Nonlinear surface electron transport over liquid helium

Transcription

1 Fizika Nizkikh Tmpratur, 9, v. 5, No., p Nonlinar surfac lctron transport ovr liquid hlium K.A. Nasydkin, V.E. Sivokon, Yu.P. Monarkha, and S.S. Sokolov B. Vrkin Institut for Lo Tmpratur Physics and Enginring of th National Acadmy of Scincs of Ukrain 7 Lnin Av., Kharkov, Ukrain nasydkin@ilt.kharkov.ua Rcivd May, 9 W prsnt xprimntal data and a thortical analysis of nonquilibrium mobility of surfac lctrons in liquid hlium. Th xprimnts ar carrid out in th tmpratur rang hr lctron mobility is limitd by lctron scattring at surfac xcitations of liquid hlium (ripplons. Holding and driving lctric filds of id rangs ar usd in masurmnts. Spcial attntion is paid to th condition of strong holding filds undr hich hot lctrons ar confind to th ground surfac lvl. Dpnding on th rlation btn th momntum rlaxation rat and lctron lctron collision frquncy, diffrnt thortical approachs ar usd to dscrib th nonlinar mobility of surfac lctrons. Th rsults obtaind allo to stimat th rang of physical paramtrs hr xprimntal data can b dscribd by th thory of nonlinar lctron transport ithin th ground surfac lvl. PACS:.. g 7.. r 7.5.+i Gas liquid and vacuum liquid intrfacs; Elctron stats at surfacs and intrfacs; Surfac conductivity and carrir phnomna. Kyords: liquid hlium, surfac lctrons, to-ripplon procss, lctron conductivity.. Introduction Surfac lctrons (SEs on liquid hlium rprsnt a rmarkabl modl systm for studying transport phnomna in highly corrlatd to-dimnsional (D lctron liquids []. Sinc th nrgy rlaxation rat of SEs inducd by scattring at hlium vapor atoms and quantizd capillary avs (ripplons is vry lo, th nonlinar transport conditions can b achivd alrady at ak driving lctric filds E mv/ cm. Usually, th aral dnsity of SEs n s is changd by varying th holding lctric fild E applid normally to th lctron sht. At th sam tim, it is vry attractiv to carry out th mobility masurmnts undr conditions hn th surfac dnsity n s and E ar indpndnt paramtrs. This allos to study th SE transport for diffrnt rlations btn th momntum rlaxation rat r and th lctron lctron collision frquncy. Morovr, kping n s small undr strong holding fild conditions prvnts vaporation of hot SEs to highr surfac lvls and allos to prform an aurat comparison ith th thory of lctron transport ithin th ground surfac lvl. In our prvious ork [] th nonquilibrium SE mobility as masurd at fixd tmpratur and surfac dnsity for diffrnt holding filds. Th rsults r in a qualitativ agrmnt ith thory though thr as a noticabl quantitativ diffrnc. That diffrnc as a rsult of th lack of th masurmnt auracy causd by difficultis in th xprimntal cll analysis. Latr [] an improvmnt of this analysis had givn bttr numrical agrmnt ith th thory at high holding filds. In this ork prsnt a systmatic xprimntal study of lctron mobility undr nonquilibrium conditions inducs by an ac driving lctric fild. Th xprimnts ar carrid out at T 5. K for n s. cm, E mv/cm, and E V/cm. Th xprimntal data of ( E ar compard ith thortical rsults obtaind using diffrnt approachs applicabl for diffrnt rangs of th paramtr / r. Such a comparison allos to mak important conclusions about mchanisms of momntum and nrgy rlaxations of SEs and to rval th influnc of lctron lctron collisions on nonlinar lctron mobility rsulting in spcific rgims of lctron transport. K.A. Nasydkin, V.E. Sivokon, Yu.P. Monarkha, and S.S. Sokolov, 9

