the solid state; studiesconcerning refractory metals have a practical signiG-cance related to their use as cathode materials.

The evaporation of tungsten was studied some timeago by Zwikker. ' More recently, tracer techniques havebeen used to investigate the volume self-diffusion ofvarious metals, e.g. , tantalum. ' The surface migration ofnumerous combinations of impurities on base metals hasbeen investigated with Geld emission microscopy. '

Recently, an intermediate temperature has been usedto maintain the electrical stability and longevity ofpulsed field emitters4; under such conditions, the tung-sten emitter may change its shape by a transport mech-anism which has been identiGed as surface migration. 'The choice of operating conditions requires a knowledgeof the diffusion constants and surface tension. Each ofthese constants may be determined with good accuracythrough the measurement of the rate of change ofcathode geometry. In an early experiment Muller' ob-tained an indirect determination of the geometricchange from the corresponding change in the fieldemission current-voltage relationship, and deducedvalues for the activation energy and the diff'usivity con-

stant; uncertainties were introduced in these measure-ments by the lack of a sensitive method for the directmeasurement of small geometrical changes, and by the

inability to determine the emitter geometry accuratelyby direct observation.

Pulsed T-F emission microscopy' provides a con-siderably more sensitive method by which the rate ofsurface migration can be directly measured during thetransport of a sufficiently small volume of material thatthe gross cathode geometry remains essentially un-changed. Other advantages of the method include acontrolled and favorable cathode geometry, excellentdepth resolution and high magni6cation, and continuousmonitoring through the emission pattern of the surfaceunder study. The present paper reports the use of thismethod to obtain improved values of the activationenergy and diGusivity constant for tungsten, and also todetermine the surface tension from the eGect on thesurface migration rate of an electrostatic surface stress;similar measurements are in progress for tantalum andmolybdenum. The method is generally applicable to allmaterials from which field emission can be obtained,i.e., metals and semiconductors, and its use is particu-larly convenient in the case of refractory metals.

IL EXPERIMENTAL TECHNIQUE

The object selected here for the study of surfacemigration of tungsten-on-tungsten is the single crystaltip of a needle shaped field emission cathode ("emitter" ).Such cathodes have an approximately spherical tipsmoothly fitted to a conical shank; a typical emitterproGle is shown in Fig. 1. Cathodes are sharpened byelectropolishing and then smoothed and cleaned bycontrolled heating in vacuum'; the heating time andtemperature can be adjusted to provide any desiredvalue of the tip radius, within the approximate range10 ' to 10 4 cm. The detailed shape of the cathode nearthe tip is obtained from electron microscope shadow-

* This work was supported by the U. S. Navy, Once of NavalResearch, and was reported, in part, at the 1955 and 1956 FieldEmission Symposia.' C. Zwikker, Physics 5, 249 (1925).

2 D. B.Langmuir, Phys. Rev. 86, 642(A) (1952}.3 R. H. Good, Jr., and E. W. Muller, in Haedbuch der Physik,

edited by S. Fliigge (Springer-Verlag, Berlin, 1956), Vol. XXI,pp. 176—231.' W. P. Dyke et al. (to be published).' J. L. Boling and W. W. Dolan, J. Appl. Phys. 29,. 556 (1958).' E. W. Miiller, Z. Physik 126, 642 (1949).

FIG. 1. Electron mi-croscope shadowgraphsof a typical 6eld emitter.J3 shows emitter rotated90' from position of A.

r W. P. Dyke and J. P. Barbour, J. Appl. Phys. 27, 356 (1956).s W. P. Dyke et al. , J. Appl. Phys. 24, 570 (1953).

SU RF ACE M I GRAT ION CON STARTS FOR

graphs, and cathodes with symmetry of revolution areselected for the measurements. The surface curvature ismaximum at the apex, and the very large curvaturegradients near the emitter tip lead to high surfacemigration rates; since the cathode has symmetryofrevolution the direction of the curvature gradient isknown, and its magnitude may be measured on thecathode shadowgraph. When the cathode is heatedtransport phenomena are activated and in the absenceof electric field material migrates from the emitter apextoward the shank; in the present work migration ratesare determined from the corresponding changes inemitter length.

