Microcalorimeter and bolometer model

Transcription

1 JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 8 5 APRIL 2003 Microcalorimtr and bolomtr modl M. Galazzi a) Physics Dpartmnt, Univrsity of Wisconsin, Madison, Wisconsin and NASA/Goddard Spac Flight Cntr, Grnblt, Maryland 2077 D. McCammon Physics Dpartmnt, Univrsity of Wisconsin, Madison, Wisconsin Rcivd 7 May 2002; accptd 4 January 2003 h standard nonquilibrium thory of nois in idal bolomtrs and microcalorimtrs fails to prdict th prformanc of ral dvics du to additional ffcts that bcom important at low tmpratur. In this papr w xtnd th thory to includ th most important of ths ffcts and find that th prformanc of microcalorimtrs oprating at 60 mk can b quantitativly prdictd. W giv a simpl mthod for doing th ncssary calculations, borrowing th block diagram formalism from lctronic control thory Amrican Institut of Physics. DOI: 0.063/ I. INRODUCION A complt nonquilibrium thory for th nois in simpl bolomtrs with idal rsistiv thrmomtrs was givn by Mathr in 982 Rf. and xtndd to microcalorimtr prformanc two yars latr. 2 Hr w us th trms bolomtr and calorimtr in th convntional sns, rspctivly indicating powr ctors and intgrating nrgy ctors. his thory shows that th prformanc of ths dvics improvs dramatically as th oprating tmpratur is rducd. Howvr, at tmpraturs blow 200 mk, it bcoms incrasingly difficult to construct a bolomtr that bhavs according to th idal assumptions. h rsistanc of th thrmomtr bcoms dpndnt on radout powr, and tmpratur and quilibration tims btwn diffrnt parts of th ctor bcom significant. hrmodynamic fluctuations btwn intrnal parts ar thn an additional nois sourc. h physical dscription for most of ths ffcts is straightforward, but combining all of thm into a ctor modl can b algbraically daunting. hortical modls that dscrib complx thrmal architcturs ar ncssary to undrstand th bhavior of ral dvics, and som groups hav alrady xtndd th idal modl dvlopd by Mathr in 982 to includ som nonidal ffcts in ordr to xplain thir xprimntal rsults. 3 6 W dvlopd a gnral bolomtr and microcalorimtr modl using th block diagram formalism of control thory. h formalism hlps with th mchanics of th problm, whil kping th physical modl rasonably transparnt. 7 9 In th modl w hav includd th thrmal dcoupling btwn th lctron systm and th phonon systm in th thrmomtr, th so-calld hot-lctron modl, th thrmal dcoupling btwn th absorbr and th thrmomtr, and nonohmic bhaviors of th thrmomtr in addition to th a Currnt addrss: Dpartmnt of Physics, Univrsity of Miami, P.O. Box , Coral Gabls, FL FAX: Elctronic mail: galazzi@physics.miami.du hot-lctron ffct. h hot-lctron modl assums that th rsistanc of th thrmomtr dpnds on th tmpratur of th lctrons, and thr is a thrmal rsistanc btwn th lctrons and th crystal lattic through which th bias powr must flow, incrasing th tmpratur of th lctrons abov th tmpratur of th lattic and, thrfor, changing th thrmomtr rsistanc. his ffct is wll known in mtals at low tmpraturs and has rcntly bn studid in smiconductors in th variabl-rang hopping rgim. 0 2 h nois analysis incorporats trms for thrmomtr Johnson and /f nois, amplifir nois, load rsistor Johnson nois, and thrmodynamic fluctuations btwn th lctron and phonon systms in th thrmomtr as wll as btwn th absorbr, th thrmomtr, and th hat sink. In th modl w also includd th ffct of thrmomtr nonohmic bhavior, i.., dpndnc of th thrmomtr rsistanc on th bias signal. his ffct is particularly important whn transition dg snsors ES s ar usd 3 and maks th modl valuabl for prdicting th prformanc of this typ of ctor. II. HE IDEAL MODEL o hlp th radr undrstand th algbra of our modl w dcidd to start our analysis with an ovrviw of th idal modl that has bn prviously dvlopd. Dspit thir diffrnt applications, bolomtrs and microcalorimtrs ar vry similar ctors and th thory of thir opration is largly th sam. h considrations of this papr apply to both kinds of ctors unlss othrwis spcifid and w will us th gnric trm ctors to rfr to both. ypically a bolomtr or a microcalorimtr is composd of thr parts: an absorbr that convrts th incidnt powr or nrgy into a tmpratur variation, a snsor that rads out th tmpratur variation, and a thrmal link btwn th ctor and a hat sink. h snsor is typically a rsistor whos rsistanc strongly dpnds on th tmpratur around th working point. An idal ctor can b rprsntd by a discrt absorbr of hat capacity C in contact with th hat /2003/93(8)/4856/4/$ Amrican Institut of Physics Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