2 Nonlinar surfac lctron transport ovr liquid hlium It is knon that momntum and nrgy rlations of hot SEs on liquid hlium ar govrnd by diffrnt scattring mchanisms. In particular, SE mobility is limitd by onripplon scattring procsss involving long-avlngth ripplons ith typical av-vctors q k ~ 5 cm (hr k is th av-vctor of lctrons. Such surfac xcitations hav vry lo nrgy quanta q T and, thrfor, giv vry small contribution into SE nrgy rlaxation. At th sam tim, nrgy rlaxation is largr for to-ripplon scattring procsss. In this cas an lctron can mit coupls of short-avlngth ripplons ith q ~ T and a small total momntum (q q k. Such a diffrnc btn momntum and nrgy rlaxations lads to intrsting nonlinar dpndncis of lctron mobility on th driving lctric fild. Th articl is organizd as follos. In th Sc., dscrib th xprimntal stup and masurmnt procdur. Thn, th thortical approachs ar dscribd. Th xprimntal rsults and comparison btn xprimnt and thory ar discussd. In Conclusion summariz th rsults of this ork.. Exprimnt Mobility masurmnts r carrid out using th xprimntal cll of circular gomtry. Th cll is dscribd in dtail in Rf.. Th principal part of th cll is a plan capacitor situatd horizontally. Th chargd surfac of liquid hlium is in th middl of th capacitor gap. Th positiv potntial is applid to th bottom lctrod to incit lctrons to b trappd at th hlium surfac. Th uppr lctrod is actually a st of masuring lctrods alloing to crat an ac lctric fild of frquncy in th lctron layr and to masur an lctron rspons to this fild as an ac output voltag. Th output voltag is a rsult of a currnt in lctron layr, and thrfor dpnds on complx conductivity of th layr. To find th rlationship btn th conductivity and th output-input signal ratio, solvd th Maxll quations for lctromagntic fild in th cll ith appropriat boundary conditions []. As a rsult, to quations r found for to componnts of th cll conductanc G Gi G.Th conductanc dpnds on ral and imaginary componnts of lctron layr invrs conductivity and : G G(,,, A; G G(,,, A G. ( Hr A dnots paramtrs dpnding on th cll gomtry, G is th mpty cll conductanc. Th solution of Maxll quations allos also to stimat th radial distribution of th lctric fild oprating in th lctron layr and dpnding on th layr conductivity. Th cll conductanc can b asily obtaind from th output-input signal ratio, taking into aount paramtrs of th masuring lin found indpndntly. Thrfor, in ordr to find th invrs conductivity of th lctron sht i.., and on nds to solv th systm of Eqs. (. It should b notd that th ral xprimntal cll is som diffrnt from th idalizd on, dscribd abov using Maxll quations. Th diffrnc appars, for xampl, hn comparing th calculatd and masurd valus of th mpty cll conductanc. Such a diffrnc is possibly arisn du to intractabl parasitic lctric links btn th lctrods and th groundd cll body. Hovr th main diffrnc btn th ral and idalizd xprimntal clls is du to grounding th cntral masuring lctrod. This grounding can b takn into aount if on multiplis th valus of G, G, and G by th Sout /( Sout Sground,hrS out and S ground ar th surfacs of th output and groundd lctrods, rspctivly. To aount xtra diffrncs, introduc a fitting impdanc btn th input and ground lctrods and us such an impdanc in all our calculations. Whn SEs ar ovrhatd by th driving lctric fild, lctron mobility is high and, thrfor, on hav to masur rathr small phas changs. To rduc th influnc of uncontrolld factors, th mpty cll conductanc as masurd undr th sam xprimntal conditions and in th sam runs as thos of th cll ith lctrons. Th G in Eqs. ( rflcts a connction btn th input and output lctrods, hich dos not dpnd on th lctron layr conductivity, and th rlvant part of th cll conductanc should b subtractd of th solution of Eqs. (. Th prsnc of th lctron layr in th cll changs, du to scrning ffcts, th connction btn th input and output lctrods. Also it changs th parasitic links btn th lctrods and th cll body. Thrfor to solv Eqs. ( proprly, a corrction to th valu of G is ncssary. Th corrction as mad for vry masurmnt basing on th critrion that th solution of Eqs. ( should giv a propr valu of th imaginary part of th invrs conductivity. Not that th valu of is proportional to th ffctiv lctron mass hich is qual to th fr lctron