The held emission cathode is mounted in a projectionmicroscope, with the emitter tip at the center of aspherical bulb coated with an aluminum-backed phos-phor. Field emission occurs at the tip when sufhcientvoltage is applied to the phosphor screen, and theemitted electrons strike the screen, yielding an emissionpattern characteristic of the crystallographic structureof the cathode material. ' Observation of this patterngives a ready determination of the crystal orientation atthe emitter tip; it permits elimination of cathodeshaving crystal boundaries near the tip, and it providesan estimate of the degree of purity of the surface understudy. Emission patterns have been extensively used forthe study of metal surfaces, and particularly for thestudy of the adsorption, migration and desorption offoreign adsorbates on a pure metal substrate, ' when suchadsorbates affect the work-function and therefore causevisible changes in the emission pattern; however, simpleobservation of the pattern does not allow the quantita-tive study of self-diffusion processes.

The usefulness of the 6eld emission projection micro-scope in the study of transport phenomena is greatlyenhanced by the use of pulse techniques. ' When thescreen voltage is applied in short pulses, a suitable com-bination of pulse length and repetition rate (e.g., one-microsecond pulses applied at a rate of 30 pulses persecond) provides visual continuity of the emission pat-tern while minimizing the disturbing effect on surfacemigration of the electric 6eld at the emitter tip. It isthen possible to draw large pulsed current densities andto obtain patterns in which the emission from latticesteps of atomic height (e.g., 2.23 A for the (110) crystalplanes of a tungsten emitter surface) can be detected bya corresponding bright ring. As surface migration pro-ceeds, atoms migrate from the apex toward the emittershank, and a repetitive process occurs in which theuppermost atom layer is dissolved and the correspondingring shrinks and disappears. It has been shown that eachcollapsing ring corresponds to the removal of a singleatom layer from the emitter tip."Thus the change inemitter length which occurs during a given period oftime can be determined with high accuracy by countingthe number of rings which collapse during the same

E. W. Muller, Physik. Z. 37, 838 (1936)."J.K. Trolan et al. , Phys. Rev. 100, 1646 (1955).

V

FIG. 2. Experimental Geld emission projection tube used insurface migration experiments (schematic illustration). A: glassenvelope; 8: aluminum back willemite phosphor anode; C: fieldemission cathode; D: cathode supporting filament. The emittertemperature is controlled through the variable voltage VJ„ thesteady-state bias Geld F0 is controlled through the variable voltageVo', a pulser supplies the low duty cycle voltage V„required toproduce a visible emission pattern on the anode screen.

period. The rate of change in emitter length and thecorresponding surface migration rates are thus accu-rately determined by measuring the "ring rate" over theperiod required for the dissolution of about 50 atomlayers at the emitter tip; since the emitter radius islarge compared with this change, the observation ismade at essentially constant geometry. This contributesconsiderably to the accuracy of the experiment in viewof the sensitive dependence of migration rate on emitterradius, as will be developed in the next section. Also aseries of successive measurements, e.g. , to determine thevariation with temperature of the migration rate andthe corresponding activation energy, may be performedwith the same emitter, thus allowing greater accuracyby eliminating the effect of small errors in the determi-nation of emitter geometry. In this case the initial andfinal values of the emitter radius are obtained directlyfrom electron microscope shadowgraphs of the emitterprofile, and intermediate values corresponding to suc-cessive measurements are determined from the relation-ship between screen voltage and emitted current.

Thus the use of pulsed 6eld emission microscopy andthe ring method allow a sensitive and accurate determi-nation of the geometrical changes resulting from trans-port phenomena, which can be applied to a quantitativestudy of self-diffusion processes in single crystals of puremetals.

The experimental tube is shown in Fig. 2. A very highvacuum is desired in order to maintain a high degree ofsurface purity throughout the experiment. There' arewell-known methods for obtaining vacuums of the orderof 10 "mm Hg in sealed off tubes, and the methods usedherein have been described in an earlier paper. "At suchpressures the emitter surface is cleaned initially by abrief heat Qash and remains clean for several hoursduring which data are obtained. Cathode temperatureswere measured at the emitter shank (within 0.5 mm ofthe emitter tip) by a micro-optical pyrometer, and the

"W. P. Dyke and J. K. Trolan. Phvs. Rev. 89, 799 (1955).