2 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon 4857 FIG.. hrmal sktch of a bolomtr or microcalorimtr. sink through a thrmal conductivity G s Fig. and a thrmomtr always at th tmpratur of th absorbr. h thrmomtr snsitivity is spcifid by dr R d, whr is th ctor tmpratur and R is th snsor rsistanc. h thrmal conductivity G is dfind as G dp d, 2 whr P is th powr dissipatd into th ctor. h conductivity G can gnrally b xprssd as a powr law of th ctor tmpratur, i.., GG 0. Notic that numrically G 0 is qual to th thrmal conductivity at K, but dimnsionally G 0 is a thrmal conductivity dividd by a tmpratur to th. In quilibrium, with no othr input powr than th Joul powr P usd to rad out th thrmomtr rsistanc, th quilibrium tmpratur of th ctor is rmind by intgrating Eq. 2 btwn th hat sink tmpratur S and th ctor tmpratur: GdP. 3 S Assuming th powr law xprssion for G introducd bfor and intgrating it bcoms S P. 4 G 0 It is important to rmmbr, whn calculating th quilibrium tmpratur, that th powr P dpnds on th valu of th snsor rsistanc and, as a consqunc, it dpnds on th tmpratur, as xplicitly indicatd in Eq. 4. o calculat th quilibrium tmpratur it is thrfor ncssary to solv th systm of quations rprsntd by Eq. 4, th P vs R curv and th R vs curv. In gnral, th systm must b solvd numrically. Of intrst from th point of viw of th ctor opration is how th tmpratur ris abov th quilibrium tmpratur dpnds on an xtrnal incidnt powr W. h powr input to th ctor (WP) is partly stord into th hat capacity of th ctor and partly flows to th hat sink through th thrmal conductivity. h quation that rmins th gnric tmpratur D of th ctor is thrfor C d D DGdWPD, S whr w xplicitly indicatd that th bias powr can b a function of th tmpratur D and whr th quantitis D, W, and P can b a function of tim t. W can xprss th gnric ctor tmpratur D as a function of th quilibrium tmpratur dfind in Eq. 4 as D. Equation 5 thn bcoms C d Gd S Gd WP. If w stay in th so-calld small-signal limit i.., w assum that is small compard to w can xpand th scond intgral to lowst ordr in /, obtaining C d FIG. 2. ypical ctor radout circuit. GdG S WPP, 7 with PP()P(). Subtracting Eq. 3 from Eq. 7 and considring that th quilibrium tmpratur dos not chang with tim, w obtain C d GWP, 8 whr for simplicity w xprssd GG(). In gnral, th bias powr will chang with tmpratur, sinc R changs, and its xprssion dpnds on th bias sourc impdanc. A typical bias circuit is illustratd in Fig. 2 whr R is th thrmomtr rsistanc and R L is a load rsistor. h most commonly usd bias conditions ar nar currnt bias (R L R) and nar voltag bias (R L R). Mor complx bias circuits ar also usd and can always b rprsntd by th circuit of Fig. 2 using hvnin quivalnc thorms. Diffrntiating th xprssion for th Joul powr PI 2 RV 2 /R and using th bias circuit of Fig. 2 w obtain P P RR L R L R. 9 his trm is gnrally rfrrd to as th lctrothrmal fdback trm and it oftn plays an important rol in th 5 6 Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

3 4858 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon rspons of a ctor. For simplicity in th small-signal analytical calculations, w writ th lctrothrmal fdback trm as PG EF, 0 whr G EF P RR L R L R, so that Eq. 8 bcoms C d GG EF W 2 or, introducing an quivalnt thrmal conductivity G ff G G EF which w rfr to as ffctiv thrmal conductivity, C d G ff W. 3 h asist way to solv this diffrntial quation is using Fourir transforms. h procdur is to us Fourir transforms to convrt th trms of Eq. 3 to th frquncy domain, solv th quation in th frquncy domain whr it bcoms a linar quation, and thn Fourir invrt transform th rsult to th tim domain. h advantag of solving Eq. 3 in th frquncy domain coms from th fact that th xprssion d(t)/ in th frquncy domain bcoms j(), whr w usd th nginring notation j. Equation 3 in th frquncy domain thn bcoms jcg ff W, whos solution is 4 W, 5 G ff j ff with ff C/G ff. h ctor systm bhavs as a low-pass systm, with tim constant ff. For ngativ lctrothrmal fdback G EF must b positiv and th ctor tim constant is shortnd. For positiv fdback G EF is ngativ and th ctor tim constant is lngthnd and, in th cas of G EF biggr than G, th ctor bcoms unstabl. h sign of G EF dpnds on th sign of and on th bias condition usd i.., th ratio R/R L ). In th small-signal linar limit considrd hr and in absnc of amplifir nois, th signal has no ffct on th ctor prformanc. Howvr, positiv fdback rducs th ffct of amplifir nois, whil ngativ fdback hlps linariz th larg-signal gain and improvs microcalorimtr rsolution for larg signals at high count rat. Sinc it can usually b arrangd that amplifir nois is ngligibl, ths practical considrations normally favor ngativ fdback. Currnt bias (RR L ) for ctors with ngativ and voltag bias (RR L ) for ctors with positiv ar thn usd. In oprating a ctor, what is rally ctd is not dirctly th tmpratur variation, but th rsistanc variation R, which is rad out ithr as a voltag or currnt variation, that is VV R L R L R, 6 II R R L R. 7 W can gnrically indicat th output signal as X and th rlation btwn th output and th tmpratur as X X A tr, 8 whr A tr is a dimnsionlss paramtr that quantifis how much th output signal is snsitiv to rsistanc changs and that w call th transducr snsitivity. Numrically A tr is dfind as A tr R dx X dr. 9 Notic that th xprssion of A tr can b asily drivd from Eqs. 8 and 6 or 7 for voltag and currnt radout, and is always smallr or qual to unity for passiv bias circuit (R L 0). h rspons of a ctor is usually quantifid by th rsponsivity S(), dfind as S X W ; 20 that is, th rsponsivity charactrizs th rspons of th ctor, X, to an input powr W. In th idal modl just dscribd w can combin Eqs. 5 and 8 to obtain X XA tr W, 2 G ff j ff and th rsponsivity is thn qual to S XA tr. 22 G ff j ff A ctor at th working point is also oftn dscribd by th complx dynamic impdanc Z()dV()/dI(). h dynamic impdanc Z() diffrs from th ctor rsistanc RV/I du to ffct of th lctrothrmal fdback. Whn th currnt changs, th powr dissipatd into th ctor changs too; thrfor, th tmpratur and th ctor rsistanc chang. It is oftn usful to xprss th ctor prformanc and charactristics in trms of th dynamic impdanc sinc it can b asily masurd xprimntally. h calculation of th analytical xprssion of th dynamic impdanc is simpl. Diffrntiating Ohm s law, V IR, w obtain dvi drr di. 23 Using Eq. 8 in th frquncy domain with W0 and th dfinition of th thrmomtr snsitivity in Eq., w obtain: dr R d R G j dp, 24 Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