mass if lctron tmpratur is highr than th Wignr solid mlting tmpratur. Th stability of solutions of Eqs. ( is limitd by th masurmnt auracy sinc. Bcaus of this inquality th valu of rathr rflcts th inauracy of masurmnts than th physically-maning imaginary part of invrs conductivity. Prviously, hn solving Eqs. ( [,] ithr th valu as disrgardd or th st of Eqs. ( as rducd to a singl quation for crtain chosn valu of. Th mthod of calculation basd on solving th st of Eqs. ( usd in th prsnt ork and that of solving th rducd singl quation giv almost th sam rsults. Nvrthlss prfr th first mthod as mor strict. To simplify th analysis of rsults initial data r approximatd by a smooth curv bfor bing usd for sti- Fizika Nizkikh Tmpratur, 9, v. 5, No. 99

3 K.A. Nasydkin, V.E. Sivokon, Yu.P. Monarkha, and S.S. Sokolov A, mv Fig.. Th amplitud A and phas shift of th mpty cll as a function of driving potntial. mation of th cll conductanc undr solution of Eqs. (. Rsults of avraging of four masurmnts sris for th mpty cll ar prsntd in Fig.. Th phas chang and output voltag amplitud ar masurd for diffrnt amplituds of th input voltag (circls. Th masurmnts ar prformd in tmpratur rang T K. In addition to quit xpctd linar dpndnc btn th input and output amplituds th incras of phas ith incrasing th input voltag is obsrvd. Th possibl rason for such nonphysical dpndnc may b a nonlinarity in th masuring circuits apparing as a rsult of diffrnc btn ral and idalizd clls. As notd abov, to tak this dpndnc into aount, th fitting impdanc as usd in data procssing. Th smoothing of th xprimntal data is dmonstratd by lins in Fig.. Not that did not obsrv th influnc of th holding potntial on th conductanc G of th mpty cll. Th approximation of th initial data undr valuation of SE mobility as also prformd. As an xampl, th data on lctron mobility at th holding fild 9 V/cm arshoninfig..fig.,a and,b illustrat, rspctivly, th rlationship btn th output voltag phas shift and amplitud and th input voltag amplitud. Points indicat th xprimntal data and lins ar rsults of th approximation usd in furthr calculations. Th stimatd lctron mobility is shon in Fig.,c (solid lin. Th dashd and dottd lins corrspond to thortical calculation in on-lctron and complt control approximations, rspctivly ( discuss ths approachs in nxt Sction. Th points ar xprimntal data obtaind ithout th initial data approximation. Thy illustrat data rrors. Th rrors incras ith dcrasing th holding potntial and incrasing th lctron mobility incras. Rgarding valuation of th driving fild oprating in th lctron sht plan should lik to not th folloing. Th solution of Maxll quations allos to calculat th radial distribution of th driving fild in th lctron layr E ( r. It is ssntial that th fild valu dpnds on th layr conductivity. Th solution of Eqs. ( as obtaind for th linar rgim hr th rlation btn local currnt and local fild is givn by Ohm s la and th conductivity dos not dpnd on E []. In prsnt ork, study th nonlinar conductivity hich dpnds on th fild. Undr such condition th solution of Eqs. ( obtaind in Rf. dos not corrspond xactly to xprimntal situation. Though thr ar to factors hich justify using th linar calculation in th nonlinar cas. First, high rat of lctron lctron rlaxation prvnts tmpratur gradints in th lctron layr so it is quit possibl that th fild distribution diffrs unssntially from th linar mod at som avragd valu A, mv 5, dg a b c V,mV V,mV E, mv/cm Fig.. Th amplitud AV ( (a, phas shift ( V (b for th xprimntal cll ith surfac lctrons, and SE mobility as a function of driving fild E (c for holding fild E 9 V/cm. Points indicat th xprimntal data and lins ar rsults of th approximation. Th dashd and dottd lins corrspond to thortical calculations in on-lctron and complt control approximations, rspctivly. 97 Fizika Nizkikh Tmpratur, 9, v. 5, No.