1454 BARBOUR, CHARBONN IER, DOLAN, D YKE, MARTIN, AN D TROLAN

true tip temperature was then obtained from the calcu-lated temperature drop along the shank. The calculatedtemperature correction is quite small. In order to checkthe accuracy of the calculation and the possible eGect ofthermomotive forces on the surface migration, one of theexperimental tubes was built with an additional cylin-drical electrode surrounding the emitter and maintainedat the cathode filament temperature. Good agreementwas found between the data obtained with both types oftubes.

While the electric Geld at the emitter tip was highduring the pulse (e.g. , 7)& 10' v/cm), the ratio of field-ontime to total time (duty cycle) was 3)&10 ', i.e., suffi-

ciently small that the pulsed electrostatic forces had noappreciable eGect on the migration rate. This wascon6rmed by the observation that an increase in dutycycle to 3X10 4 produced no detectable change in thering rate.

III. THEORY

Herring" has applied the principles of thermody-namics to the theoretical study of transport phenomenaand of the associated geometric changes in heated crys-tals of pure metals. The resulting general expressionshave then been applied to the conditions prevailing infield emission experiments, to provide the basis for theinterpretation in terms of physical constants of the ex-perimental measurements presented in the next section.

A. General Expressions

Starting from energy considerations, Herring hasestablished the condition for the geometrical equilibriumof the solid-vacuum interface at the surface of anisothermal single crystal of arbitrary shape, The equi-librium condition is simply that the difference p~ of thechemical potentials for atoms and lattice vacancies,evaluated at some point 3f just beneath the surface, beindependent of the position of point M. Herring hasderived the following expression for p~ ".

( 1 1 ) 1 8'y 1 8'yttsr=t p+"»I —+—I+—,+-

(21 E2) El r)stl E2 BS2

where Qp is the volume per atom in cm', y is the surfacetension in dynes/cm, Rt and Es are the principal radiiof curvature of the metal surface just about the point3E, n~ and e2 are the surface directions associated withthe principal radii of curvature, and I', represents thenormal component of any externally applied surfacestress in dynes/cm', tsar is expressed in ergs/atom, and

pp is an arbitrary constant which represents the valueof p below a stress-free plane surface.

Conversely, if p~ is not a constant everywhere below

» C. Herring, in Stractare artd ProPerties of Solid Slrfaces, editedby R. Gomer and C. S. Smith (University of Chicago Press,Chicago, 1953).

» C. Herring, in The Physics of Powder Metallurgy, edited by W.E. Kingston (McGraw-Hill Book Company, New York, 1951).

the solid-vacuum interface the metal surface will alterits shape when heated, as a result of one or several atomtransport processes such as volume diffusion or surfacemigration. Scaling laws, i.e., a comparison of the timedurations required for the same relative geometricalchange to take place in two geometrically similar aggre-gates, other parameters being equal, have been derivedby Herring" for four possible processes. As was notedearlier, these laws have been used at this laboratory5 toshow experimentally that surface migration is the pre-dominant transport process near the tip of tungsten Geldemission cathodes, under the conditions prevailing inthe experiments described below; this results from theunusually small size and therefore high surface-to-volume ratio of the Geld cathode tip, and it permits adirect study of surface migration processes which inother experiments are often masked by volume diffusion.

In cases of small departures from equilibrium, thebasic assumption is made that the rate of kinetic changeof the surface is everywhere proportional to the gradientof the chemical potential, and Herring derives the basicexpression for the flux of atoms Jsr (in atoms/cm sec) atany point M on the surface:

D g—Q/RT

(vt )~,A pkT

where D, is the diffusivity constant in cm /sec for sur-face migration, Q is the activation energy in calories per'mole, T is the crystal temperature in degrees Kelvin, A p

is the surface area per atom and (Vts) ~ is the gradient atpoint 3I of the quantity ted given by Kq. (1).

B Application to Field Emission Cathodes

The purpose of this section is to apply the generalexpressions to the case of field emitters, and morespecifically to relate the time rate of change ds/dk of theemitter length to basic physical constants. As notedpreviously the geometry of the field cathode is such thatsurface migration predominates, other transport phe-nomena such as volume self-diffusion having only anegligible effect on the geometrical changes of theemitter for the range of temperatures investigated.