4 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon 4859 FIG. 3. Block diagram rprsntation of a systm with transfr function H(). with C/G. his is calld th intrinsic or thrmal tim constant of th ctor. Diffrntiating th xprssion of th Joul powr dissipatd into th thrmomtr PVI, w obtain dpv dii dv, 25 which, combind with Eqs. 23 and 24, givs dvr dii R IdVV di. G j 26 Notic that in Eqs most of th trms ar function of th frquncy. Solving Eq. 26 w obtain Z dv di R P j G P G j whr w usd th xprssion Z 0 R j 2Z 0 Z 0 j Z 0R 2R, 27 P G Z 0 Z0R P. 28 G Notic that whn, Z R. h dynamic impdanc Z()dV/dI is asily masurd xprimntally. It can b rmind most radily by adding a small ac signal to th bias voltag and masuring th transfr function of th ctor F(). his is th ratio of amplituds and rlativ phas btwn changs in th ctor voltag and changs in th bias voltag as a function of frquncy. Most spctrum analyzrs hav th capability of masuring th complx ratio btwn two signals as a function of frquncy and can do this simultanously ovr th frquncy rang of intrst using a band-limitd whit nois sourc. Signal avraging allows vry prcis masurmnt to b mad whil rmaining in th small-signal limit. h dynamic impdanc is asily drivd from th transfr function using th valu of th load rsistanc and making appropriat corrctions for stray lctrical capacitanc or inductanc in th circuit. For xampl, in th bias circuit of Fig. 2, without stray capacitanc, th impdanc is qual to Z() R L F()/F(). It is thn possibl to rmin valus for many of th important paramtrs of th ctor by fitting th ral and imaginary parts of th transfr function by adjusting th thrmal and lctrical paramtrs in th xprssions givn in this papr. his is vry valuabl for diagnosing prformanc problms or improving th dsign of ctors. Not that whn th thrmomtr tmpratur cofficint is positiv, FIG. 4. Som gnral oprations with th block diagram algbra. th impdanc can bcom infinit. It is thn mor convnint to work with th invrs quantity /Z()dI/dV. In th cas of a ctor whos signal is rad out as a voltag chang, whr th rsponsivity S is dfind as S()dV()/dW(), w can also writ S Z 0 /R 2I Z 0 /R L j ff. 29 At this point w want to introduc a usful tchniqu for analyzing th rspons of a bolomtr or a microcalorimtr: block diagram algbra. his tchniqu is gnrally usd in lctrical nginring to analyz fdback systms and it is vry usful whn xtnding th thory of bolomtrs and microcalorimtrs to mor complicatd ralistic systms. h algbra of block diagrams and th languag of control thory hav bn succssfully usd bfor in th analysis of microcalorimtrs and bolomtr. 7 9 h basic ida is that a systm with transfr function in th frquncy domain qual to H() is rprsntd by th diagram of Fig. 3. If an input In is applid to th systm, th output is Out() H() In(). Complicatd systms can always b rducd to th systm of Fig. 3 using th block diagram algbra. Figur 4 shows som of th common oprations that will b usd in this papr. h procdur to solv th rspons of a systm using th block diagram algbra is thn th following: i Writ th diffrntial quations that dfin th systm rspons. ii Convrt th quations to th frquncy domain and, for ach quation dfin th individual systm rspons and th input to that systm. iii Lay out th block diagram that dscribs all th quations togthr. iv Us th block diagram algbra to rduc th block diagram to th form of Fig. 3 that rprsnts th systm rspons in th frquncy domain. his rprsntation is particularly usful to dal with fdback systms, i.., systms whr th output is combind to th input through a transfr function G() as in Fig. 5. In Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