4 Nonlinar surfac lctron transport ovr liquid hlium of th driving fild. Scond, th driving fild influnc on th conductivity is not drastic vn though bing ll-pronouncd. Thrfor th linar solution can b a quit rasonabl approximation to th actual xprimntal conditions though th lvl of rrors in this cas is not clar apriori. In our ork, th mobility of lctrons is masurd as a function of driving lctric fild hich is a rsult of som avragd radial driving lctric fild across th cll taking into aount its dpndnc on layr conductivity. W stimatd such an avragd fild as Rl E E ( r dr. R l Hr R l is th radius of lctron layr.. Thory.. Mobility as a function of th ffctiv lctron tmpratur W shall confin ourslvs to lctron scattring ithin th ground surfac lvl only. In this cas th av function of a SE is usually rittn as, k ( z, r / zxp( z ikr, hr th paramtr is found by variation to minimiz th ground nrgy. For ( zro holding fild, m /, hr ( / ( º /(º +,º is th liquid hlium dilctric constant, and m ar lctron charg and mass, rspctivly. Th abov givn av-function is found assuming that th lctron potntial barrir at th intrfac V. For th ral valu of V V, pntration of th lctron av-function into liquid hlium is dscribd by th paramtr mv / 5 cm 7 hich is much largr than. SEs on liquid hlium rprsnt a highly corrlatd lctron liquid of hich th avrag Coulomb intraction nrgy is much largr than th avrag kintic nrgy. This mans that an lctron is frquntly rflctd by a rpulsion potntial barrir of othr lctrons. Th frquncy of such lctron lctron collisions can b stimatd as a frquncy of harmonic oscillations in th fild of othr lctrons fixd in a D triangular lattic /. /mn s []. This frquncy is alays much highr than th lctron nrgy rlaxation rat ~ r hich is about s at T ~K. Thrfor, in th nonlinar transport xprimnt SEs can b charactrizd by an ffctiv lctron tmpratur T hich can b substantially highr than th ambint tmpratur T. Rgarding momntum rlaxation hich is much highr than nrgy rlaxation, to opposit xtrm rgims can b ralizd for SEs on liquid hlium: rand r. In th first rgim, lctron momntum rlaxation is govrnd by lctron ripplon scattring, and th momntum distribution function is found as a solution of th kintic quation for nonintracting SEs. Th ffctiv lctron tmpratur is just a paramtr found from th nrgy balanc quation. In th scond rgim, momntum rlaxation is govrnd (controlld by lctron lctron collisions, and th momntum distribution function is just th quilibrium distribution function f [( k ku /T] shiftd aording to th drift vlocity u (hr k k / m. Th drift vlocity is found from th total momntum balanc quation. This rgim is somtims calld as th complt control rgim. Bcaus of xtrmly lo nrgy rlaxation, th main nonlinar ffct coms from lctron hating (T T. Th othr nonlinar paramtr ku / T is assumd to b small. In this cas, can xpand th distribution function in ku / T only up to th linar trm, kping in mind that th ffctiv collision frquncy (momntum rlaxation rat no dpnds on th lctron tmpratur T( E. Th abov dscribd to approachs lad to diffrnt mobility quations. Th mobility found in th singl-lctron tratmnt is givn by hr r ( ( x ( T T xxp( x m ( dx, ( ( x J( x ; x r T ; m. Th dimnsionlss function J( xis dfind by lctron ripplon coupling T hr / T x J( x x sin sin d / / x ( y y T x sin sin ( y / d y ln y T, (, ( ( (/( m and introduc th notation /( m hr me /(. Th function ( y dscribs th contribution from polarization intraction btn SEs and intrfac displacmnts [5]. If on applis th approximation ( y / ( y numrically justifid for y ~, th xprssion for J( xsimplifis considrably and ( r ( x is no rads as Fizika Nizkikh Tmpratur, 9, v. 5, No. 97