The previous equations show that the chemical po=tential p~, and therefore also the migration rates, maybe changed by application at the cathode surface of anexternal stress which varies with position. The experi-mental arrangement used makes it possible to apply aknown electrostatic stress, simply by application of arelatively small dc bias voltage at the phosphor screen;the larger Geld required for emission is then super-imposed in the form of low duty cycle pulses.

Although in a strict sense some of the physicalquantities appearing in Eqs. (1) and (2), such as A p, Dp,

Q or y, are anisotropic because of the crystalline struc-ture of the emitter material, the eGectsof this anisotropy

' C. Herring, J. Appl. Phys. 21, 301 (1950).

&455AT ION CON STAN TS FOR KSURFACE M ANGRA

ered here are relatively small exceptf r the metals consids ecific orientations such as

in e ts where over-a, ll migra-in the resent experimen s w

orientations, da ood ap roximation inma be neglected to a goo aptension y may1 tential assuming an,'1, for the chemica po en

'expression ~

~~due onl to e ectros a i1 t t tic forces one ob-external stress du y

tains the si psim ler expression or t e sur aceint on the emitter surface:current JM at some point on e em

1 F'O'De &'~" — (1 1 )JAp kT Rg p 7r

s the volume of material, in cm'/cm sec,where JM denotes e voline of unit lengths er unit time across a ine op

lar to the direction o eperpendiculard t of the quantity inre resents the gra ien o

'-e s 3f. This gradient, andbracke s,kets evaluated at point . id' t d 1 ngmeridiancurveson

itt h tiso J are directe a on

tion' '

convenient alternateitter surface since the emit er a

tion. E uation 4 e ow is a conv'

h th radius of curvature ofi Ii is the applied electric field atf E . (3), in whic hr s st e ra

d t d' ensionless coefficientsitter at the apex, Ii p is t e app i

'1 f th ittdc re resent imen

'

de end only on the etai s otM. d d

IIIII

I

IIIII

III

l

l

I

I

I

I

tII

III

II

IIIIIIII

III

II

IIIIIIII

III

IG. . u' ' '

~ and O2 of the Geld emitter surface.FIG. 3. Successive positions O~ an 2 o

(dki

&dt I p Ap

pe2 D —QIRT g 2a

r3f

(dz 'l &M=-Idt) p rM Ap

geometry and th pthe articular p

Thus a value of the activation energy—ai

1—— i. () nenerkT ' &,. 16'. migration 0 ungs

red temperature dependence of thep

termination o oes noemitter receding rate; the determ'k ld fh

1. 3figratioe ze e

require precise now e

ents have

Therma/ Etchddition raphical curvature measuremen s a

microsco e shadow-f.eld forces the surface

erformed on the electron mic pIn the absence of electric xie 0

db t es ace tensionforces, anmigration is controlled y e s nthe a ex toward the s a,n oatoms migrate from e p

i ration inu f the oint M considered at eas up

adil understood since migra'

e emitterFigduce the surface energy. 0=90, an s ows ot.osition of the emitter sur ace a

to another. us orr —1.25 r

illustrates the position ot s sur ace migra ion p o

tant Dp an average value 2athe final

h mitte aemitter enges. It is observed experimentally that tht e rate of

expression:c'

h ds, dt is much larger than the~p~ ape-~I~~ ~.25

kT r'

1 t the emitter axis, whichroximation the rate of Row o materia

Z. Migration ie the Presence oe o A lied Pzeldt 1 t 2 J 1

lane P erpendicu ar o eby definition of J~ must e equa orelate o sd t dz/dt by the expression:

(5)(d U/dt) =mrM'(dz/dt), .

n the a "nce of appliu deere se to d th hw ic

forces wi oelectric fieM:

( )'

(3) and a straightforward derivation ea s to6 basic Eq. , anf th emitter receding rate in t e pres-the expression or e em

BARBOUR, CHARBON N I ER, DOLAN, D YKE, MARTIN, AN D TROLAN

where co is the value of the coeKcient c~ at the apex.Thus the theory predicts that the emitter receding ratewill decrease with dc applied field, by a relative amountproportional to the square of the held but independentof the 6eld polarity. " The field distribution at theemitter surface can be calculated by electrostatictheory' for a given emitter geometry, and an averagevalue co——0.5 is then obtained by comparing the curva-ture and 6eld gradients near the emitter apex. Thus thesurface tension for the emitter material is obtained fromthe expression:

y=rppis/8' (9)

in terms of the minimum held Iio& required at theemitter apex to reduce the receding rate to zero. Apractical significance is that when a steady-state field ofthis magnitude is applied the emitter may be heated tomaintain electrical stability while avoiding emitterdulling.