5 4860 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon FIG. 5. Block diagram rprsntation of a fdback systm. this cas, whnvr an xtrnal input In is applid to th systm th output is H Out HG InH c-lin, 30 whr H c-l () is calld th closd-loop transfr function. Going back to th thory of bolomtrs and microcalorimtrs, w can writ Eq. 2 as C d G WG EF, 3 which, in th frquncy domain, bcoms jc G WG EF. 32 W now want to gnrat th block diagram dscribing this quation. h lft part rprsnts th rspons of th systm that w ar analyzing th output as a function of an input powr, whil th right part rprsnts th input to that systm. h fact that th input dpnds on th output is a consqunc of fdback. h lft part of th quation rprsnts a low pass systm with transfr function H G j, 33 h input consists of an xtrnal input W minus th output itslf modifid by th transfr function G EF. his is a typical fdback systm rprsntd by th block diagram of Fig. 6, whr w also includd th convrsion of to X. If w now solv th block diagram using th block diagram algbra and Eq. 30, w obtain X GG EF j C GG EF XA tr W, G ff j ff which is th sam xprssion of Eq. 2. III. HO-ELECRON MODEL XA tr W 34 FIG. 6. Block diagram rprsntation of a ctor. FIG. 7. hrmal sktch of a bolomtr or microcalorimtr in th hotlctron modl. A first-ordr corrction to th standard thory of bolomtrs and microcalorimtrs is th introduction of th hotlctron modl. h modl assums that th thrmal coupling btwn lctrons and lattic in th snsor at low tmpratur is wakr than th coupling btwn lctrons, so that th lctric powr applid to th lctrons riss thm to a highr tmpratur than th lattic. his bhavior is a known proprty of mtals and has rcntly bn quantifid in dopd silicon, 2 so that it affcts both ES s and smiconductor snsors. h ctor can thrfor b dscribd as composd of two diffrnt systms th lctron systm and th phonon or lattic systm and th two ar thrmally connctd by a thrmal conductivity G -l. W assum for modls drivd in this papr that th ctor rsistanc rsponds to th tmpratur of its lctron systm and that th Joul powr of th bias is dissipatd thr. For conomy of prsntation, th modls drivd hr assum th input powr ntrs through th absorbr phonon systm, which is thn thrmally connctd to th thrmomtr lattic and furthr to th hat sink through th thrmal conductivity G s Fig. 7. hr ar important classs of ctors whr signal powr is absorbd dirctly in th lctron systm of th thrmomtr or absorbr, and th primary thrmal path to th hat sink could b from th absorbr lattic or ithr lctron systm. In th gnral cas, ths all rsult in diffrnt thrmal circuits, and th block diagrams must b modifid accordingly. In th approximation of this sction, th phonon systm includs both th absorbr and th phonons in th thrmomtr w will discuss latr th cas of a dcoupld absorbr. In quilibrium with no xtrnal powr, th lctron systm is at a highr tmpratur than th phonon systm is and th Joul powr flows from th lctron systm to th phonon systm and from thr to th hat sink. h quilibrium tmpratur of th two systms without any signal powr applid can b calculatd in a way similar to that usd for th simpl modl dscribd in th prvious paragraph. As rportd in th litratur, th thrmal conductivity btwn lctrons and phonons can b dscribd as a powr law of th lctron tmpratur Rfs. 0 and : G -l G From th dfinition of thrmal conductivity, w also hav Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

6 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon 486 G -l dp, 36 d and if w combin th two and intgrat from th lattic tmpratur l to th quilibrium lctron tmpratur, w obtain P G l, 37 0 whr w xplicitly indicatd th dpndnc of th powr P on th lctron tmpratur. h quilibrium tmpratur of th lattic systm is still rmind by Eq. 4 which, in this cas, can b writtn as l S P G Equations 37 and 38 rprsnt a systm with two variabls and l that can b solvd numrically. Hr w ar considring ctors whr th xtrnal powr is absorbd in th phonon systm, and th snsitivity of th ctor can b strongly affctd by th rducd snsitivity of to changs in l introducd by th quilibrium diffrnc of ths tmpraturs and nonlinar natur of G -l. W considr ths ffcts in two stps. A first approximation is to assum that th hat capacity of th lctron systm is ngligibl. his cas can b solvd asily, and it is sufficint in many cass. W will thn driv th gnral rsult for C 0. A. Hot-lctron modl with C Ä0 If th lctron systm hat capacity C can b nglctd, th dpndnc of th lctron tmpratur on th lattic tmpratur is simply rmind by Eq. 37. Whn th tmpratur of th lattic systm changs by l th tmpratur of th lctron systm will instantly chang by and Eq. 37 bcoms P G l l If w subtract Eq. 37 from Eq. 39 w obtain P G P 0 l l l. 40 Assuming that and l l w can xpand Eq. 40 to lowst ordr in / and l / l, obtaining G 0 which rducs to P l l P G 0 l l. l l, 4 42 W alrady calculatd th chang in Joul powr P in Eqs. 9 and 0, which in th hot-lctron cas dpnds on th chang in lctron tmpratur : PG EF, 43 and thrfor l G l EF G 0 G -l l G -l G l EF G -l l G -l G EF G -l l A -l l, 44 whr G -l ( ) is th lctron-lattic thrmal conductivity calculatd at th lctron tmpratur, G -l ( l ) is th lctron-lattic thrmal conductivity calculatd at th lattic tmpratur, and A -l G -l l l G -l G. 45 EF G -l h quantity A -l is adimnsional and rprsnts th tmpratur snsitivity of th thrmomtr. Whn A -l, th thrmomtr is compltly snsitiv to tmpratur changs in th lattic systm, whn A -l 0, th thrmomtr is compltly insnsitiv to tmpratur changs in th lattic. W now want to rprsnt th ctor using th block diagram algbra. h ctor bhavior is dscribd by Eqs. 8, 3, and 44, which in th frquncy domain can b writtn as jc l G l WG EF, 46 A -l l, and 47 X XA tr. 48 Convrting ths thr quations in block diagram algbra and conncting th blocks of th algbra togthr w obtain th rprsntation of Fig. 8a. With som simpl algbra, th diagram is quivalnt to that of Fig. 8b, and considring that G EF, th hot-lctron modl with ngligibl hat capacity of th lctron systm is thn quivalnt to th standard modl with th substitutions ff A -l, 49, 50 and thrfor th rsponsivity of th ctor bcoms A -l XA tr S 5 GA -l G EF j ff with ff C l /(GA -l G EF ), whr C l is th lattic hat capacity. Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