5 K.A. Nasydkin, V.E. Sivokon, Yu.P. Monarkha, and S.S. Sokolov ( r / T EpE T x E p T ( x T E x ; mt Ep. (5 At T 5. K th typical valu of E p is about V/cm. For th complt control rgim ( r th many-lctron xprssion for SE mobility can b found as ( m, ( ( m m r ( m hr th ffctiv collision frquncy r is dfind by m T ( r / T x x T T / / / x xp( xdx T x xp( xdx. 9 T For ( y / ( y, th Eq. (7 is rducd to ( m r E p T T / E E p T T E. ( Considr asymptotic limits of SE mobility as a function of E givn by Eqs. ( (. In th limit E E p, hr lctron ripplon scattring is du to polarization intraction btn SEs and intrfac displacmnts, on obtains ( ( m 9 m (7. (9 T Th asymptot (9 dscribing SE mobility ithin th ground lvl l dos not dpnd on T and hnc on th driving lctric fild vn for hot lctrons. This proprty as firstly found using on-lctron approach []. For th quilibrium cas (T T, th rlationship btn ( and ( m givn by Eq. (9 as obtaind and obsrvd xprimntally for E E p and diffrnt lctron dnsitis [7,]. Of cours, hating of SEs by th driving lctric fild can mak T comparabl ith th nrgy diffrnc btn th ground and xcitd surfac lvls. This lads to vaporation of SEs from th ground surfac lvl and to a sharp incras of thir mobility [9,] vn at lo driving filds. Undr such conditions, instad of Eq. (, hav a mor complicatd dpndnc of lctron mobility on T, hich as found qualitativly using crtain approximations and simplifications. In th opposit limit E E p, th contribution from th holding fild trm of th lctron ripplon coupling prdominats, and SE mobility is givn by ( ( m T. ( met In this cas, SE mobility ithin th ground surfac lvl is a linar function of T in agrmnt ith Rf., and ( m found for ris half as much th singl-lctron mobility (. Th xprimntal confirmation of mobility xprssions of Eqs. (9 and ( undr quilibrium conditions (T T for diffrnt holding filds had givn th vidnc of ralization of th complt control rgim in th systm of SEs on liquid hlium [7,]. Giving corrct thortical dscription of SE mobility in th xtrm rgims ( rand r th rsults of Eqs. ( ( ar not suitabl in th intrmdiat cas hn r. On can xpct a smooth transition btn ths to asymptots hn varying th paramtr / rby changing lctron dnsity or th holding lctric fild. A thortical dscription of SE mobility applicabl for arbitrary rlation btn and r as found in Rf. : ( m r ( r ( m, ( ( hr ( r /m. This xprssion rproducs Eqs. m ( and ( as th opposit limiting cass ( r, ( m r and ( r, ( r, rspctivly. Fitting xprimntal data by Eq. ( on can stimat for diffrnt holding and driving lctric filds... Enrgy rlaxation In our xprimnt, th SE mobility ( E as masurd as a function of th driving lctric fild. In ordr to compar data ith thortical valuations of ( T, us th nrgy balanc quation hich stablishs th rlationship btn E and T : ( T E Q, ( hr Q is th nrgy transfrrd by an lctron to ripplons pr unit tim. For lctron scattring hich involvs to ripplons ith q q q, q, it is givn by [] m Q W ( q q ( N q q q xp xp T T q dq, ( 97 Fizika Nizkikh Tmpratur, 9, v. 5, No.