If the dc applied field is further increased, electrostaticforces predominate over surface tension forces, the netsurface migration currents are reversed and are now

directed toward the emitter apex. According to Eq. (8)one might expect the emitter to extrude under suchconditions; however, simple extrusion does not occurbecause of the difhculty of nucleating new atom layersin certain crystallographic directions, which preventsgrowth of the emitter in these directions. As a result, amore complicated process called "build-up" tak.es place,in which the initially rounded emitter gradually as-sumes a polyhedral shape. A discussion and experi-mental study of the build-up process is forthcoming inanother paper"; in this case the simple theory given inthis section, rvhich treats the emitter material as beingisotropic, does not apply.

"It may be noted that Eq. (8) is not strictly correct, becauseapplication of an electric field at the emitter surface causes achange not only in the chemical potential p but also in the activa-tion energy Q for surface migration. The latter effect arises becausethe activation energy is a measure of the difference between thepotential energies of the migrating atom at a stable site and at thesaddle separating two stable sites, and the migrating atom experi-ences a different electric field at the stable and saddle positions.This effect was Grst pointed out by J. A. Seeker LBell SystemTech. J. 50, 907 (1951)g and was studied in more detail by M.Drechsler [Z. Elektrochem. 61, 48 (1957)j. This variation of Qwith applied field does not change the field which reduces theemitter receding rate to zero, and it does not invalidate the Eq. (9)from which the surface tension is obtained; however it will causea slight departure from linearity in the relationship between ds/dtand Il2. Calculations of the field effect on activation energy by amethod similar to that of Drechsler show that the maximum varia-tion from the zero field activation energy is less than 0.02 ev/atomor 0.6% for the fields of interest in the present work (Fp &~10'v/cm), and may therefore be neglected without significant error inthe derivation of Eq. (8); this is further confirmed by the experi-mental results shown in Fig. 5.

"'P. C. Settler and F. M. Charbonnier (to be published).

ence of a dc electric field having the value Po at theemitter apex:

rJ p'y t ifs

Kdf) rp 4 cp 16''7) (df) p

IO I tOI

I

II

I

I5 I- —------- I-

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

Icoo

I

I

I

I

I

I

I

I

I

50

COCIXOCJ 2'CO

a

aO

IL

WOl

lalX

0.250

~ ~

i'

T2I

II

I +o

, I~ ~ I~ ~ ~ ~ ~ ~

J%t ~

60NUMBER OF RINGS

4

~0~ ~

~a1 ~e~ ~ o~e~ ~ ~0

~ ~~0

I20

FIG. 4. Typical plot of the time separation s between the collapseof successive rings. At time to the emitter tip temperature is re-duced from T&=2750'K to T~=2315'K. Disregarding statisticalfluctuations in the individual counts, the time separation s followsthe curve ABCD.

IV. EXPERIMENTAL RESULTS AND DISCUSSION

The receding rate of heated tungsten 6eld emissioncathodes has been measured as a function of tempera-ture and dc applied field. The axis of the emitters usedin these experiments coincides with a (110) crystallo-graphic direction, and the emitter receding rate is de-termined from the rate at which bright emission ringscollapse at the center of the emission pattern. As notedin Sec. II, the collapse of each ring corresponds to thedissolution by surface migration of one atom layer at theemitter apex; the corresponding reduction in length is(V2/2)&&3. 16=2.23 angstroms, since tungsten has abody-centered cubic structure with a lattice parameterof 3.16 A.