7 4862 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon FIG. 9. Rprsntation of th lctron systm in th hot-lctron modl with C 0. FIG. 8. Block diagram rprsntation of a ctor using th hot-lctron modl with C 0. a Block diagram as drivd from th quations that dscrib th ctor. Notic that th rprsntation of th EF as acting on th lattic systm of th snsor is du to th fact that w ar assuming that C 0. In gnral, if C 0, th EF is an lctric ffct and acts on th lctron systm. b Equivalnt rprsntation to highlight th ffct of th trm A -l. B. Hot-lctron modl with C Å0 If th hat capacity of th lctron systm is not ngligibl, th lctron tmpratur is dfind in analogy to Eq. 5 by d C G-l dp, l 52 with G -l dfind by Eq. 35. What w ar intrstd in is th ris of th lctron tmpratur abov quilibrium whn th lattic tmpratur riss by l. Equation 52 thn bcoms C d G-l dp. l l 53 Subtracting Eq. 52 from Eq. 53 w obtain d C G-l d l l lg-l dp, 54 which, using Eq. 35 and xpanding th rsult to lowst ordr in / and l / l, bcoms d C G -l G -l l l G EF, 55 with G EF dfind by Eq. 43. h systm of Eq. 55 is rprsntd by th block diagram of Fig. 9 with -l C /G -l ( ), which has th solution A -l j l, 56 with C /G -l ( )G EF and A -l dfind by Eq. 45. Notic that Eq. 56 rducs to Eq. 44 if C 0. h bhavior of th lattic systm is still rgulatd by Eqs. 5, 7, and 8, with th substitutions of C l for C and of l for, i.., d l C l G l WP l. 57 h powr P l is th powr flowing from th lctron systm to th lattic systm through th thrmal conductivity G -l : G-l d. 58 P l l hrfor, P l G-l d l lg-l d, l 59 which, considring th xprssion of Eq. 35 for th thrmal conductivity and xpanding th rsult to lowst ordr in / and l / l, bcoms P l G -l G -l l l. 60 Equation 57 thn bcoms d l C l G l WG -l G -l l l. 6 Equations 55 and 6 can b writtn in th frquncy domain as jc G -l G -l l l G EF 62 and jc l l G l WG -l G -l l l, 63 and ar rprsntd by th block diagram of Fig. 0a. h diagram can b solvd to obtain an analytical xprssion for th ctor rsponsivity. In Figs. 0b and 0c w show two intrmdiat stps in th solution of th block diagram algbra. h ctor rsponsivity is thn qual to S C G EF A -l j G EF G j l j A -lxa tr. 64 with l C l /G. Notic that in th cas of C 0 this xprssion rducs to Eq. 5, i.., th hot-lctron modl with ngligibl lctron hat capacity, as xpctd. Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

8 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon 4863 FIG. 0. Block diagram rprsntation of a ctor using th hot lctron modl with C 0. a Block diagram as drivd from th quations dscribing th systm. b,c Intrmdiat stps for th solution of th block diagram rprsntation. Morovr, in th cas of G 0, i.., whr lctrons and phonons can b thrmally considrd as a singl systm, th rsponsivity bcoms S GG EF j C lc GG EF XAtr. 65 his is just th idal rsponsivity of a bolomtr or microcalorimtr with thrmal conductivity G, tmpratur, and hat capacity CC l C. In analogy to Eqs , w can also calculat th dynamic impdanc of th ctor. W can writ Eqs. 62 and 63 without xtrnal powr W and xplicitly using th symbol P for th chang in Joul powr. and jc G -l G -l l l P jc l l G l G -l G -l l l Combining Eqs., 23, 25, 66, and 67, w thn obtain GG -l l jc l ZR GG -l l jc l G -l jc P G -l jc P G -l G -l l. 68 G -l G -l l IV. NOISE SOURCES hr ar svral nois sourcs that affct th prformanc of bolomtrs and microcalorimtrs, most of which hav alrady bn takn into account by Mathr in 982. hs includ th Johnson nois of th snsor, th thrmal nois du to th thrmal link btwn th ctor and th hat sink also rfrrd to as phonon nois, th Johnson nois of th load rsistor usd in th bias circuit, and th nois of th radout lctronics amplifir nois. In his papr Mathr also mntions a /f nois contribution that sms to b mor rlatd to th snsor charactristics. his nois was studid and quantifid for silicon-implantd thrmistors by Han t al. in h ffct of th nois on th ctor prformanc is gnrally quantifid by th nois quivalnt powr NEP. h NEP corrsponds to th powr W() that would b ncssary as input of th ctor to gnrat an output X() qual to th output gnratd by th nois. h NEP is calculatd as th ratio btwn th output X() gnratd by th nois and th rsponsivity of th ctor S(). In th cas of bolomtrs, th NEP dirctly quantifis th limit of th bolomtr in cting a powr signal at frquncy. In th cas of microcalorimtrs th NEP is rlatd to th bst possibl nrgy rsolution of th microcalorimtr by th xprssion 2 E rms 0 2d NEP Hr w want to analyz th ffct of th nois on th ctor prformanc in th pictur of th hot-lctron modl. h introduction of th hot-lctron modl has two main ffcts: it changs th NEP of th nois sourcs and it Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