6 Nonlinar surfac lctron transport ovr liquid hlium hr W( q is th lctron to-ripplon coupling function hich plays a crucial rol in finding Q. Diffrnt approachs to finding W( qar discussd in litratur. Aording to th prviously givn analysis [5], for lctron scattring ithin th ground surfac lvl can rprsnt W( q as a sum of long-avlngth and shortavlngth parts: W( q W( ( q ( q q W( ( q ( qq. ( Hr ( x is th unit stp-function, q 7. cm,and W( ( q W m ; q q W( ( q m. Th function ( x has a rathr cumbrsom form: xp( ( K ( x d, x hr K n ( x is th modifid Bssl function of th scond kind. Placing Eq. ( into Eq. (, arriv to th folloing xprssion for th nrgy transfrrd to ripplons pr unit tim: ith y y y ~ T Q T ( T (5 T ( x ( 5 q E y y T q E y xy / ( y / ( xy dy y xy ( xy dy ; ( / ( q T, q E 5 ( / ~ mw ( T T. / 5 / / / Th function ( T /T is plottd in Fig. for T 5. K (solid lin. Th function ; T/T Fig.. Th functions ( T / T (, F ( T / T(, and F ( T / T ( fort 5. K. y xy ( F( x y dy ( 5 xy/ ( prviously obtaind in th Rf. is also plottd in this figur as a dashd lin. It is intrsting to not that th solid lin rprsnting ( T / T is numrically clos to th intuitivly introducd function F( T / T T / T (dottd lin hich as idly usd in prvious thortical tratmnts of hot SEs. Still, at x ~th diffrnc btn functions ( x and F( x is quit noticabl. In prsnt ork us Eq. (5 ith ( x of Eq. (. Strictly spaking, for high nough E, th on-ripplon procss contribution into nrgy rlaxation frquncy should b considrd in addition to that of to-ripplon procsss. Th stimats [] sho hovr that on can disrgard th on-ripplon contribution to nrgy rlaxation for holding filds E 5 V/cm of th rang applid in th actual xprimnt. Only for E clos to V/cm th on-ripplon procsss do contribut in nrgy rlaxation. For this rason rstrict ourslvs to nrgy rlaxation limitd by to-ripplon procsss.. Rsults and discussion Typical xprimntal rsults for SE mobility as a function of driving lctric fild ar plottd in Fig. for T 5. K and diffrnt holding filds. Th lctron dnsity is. cm. Similar data r obtaind for othr tmpraturs and lctron dnsitis. Th xprimntal data (points ar compard ith thortical curvs calculatd for singl-lctron approximation (solid lin and th abov givn many-lctron tratmnt (dottd lin assuming that SEs populat only th ground surfac lvl (l. Figur displays rsults obtaind for diffrnt holding filds. Strong discrpancy btn xprimntal data and thortical curvs obsrvd at lo holding filds Fizika Nizkikh Tmpratur, 9, v. 5, No. 97

7 K.A. Nasydkin, V.E. Sivokon, Yu.P. Monarkha, and S.S. Sokolov Fig.. Th SE mobility as a function of driving lctric fild for diffrnt holding filds. Points ar th xprimntal data. Th solid and dottd lins corrspond to thortical calculations in on-lctron and complt control approximations, rspctivly. is attributd to ffct of SE «vaporation» from th ground lvl to highr surfac lvls (l. Sharp incras in ith E, a local maximum and th folloing slo dcras ar in qualitativ agrmnt ith calculations prformd prviously for th all-lvl tratmnt undr conditions T. KandE V/cm []. Still, 97 Fizika Nizkikh Tmpratur, 9, v. 5, No.