Figure 4 shows a typical plot of the time separationbetween the collapse of successive rings. At time $0 theemitter temperature was reduced from Tj to a smallervalue T2, with a corresponding increase in the timeseparation between rings. Apart from small statisticalfluctuations due to the random nature of diffusionprocesses the ring rate (i.e., the reciprocal of the timeinterval separating the collapse of 2 successive rings) hasa well-defined and reproducible steady-state value, for agiven temperature T and dc bias field Ii0 at the emittertip; this steady state value (corresponding for instanceto the segments AB or CD of the curve ABCD shown inFig. 4) is used, in conjunction with Eqs. (7) to (9), forthe determination of the surface migration constantsand the surface tension. In addition, a transient BC isobserved following the time 30 at which a change is madein either T or Ii 0. The transient has a duration ht which

SURFACE M IGRATION CONSTANTS FOR W

is too long to be explained by the thermal inertia of theemitter tip; the transient corresponds to a number ofrings which depends on the magnitude of the tempera-ture or Geld change but is otherwise reproducible. Thesecharacteristics of the transient may be explained bynoting that the detailed shape of the emitter surfacedepends on the relative values of the surface energy atdiferent points; since the emitter material is slightlyanisotropic, changes in temperature or surface Geld will

modify the relative values of the surface energy andcause local rearrangements of the surface to adjust to anew equilibrium geometry. Such rearrangements are ex-pected to be small in terms of the overall emittergeometry, and have therefore been ignored in thederivation of the expressions (7) and (8) for the steady-state migration rates (this view is supported by the goodagreement between the functional relationships pre-dicted by the theory and those measured experimentally,particularly with respect to the eGect of a surface Geldon the steady state migration rate). On the other handthese temporary local rearrangements of the surfacewould be expected to aGect the ring rate appreciably,because of their effect on the size of the small (110)plane facet at the emitter apex, and the transient ob-served in the ring rate is thought to arise from this eGect.

The dependence of the receding rate on the magnitudeof the dc applied Geld is shown in Fig. 5 for a typicalemitter. The percentage of zero-Geld receding rate isplotted vs the square of the dc electric Geld at theemitter apex. The experimental points fall on a straightline, in agreement with Eq. (8) derived from the theory.

Using r=5.5&(10 ' cm as obtained from electronmicroscope shadowgraphs of the emitter and the valueFe& ——1.08&& 10 v/cm determined by the intercept of theexperimental line with the x axis, one may calculate pfrom Eq. (9).This procedure has been used with severalemitters, and yields values of p in the range 2700 to 3100

LU

+~ IOO&

o 4O ~ gI 80Z \ ~OLU

0%

~ 60'O ~ ~

4J aN

40-UJ 0cCI-ZLU

20.uJCL ~~

2., Fol

0 0.5 I 1.5SQUARE OF THE dc BIAS ELECTRIC FIELD F AT THE EMITTER APEX IN (IO V/cfn) UNITS0

FtG. S. Dependence of the emitter receding rate on the magni-tude of the dc electric Geld applied at the emitter apex, for a fixedtip temperature (2030'K). O—Run 1: bias Geld gradually in-creased from zero to a maximum value slightly larger than F01.~—R.un 2: bias field gradually decreased from this maximumvalue down to zero.

-I6 ~

-l8.3'

-20 '

iCh~ ~

~ 5, ,~ 4

j~ + ~

~ l ~

~ l ~5 ~

~5

h~LEAST SQUARES

LINE

-22.

dynes per cm. Thus the value:

p =2900 dynes/cm (10)

is proposed for pure tungsten in the solid phase at2000'K; the uncertainty on this determination of p isestimated at &10%, and results mainly from possibleerrors in ro and Foj due to microscope calibration and toapproximations in the calculation of Po~. Due to thecrystalline structure of tungsten, the value of y is ex-pected to vary with crystallographic orientation and tobe lowest for the (110) plane in the present case of abody-centered cubic crystal; however both theoreticaland experimental considerations indicate that the varia-tions with crystallographic direction of the surface ten-sion probably do not exceed 10% of the average value.It is of interest to compare our value of y with the value2300 dynes/cm recently obtained'r by the conventionalliquid drop technique for tungsten at 3380'K. Thevalues obtained by the two methods appear to be inreasonably good agreement, in view of the expected de-crease of the surface tension with increasing temperature.

The experimental dependence of the emitter recedingrate on temperature is shown by Fig. 6, which corre-sponds to several successive runs of a typical emitter.The coordinates have been selected to permit the de-termination of the surface migration constants Do andQ. For this purpose each data point (corresponding to ameasurement of ds/df at a given temperature T) is con-verted to the corresponding value of D= Doe @I~ bymeans of Eq. (7); the appropriate values of r and T areused for each experimental point, a value y=2900dynes/cm is used, Qs = 1.57X10 "cm'/atom is obtainedfrom the lattice parameter, and an average valueAs ——10 " cm'/atom is used for the surface area pertungsten atom. A plot of lnD vs 1/T shows the straightline expected from the theory; the slope of the experi-mental line yields the activation energy Q, and the

"A. Calverley, Proc. Phys. Soc. (London) B70, 1O4(I (&957).