9 4864 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon FIG.. Block diagram rprsntation of th nois in a ctor using th hot-lctron modl a and quivalnt rprsntation for th idal modl b. Notic that if th output X is a currnt, th load rsistor nois that adds to th output is rprsntd by i RL. introducs a nw nois trm, which is th thrmal nois du to th thrmal fluctuations btwn th lattic and lctron systms. A. Effct of th hot-lctron modl on th nois h diffrnt nois contributions affct th ctor in diffrnt ways. In particular, th thrmal nois corrsponds to a powr nois on th lattic systm. h Johnson nois is calculatd as a voltag fluctuation but can b introducd in th modl as an lctron tmpratur nois trm. h /f nois is calculatd as a fluctuation in th valu of th rsistanc but can b dscribd as lctron tmpratur nois trm as wll. h load rsistor nois can b dscribd as a nois that adds to th output signal and also gnrats a Joul powr nois on th lctron systm. h amplifir nois adds dirctly to th output signal. In Fig. a th contributions of th diffrnt nois trms in th microcalorimtr ar shown. h sam nois sourcs in th idal modl scnario ar shown in Fig. b. 9 Dimnsionally, th thrmal nois P th is a powr spctral dnsity in units of W/Hz), th Johnson nois J and th load rsistor nois RL ar voltag spctral dnsitis (V/Hz), th /f nois (R/R) /f has dimnsions Hz /2, and th amplifir nois amp has th dimnsion of th transducr output X dividd by squar root of frquncy (V/Hz or A/Hz). h thrmal nois was calculatd quantitativly by Mathr in 982 assuming diffusiv thrmal conductivity and is 2 qual to l k 2 /2 S l k l d 2 P th 4k b G l, 70 lk k l d S whr k b is th Boltzmann constant and k() is th function dscribing th tmpratur dpndnc of th thrmal conductivity of th hat link matrial. h Johnson nois of th snsor rsistanc is simply dscribd by J 4k b R. 7 h load rsistor nois can b rprsntd by a voltag nois across th ctor, qual to RL 4k b S R L R R L R, 72 whr w assumd that th lctrical circuit is hat sunk at th tmpratur S. his nois adds dirctly to th output signal as a voltag RL or as a currnt i RL RL /R, and gnrats Joul powr nois in th lctron systm P RL 2I RL s Fig.. h /f nois is, by dfinition, frquncy dpndnt, and it is usually dscribd as a fluctuation in th valu of th rsistanc: R R /f. 73 Solving th block diagram of Fig. indpndntly for ach nois contribution and using th xprssion of S() of Eq. 64 w obtain Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

10 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon 4865 NEP th P th, 74 NEP J 4k b P 2 l G j l j -l NEP RL l jc, RL S 2I RL G -l l GG -l l jc l, NEP /f R R /f l l jc, NEP amp amp S. G j l j -l FIG. 2. hrmal sktch of a bolomtr or microcalorimtr in th cas of absorbr dcoupling and hot-lctron modl. G NEP h P h G -l l j l, 82 Notic that th NEP du to th radout lctronics and to th load rsistor ar th only trms that dpnd on th lctrothrmal fdback. hrfor, if ths trms ar small compard to th othr contributions, as is usually th cas, th lctro-thrmal fdback changs th tim constant of th ctor, but dos not affct NEP. h xprssion of th NEP in th cas of ngligibl lctron hat capacity is asily drivd using C 0. Notic that in th limit of G 0, Eqs. 74 and 75 rduc to th idal xprssions calculatd by Mathr: NEP th P th, 79 NEP J 4k b G P j C lc G. 80 B. hrmal nois du to hot-lctron dcoupling h hot-lctron modl also introducs an xtra nois trm in addition to thos just considrd. his is du to powr fluctuations btwn th lattic and lctron systm. h magnitud of ths fluctuations dpnds in part on th physics of th lctron-phonon dcoupling. A simpl xprssion appropriat for radiativ nrgy transfr was calculatd by Boyl and Rodgr in 959 Rf. 5: P h 2k b G -l 5 5 l. 8 3 A mor rigorous xprssion for lctron-phonon dcoupling was also calculatd by Golwala t al. in 997 Rf. 6. Notic that ths fluctuations transport powr from th lattic systm to th lctron systm and vic vrsa; thrfor, if a powr P h adds to th lctron systm, th sam powr P h is subtractd from th lattic systm. h ffct is shown in th block diagram of Fig.. Solving th block diagram for th hot lctron nois w obtain whr l has bn prviously dfind as l C l /G. his xprssion dos not dpnd on C and thrfor is valid also for th cas C 0. Morovr, if G 0, this trm is zro, as xpctd. V. ABSORBER DECOUPLING Anothr aspct that may affct th prformanc of bolomtrs and microcalorimtrs that w want to study is th ffct of th absorbr thrmal conductivity. Most of th ctors ar built with absorbr and snsor as diffrnt ntitis connctd by poxy or othr matrial with a thrmal conductivity G a. Dpnding on th xprimntal stup, thr ar diffrnt configurations that must b usd to dscrib th thrmal systm. For xampl, th thrmal link to th hat sink can b through th absorbr or th thrmomtr and th absorbr can b in thrmal connction with th lattic systm whn an lctrical insulating matrial is usd or th lctron systm whn a conducting matrial is usd. What w want to analyz hr is th cas in which th ctor is connctd to th hat sink through th thrmomtr lattic systm and th absorbr is connctd to th lattic systm of th thrmomtr. In this cas, th xtrnal powr hits th absorbr and is rlasd to th lattic systm and thn ctd in th lctron systm s Fig. 2. W assum that th absorbr has a hat capacity C a. Notic that th analytical tools that w giv hr can b asily usd to quantify th bhavior of any othr configuration. A. Rsponsivity and dynamic impdanc In quilibrium, with no othr powr input than th Joul powr in th snsor, thr is no powr flow through th thrmal link G a and thrfor th tmpratur of th absorbr is qual to th lattic tmpratur a l. If an xtrnal powr W is applid to th absorbr, th ctor is dscribd in th frquncy domain by th st of quations Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