8 Nonlinar surfac lctron transport ovr liquid hlium E =5mV/cm E =7mV/cm E =9mV/cm Fig. 5. Th SE mobility as a function of holding lctric fild for thr valus of driving fild. Points ar th xprimntal data, dash-and-dottd lins ar a rsult of th approximation. Th solid and dottd lins corrspond to thortical calculations in on-lctron and complt control approximations, rspctivly. -,s 7mV 5mV 9mV... Fig.. Th frquncy of intr-lctron collisions as a function of holding lctric fild for thr valus of driving fild. in thos calculations thr as also a substantial dcras in th ground lvl mobility of hot lctrons causd by mploymnt of a lo-tmpratur approximation for lctron ripplon coupling ( y. Similar sharp incras in th nonlinar mobility and complicatd nonohmic curvs inducd by SE vaporation r thortically obtaind in Rf. 9. Th discrpancy notd abov could also b attributd to th lack of xprimntal auracy undr high lctron mobility conditions lading to vry small valus of th phas shift hich ar of th sam ordr as th snsitivity of xprimntal procdur. Whn incrasing th holding fild th xprimntal data bcom closr to thortical curvs obtaind for th on lvl modl. For E 7 V/cm, data ar clos th thortical curv calculatd in th singl-lctron approach. Th most intrsting bhavior of mobility is obsrvd at highr E. Th xprimntal data ntr th intrval btn to thortical curvs and coincid ith th complt control curv at highst holding filds. A similar tndncy as obsrvd in Rf. 7 hr SE mobility as masurd in th linar transport rgim. To mak a situation mor clar, in Fig. 5 dpictd th dpndncis ( E masurd for fixd valus of E.Th graphics illustrat th transition btn to scattring rgims undr incrasing th holding potntial. On should mphasiz th transition is obsrvd for th rang of holding fild of V/cm for all thr driving filds. As mntiond abov, th gnral xprssion for SE mobility as obtaind in Rf. for arbitrary rlation btn th frquncy of intr-lctron collisions and rlaxation frquncis in on-lctron and complt control rgims. Adjusting our xprimntal data to Eq. ( can stimat th frquncy. Th rsults ar prsntd in Fig. for V/cm E V/cm. Not that th auracy of xprimnt is not satisfactory for holding fild clos or smallr than V/cm hr th xprimntal points ar rathr clos to th thortical curv of on-lctron approach. For E highr than V/cm, obsrvd th hich dpnds akly on holding fild (dcrass a littl for holding filds clos to V/cm and thn incrass in th rang of holding fild hr xprimntal auracy is satisfactory and dcrass undr incrasing E. Th valus of chang in th rang ~( s. To lucidat th problm of nrgy balanc in th SE systm, applid th approach hr xprimntal data for ( E r compard ith thortical curvs calculatd using to approachs for ( T (opn and solid circls rprsnt th singl-lctron and complt control approachs, rspctivly. Such a comparison givs a Fizika Nizkikh Tmpratur, 9, v. 5, No. 975

9 K.A. Nasydkin, V.E. Sivokon, Yu.P. Monarkha, and S.S. Sokolov 5 7 T,K 5 7 T,K T,K T,K T,K T,K Fig. 7. Th driving lctric fild as a function of ffctiv lctron tmpratur. Points ar th rsult of calculation using xprimntal data and on-lctron (opn circl and complt control (solid circl approachs. Th solid lins ar th thortical curvs T ( E. possibility to obtain th dpndnc E ( T (Fig. 7. Th sam figur contains th thortical curvs T( E of to mobility rgims obtaind from th nrgy balanc quation ( using diffrnt approximations for ( T /T prsntd in Fig.. On can s that th on-lvl thory and xprimnt disagr strongly at lo holding filds. Th sam bhavior is obsrvd in Fig.. At highr holding filds th agrmnt btn xprimntal points and thortical curvs bcoms bttr. Not that th bst agrmnt is attaind for th function ( T /T of Eq. (. In Fig., plottd xprimntal dpndncis E ( T obtaind undr assumption that T is calculatd using ithr ( ( T or ( m ( T. At lo holding filds hr th xprimntal auracy is not high obsrv th practical indpndnc of driving fild in th hol rang of ffctiv lctron tmpratur. In othr ords th ffctiv lctron tmpratur incrass sharply ith th driving lctric fild. It mans that th possibility to ovrhat th lctron sht is vry high. Whn incrasing th holding potntial ntr th rang of holding filds hr SEs mostly oupy th ground surfac lvl and th on-lvl thory givs th adquat dscription of SE mobility. Undr such a condition th dpndncis E ( T and consquntly T( E ar incrasing functions in agrmnt ith thortical stimation of Rf.. 97 Fizika Nizkikh Tmpratur, 9, v. 5, No.