I04iT

FIG. 6. Natural logarithm of the surface migration coefficient Din cm'/sec, as a function of reciprocal temperature in 'I, forseveral runs of a typical tungsten Geld emitter. The least squaresline D =Dec ols is used to determine the values of De and Q.

1458 BARBOUR, CHARBON N I E R, DOLAN, D YKE, MARTIN, AND TROLAN

intercept of the line with the y axis (1/T=O) yields the

diffusivity constant Do. Such data has been obtainedfrom several emitters, leading to the following values:

Q= 3.14+0.08 ev/atom, or 72 kcal/mole,

Ds 4c——ms/sec.

These values correspond to pure tungsten in the solid

phase at an average temperature of 2000'K. The errorshown for Q is simply the statistical probable errormeasuring the scatter in determinations of Q for variousemitters. The value of Do is probably reliable within afactor 2, the main uncertainty in Do resulting from theuncertainty in Q.

The values obtained by the present technique for thephysical constants Do, Q, and p are well defined andreproducible. However they require interpretation sincethe crystalline structure of tungsten leads one to expecta variation of its physical properties with crystallo-graphic direction. In the course of an experiment theoverall recession of the emitter tip may exceed onethousand atom layers, and atoms removed from theapex of the rounded emitter must migrate over largedistances and on surfaces corresponding to a variety ofcrystallographic orientations. The migration rate is asensitive function of the activation energy and the rateof the overall process must therefore be controlled bythat portion of the total surface migration path with thehighest activation energy, at least for the condition ofsteady Qow during which the emitter receding rate ismeasured. In body-centered cubic crystals directions ofthe L111]family and planes of the (110) family play aspecial role as they correspond to the densest packing ofatoms. It follows that surface migration in the L111jdirection corresponds to an exceptionally low activationenergy and occurs most readily; however this directionof migration is possible only on crystal planes such as(110)or (211) for which the sum of Miller indices is zero.In general migration must proceed in a general directionother than L111jor over surfaces which do not containL111]directions, and a higher activation energy would

be expected.Stranski and Suhrmann' have proposed a simple

method for estimating the potential energy of surfaceatoms as a function of location: it is assumed that inGrst approximation the binding energy between twoatoms is a function of their distance only, and is inde-

pendent of the presence of other neighboring atoms. IfP& represents the binding energy between nearest neigh-bors (separated by rr ——wE/2, where a is the latticeparameter, for a body-centered cubic crystal), @& repre-

sents the binding energy between second nearest neigh-

bors (rs=a), etc., Stranski and Suhrmann have found

@s—0.5&r and Ps(0.05&I for tungsten. Similar values

have been obtained by Muller assuming a Van der

Waals type of interaction obeying a (1/r') law. The

' L ¹ Stranski and R. Snhrmann, Ann. Physik 1, 155 (1947).

known heat of sublimation of tungsten (8.7 ev/atom at2000'K) may then be used to obtain P&—1.6 ev. Thesurface migration of an atom occurs in a succession ofindividual steps, in each of which the atom leaves astable site where its potential energy has a minimum E,and must go over a saddle where its potential energy isE, in order to reach the next stable site. The corre-sponding activation energy is simply Q=2(E, —E&).

This method has been used to compare the activationenergies over the various migration paths in the presentexperiments where tungsten atoms migrate from theapex toward the shank of a rounded field emitter withits axis parallel to the (110)direction, assuming ps—0.5&rafter Stranski and Suhrmann and p„=ps(a/r )' fore&2. It is found that the initial step in the over-allmigration, in which an atom is removed from the edgeof the uppermost (110) atom layer, corresponds to anactivation energy Qo—2P&, the atom then migratesreadily over the underlying (110) layer (with Q—g&)until it becomes attached to the edge of this secondlayer. However, as the atom proceeds further away fromthe apex it must migrate over a large number ofcrystallographic plane facets, such as (100) and (211),for which the activation energy Q is of the order of 2.3p&and therefore somewhat larger than Qs. Such regionsdetermine the over-all migration rate and therefore theactivation energy measured for the process, since ma-terial cannot accumulate anywhere under the conditionof steady-state migration prevailing during the meas-urements.