11 4866 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon FIG. 3. Block diagram rprsntation of a ctor with a finit thrmal conductivity btwn absorbr and lattic systm. W hav usd th notation G l GG a G -l ( l ). Notic that this implicitly intgrats th hat rlif for th lattic systm providd by th lctron and absorbr dcoupling into th lattic rspons function. his is diffrnt from what was don bfor in Fig. 0, whr th hat rlif was xplicitly rportd in th block diagram as a fdback ffct. h two dscriptions ar quivalnt. W usd th implicit dscription hr to compact th block diagram algbra. jc a a G a a WG a l, jc l l GG a G -l l l G a a G -l, jc G -l G -l l l G EF If w want to build th block diagram associatd with ths thr quations, w can considr th lft sid of th quations as th rspons function of th thr systms absorbr, lattic, lctrons and th right sid as th input to ach systm. Conncting th thr systms givs th block diagram of Fig. 3. h diagram can b solvd to obtain th ctor rsponsivity S jc a j j j a GG -l l jc l G -l A -l j a A -l XA tr, 86 with a C a /G a. Notic that if G a, this xprssion rducs to th on without absorbr dcoupling for a ctor with lattic hat capacity C l C l C a. Using Eqs., 23, 25, and 83 85, w can also calculat th ctor dynamic impdanc GG -l l jc l j a jc a G -l jc P G -l G -l l j a ZR GG -l l jc l j a jc a G -l jc P. 87 G -l G -l l j a B. Nois contribution As in th hot-lctron modl of th thrmomtr, thr ar two ffcts introducd by th thrmal link btwn th absorbr and lattic systm. h first ffct is that th rspons of th ctor is diffrnt; thrfor, th NEP s du to thrmal, Johnson, /f, load rsistor, amplifir, and hotlctron nois ar diffrnt. h scond ffct is th introduction of an xtra nois trm du to th powr fluctuations btwn absorbr and lattic. Figur 4 shows th block diagram of th ctor with th nois sourcs vidnt. As in th hot lctron modl, th nois du to th link btwn absorbr and lattic can b dscribd as a powr flow out of th absorbr and into th lattic or vic vrsa. his powr has sam valu P a, but opposit sign at th two nds of th link. Sinc th tmpratur of absorbr and lattic systms ar qual, th valu of P a is simply FIG. 4. Block diagram rprsntation of nois in a ctor with a finit thrmal conductivity btwn absorbr and lattic systm. Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

12 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon 4867 P a 4k b G a 2 l. 88 Solving th block diagram of Fig. 4 indpndntly for ach nois sourc w obtain NEP th P th j a, 89 NEP J 4k b P 2 j -l jc a G l l j a j l j a jc, 90 NEP /f R R j -l jc a G /f l NEP RL l j a j l j a jc, RL 2I S RL G -l l jc a j a G G -l l jc l, NEP amp amp /S, P h NEP h G -l l jc ag j a j l, 94 NEP a P a j a. 95 Notic again that if G a, ths xprssions ar qual to th hot-lctron xprssions for a ctor with lattic hat capacity C l C a and th absorbr NEP is qual to zro. VI. NONOHMIC BEHAVIOR OF HE HERMOMEER Anothr ffct that may chang th prformanc of a ctor is th nonohmic bhavior of th thrmomtr; i.., th thrmomtr rsistanc may not dpnd only on th thrmomtr lctron tmpratur, but also on th currnt or voltag that is usd to radout th tmpratur chang: R R(,I). 7 his ffct is particularly strong whn ES thrmomtrs ar usd. 3 h rsponsivity of a ctor with nonohmic thrmomtr has alrady bn calculatd by Mathr in 984 Rf. 7 and its ffct on ES microcalorimtrs was studid in ail by Lindman in 2000 Rf. 3. A nonohmic thrmomtr can also b asily includd in our modl. If th rsistanc of th thrmomtr dpnds on th radout signal, w can writ dr R I d R I IdI or, quivalntly, dr R V d R V VdV, whr I R V V R R I R V., I I R R I, V R R V, 98 Using Eq. 96 or 97 is quivalnt, and it is always possibl to go from on notation to th othr using Ohm s laws: I V, V I. 99 I I h only trms in our modl that ar affctd by th nonohmic bhavior ar th lctrothrmal fdback trm G EF and th transducr rsponsivity A tr. W can calculat thm assuming th bias circuit of Fig. 2: PI 2 R P2IR II 2 R, 00 I V bias R L R I I R L R R, 0 and VV bias IR L VR L I. Using Eq. 96 w obtain RR L R L R I I, P P V V R L I R L R I and I I R I R L R I 02 03, h modl dscribing a nonohmic thrmomtr is thrfor idntical to that dscribing a linar on, with th substitution and I, G EF P RR L R L R I I, R L A tr R L R I, 08 for voltag radout, or R A tr R L R I, 09 for currnt radout. With this substitution in th quations that w drivd prviously in th papr, it is possibl to prdict both rsponsivity and nois in th ctor. W can also us Eq. 96 to calculat th dynamic impdanc of th ctor. In th cas of absorbr and hotlctron dcoupling w obtain Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