10 Nonlinar surfac lctron transport ovr liquid hlium E, V/cm E, V/cm T,K T,K Fig.. Th sam as in Fig. 7 for nrgy balanc quation solvd ithr for ( T or m ( T. 5. Conclusion W prsntd th rsults of xprimntal and thortical studis of nonlinar mobility of SEs along th liquid vapor intrfac limitd by lctron ripplon scattring undr conditions hr th ffctiv lctron tmpratur is substantially highr than th ambint tmpratur. It is shon that for high holding lctric filds, mobility of SEs oupying th ground surfac lvl incrass ith th driving lctric fild in agrmnt ith th thortical modl of highly corrlatd lctron gas. For lo holding filds, lctron vaporation to highr surfac lvls inducd by hating bcoms important hich complicats xprimntal curvs and thortical analysis. Enrgy rlaxation of SE systm is shon to b in agrmnt ith th modl of SE scattring by pairs of short-avlngth ripplons qq, k, hich taks into aount th rduction in lctron to-ripplon coupling inducd by transition to th adiabatic scattring rgim [Eq. (]. Th xprimntal tchniqu and thortical formalism dvlopd can b usd for furthr studis of nonlinar transport of SEs on liquid hlium. Th ork is partially supportd by STCU through th Projct 7.. Yu.P. Monarkha and K. Kono, To-Dimnsional Coulomb Liquids and Solids, Springr, Brlin (.. V.E. Sivokon, V.V. Dotsnko, S.S. Sokolov, Yu.Z. Kovdrya, and V.N. Grigor v, Fiz. Nizk. Tmp., 75 (99 [Lo Tmp. Phys., 59 (99].. V. Syvokon, Y. Monarkha, K. Nasydkin, and S. Sokolov, J. Lo Tmp. Phys., 9 (7..C.L.Zipfl,T.R.Bron,andC.C.Grims,Phys. Rv. Ltt. 7, 7 ( V.B. Shikin and Yu.P. Monarkha, J. Lo Tmp. Phys., 9 (97.. Yu.P. Monarkha and S.S. Sokolov, Fiz. Nizk. Tmp., 5 (97 [Sov. J. Lo Tmp. Phys., 7 (97]. 7. V.A. Buntar, Yu.Z. Kovdrya, V.N. Grigor v, Yu.P. Monarkha, and S.S. Sokolov, Fiz. Nizk. Tmp., 79 (97 [Sov. J. Lo Tmp. Phys., 5 (97].. V.A. Buntar and S.S. Sokolov, Fiz. Nizk. Tmp., 5 (99 [Sov. J. Lo Tmp. Phys., 97 (99]. 9. Yu.P. Monarkha and S.S. Sokolov, Fiz. Nizk. Tmp. 5, (979 [Sov. J. Lo Tmp. Phys. 5, 5 (979].. M. Saitoh and T. Aoki, J. Phys. Soc. Jpn., 7 (97.. Yu.P. Monarkha, Fiz. Nizk. Tmp. 5, 99 (979 [Sov. J. Lo Tmp. Phys. 5, 7 (979].. I.N. Adamnko, A.V. Zhukov, and K.E. Nmchnko, Fiz. Nizk. Tmp., ( [Lo Tmp. Phys., 59 (].. Yu.P. Monarkha, Fiz. Nizk. Tmp., 9 (97 [Sov. J. Lo Tmp. Phys., 55 (97].. Yu.M. Vil k and Yu.P. Monarkha, Fiz. Nizk. Tmp. 5, 5 (99 [Sov. J. Lo Tmp. Phys. 5, (99]. 5. Yu.P. Monarkha, Fiz. Nizk. Tmp. 9, 5 (99 [Lo Tmp. Phys. 9, (99]. Fizika Nizkikh Tmpratur, 9, v. 5, No. 977