The measured activation energy Q=3.14&0.08 ev/atom thus represents an effective activation energy forthe surface migration of tungsten on its own lattice,which is applicable in the general case where the direc-tion of migration does not coincide with a L111jdirection and where the crystal surface includes severalcrystallographic orientations; it also holds approxi-mately for surfaces consisting of a single plane withorientation other than (110).The measured activationenergy is somewhat smaller than the calculated valueQ—2.3p,—3.7 ev/atom. This discrepancy may be ex-plained by the fact that the calculated value correspondsto the case of an isolated atom migrating over the sur-face of an ideal crystal, whereas in the experimental casethe migration is facilitated by the fact that a largenumber of atoms participate at a given time, and fur-thermore experimental evidence indicates that the ac-tivation energy for surface migration is reduced bythe presence of crystal imperfections such as screwdislocations.

Field emission techniques have been applied to thedetermination of the physical constants for surfacemigration. The same techniques may be used in thestudy of other transport phenomena such as volumeself-diGusion, under experimental conditions where vol-ume diffusion predominates. The same general con-siderations apply, and an expression can be derived for

SU RF ACE M I GRAT ION CON STAN TS FOR K

FIG. 7. Comparison of surfacemigration and volume diftusionrates, for tungsten Geld cathode.

3000

Ch

lL'

I-cfK4J

2000

x &ISURFACE MIGRATION PREDOMINATES

Y/1/P

EXPERIMENT

g,(+t) S.M

IOOO-5

IO-4

10-3 -2

IO IO

EMITTER TIP RADIUS r, IN cm

-IIO

(der ' Dp'e @'I"r 1.25—

]—

f

= —yap(dt J kT r'

(12)

The relative magnitude of surface migration andvolume diffusion may be estimated from the ratio x ofthe respective emitter receding rates, given by Eqs. (7)and (12):

(«/dt) s.M. Oo Dog= g(Q' —0)/&

(ds/dt)v. n. Ap Dp r

where the primed quantities correspond to volume self-diffusion. It is seen that the relative importance ofvolume self-diffusion increases as the emitter tempera-ture or radius are increased, since the activation energyQ' for volume diffusion is of course larger than Q. Usingthe approximate values Dp' 11.5 cm'/sec and Q—'—= 140kcal/mole=6. 1 ev/atom obtained by Van Liemptro forthe volume self-diffusion of tungsten, the curves ofFig. 7 are obtained for the ratio x of surface migration tovolume diffusion rates. The shade area represents therange of values of r and T used in the present experi-ment; it falls as expected in the region where surfacemigration predominates. The study of volume diffusion

by the present method would be dificult in the case oftungsten, as it would require both very large cathodes(with correspondingly high emission voltages) and very

"J.A. M. Varr Liempt, Rec. trav. chim. 64, 239 (1945).

the emitter receding rate due to volume self-diffusion: high surface temperatures, for which evaporation rateswould become signi6cant; however the study of volumediffusion by field emission techniques would be possiblefor materials having a lower ratio of volume diffusion tosurface migration activation energy. As noted previ-ously, the process which predominates under given con-ditions can be identified in a preliminary experiment bythe use of Herring's scaling laws.

V. CONCLUSIONS

The pulsed field emission projection microscope is aneffective tool for the quantitative study of self-diffusionprocesses occurring in single crystals of pure metals. Themethod can be used for all materials from which field

emission can be obtained, and it has the advantage ofproviding a well-known and controllable geometry aswell as criteria for the evaluation of the degree of purityand the crystallographic orientation of the surface understudy. Surface migration will usually be the dominantprocess under the experimental conditions most readilyachieved. From the measurement of the receding rate ofthe field cathode, which can be obtained rapidly andaccurately by the ring method, the surface tension p andthe surface migration constants Q and Dp can be deter-mined with satisfactory accuracy.

ACKNOWLEDGMENT

The authors are grateful to L. M. Perry, F. M.Collins, J. L. Holing, and R. W. Strayer for valuableassistance extended daring this work.