13 4868 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon ZR I GG -l l jc l j a jc a G -l jc P I G I -l G -l l j a GG -l l jc l j a jc a G -l jc P I G -l G -l l j a his rducs to GG -l l jc l G -l jc P I G I ZR I -l G -l l GG -l l jc l G -l jc P, I G -l G -l l. 0 for hot-lctron dcoupling only, and to P I j G I ZR I P, 2 I j G for th idal modl. W do not know of a rigorous gnral mthod for driving th Johnson nois in a nonohmic rsistor. Nor dos thr sm to b a singl dfinit schm for rmining th nt rspons of th ctor to this fundamntal thrmal nois, sinc it is an intrnal nois gnratd in th nonohmic rsistor, and it is not clar how it should itslf affct th nonohmicity of th rsistor. W ar invstigating this furthr, but for th prsnt hav assumd that th Johnson nois can b rprsntd as a random voltag sourc with powr spctral dnsity 4k b R in sris with th nonohmic rsistanc and that th Johnson fluctuations in th sourc caus th rsistanc to fluctuat du to th currnt dpndnc of th rsistor. his rsults in th sam supprssion of th Johnson nois du to th currnt dpndnc of th rsistanc as occurs for xtrnal signals and nois if th nonohmic rsistanc is xprssd as R(,I). his uncrtainty or dpndnc on th ails of th physics applis only to th Johnson nois of th snsor. Small-signal rsponsivitis to all xtrnal sourcs of signal and nois ar unambiguous, so it is only th ctor Johnson nois contribution to th NEP that is uncrtain. VII. RESULS o vrify our rsults w simulatd th prformanc of an xisting microcalorimtr and compard th rsults with data from th ctor. W considrd a microcalorimtr usd in th dvlopmnt phas of th X-Ray Spctromtr XRS for th Astro-E satllit. 8 h ctor that w usd for th comparison was part of a 66 tst array of microcalorimtrs with silicon-implantd thrmistors and Hg absorbrs. W chos this ctor bcaus th array has bn studid in grat ail and th charactristics of th pixls ar wll known. W first usd Eqs. 37 and 38 to calculat th xpctd quilibrium tmpratur of th ctor. W thn usd Eqs to calculatd th xpctd nois spctra. h sum of ths can b compard with th masurd nois spctrum, as shown in Fig. 5. In th modl all th input paramtrs ar fixd to th valus masurd xprimntally. h only valu that was not availabl and that has bn adjustd during th calculation of th thortical nois is th stray capacitanc btwn gat and sourc of th fild ffct transistor FE lctronics. h valu of 5 pf obtaind for th stray capacitanc is in good agrmnt with typical valus for th FE amplifirs usd in th masurmnt. h agrmnt btwn th modl and th masurmnt is vry good. h data st has bn acquird at a hat sink tmpratur of 65 mk. h modl prdicts an quilibrium tmpratur of 77 mk and, through Eq. 69, an nrgy rsolution of 8.4 V, to b compard with th masurd valus of 78 mk and 8.65 V. h agrmnt is wll within th accuracy of th input paramtrs in th modl and dmonstrats th powr of th modl in prdicting ctor prformanc. FIG. 5. Comparison btwn th nois from a 66 XRS array pixl courtsy of Carolin K. Stahl and our modl. h modl includs th ffct of th dcoupling btwn hot lctrons and phonons in th snsor and btwn absorbr and snsor. h nois sourcs that ar includd ar Johnson nois of th snsor, thrmal nois du to th link btwn ctor and hat sink, thrmal nois du to th link btwn phonons and lctrons in th snsor, thrmal nois du to th link btwn absorbr and snsor, Johnson nois of th load rsistor, /f nois, and nois of th radout lctronics. Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s

14 J. Appl. Phys., Vol. 93, No. 8, 5 April 2003 M. Galazzi and D. McCammon 4869 Our analytical modl has also bn compard with a modl that uss matrix notation to numrically solv th linarizd diffrntial quations of th microcalorimtr. 9 h numrical modl was dvlopd indpndntly at th NASA/ Goddard Spac Flight Cntr to prdict th prformanc of mor complx ctors. 9 Using th sam paramtr valus, th agrmnt btwn th two is within th numrical rror in th implmntation of th modls. 20 VIII. CONCLUSIONS W hav dvlopd an analytical modl that prdicts th bhavior of microcalorimtrs and bolomtrs. h modl includs th ffct of hot lctrons in th ctor snsor, a thrmal dcoupling btwn absorbr and snsor, and th ffct of a nonohmic thrmomtr. h modl analytically prdicts th ctor rsponsivity and xpctd nois undr ths conditions. h nois sourcs that ar includd in th modl ar th Johnson nois of th snsor, th thrmal nois du to th link btwn ctor and hat sink, th thrmal nois du to th link btwn phonons and lctrons in th snsor, th thrmal nois du to th link btwn absorbr and snsor, th Johnson nois of th load rsistor, th /f nois as thrmal nois, and th nois of th radout lctronics. A comparison btwn th prdictions of our modl and data from a ctor dvlopd for th XRS instrumnt shows good agrmnt. W also dscribd a diffrnt way to analyz th prformanc of bolomtrs and microcalorimtrs, using block diagram algbra. h formalism that w introducd can b applid to th dscription of diffrnt ctor configurations. ACKNOWLEDGMENS W would lik to thank Carolin Stahl and Enctali Figuroa-Fliciano for a usful discussion in th dvlopmnt of th modl and for supplying th XRS data and th comparison with th numrical modl. W also would lik to thank th othr mmbrs of th microcalorimtr groups at th Univrsity of Wisconsin and th NASA/Goddard Spac Flight Cntr for th usful discussion and suggstions. his work was supportd by NASA Grant No. NAG J. C. Mathr, Appl. Opt. 2, S. H. Mosly, J. C. Mathr, and D. McCammon, J. Appl. Phys. 56, H. F. C. Hovrs t al., Appl. Phys. Ltt. 77, J. M. Gildmistr t al., Appl. Opt. 40, A. Flischmann t al., AIP Conf. Proc. 605, M. Piat t al., AIP Conf. Proc. 605, A.. L t al., IEEE rans. Appl. Suprcond. 7, M. J. M. E. d Nivll t al., J. Appl. Phys. 82, M. Galazzi, Rv. Sci. Instrum. 69, Ning Wang, F. C. Wllstood, B. Saudolt, E. E. Hallr, and J. Bman, Phys. Rv. B 4, J. Zhang t al., Phys. Rv. B 57, D. Liu t al., AIP Conf. Proc. 605, M. Lindman, Ph.D. thsis, Univrsity of California at Davis, S-I Han t al., Proc. SPIE 3445, W. S. Boyl and K. F. Rodgrs, Jr., J. Opt. Soc. Am. 49, S. R. Golwala, J. Jochum, and B. Saudolt, in Procdings of th Svnth Intrnational Workshop on Low mpratur Dtctors, Munich, Grmany, 27 July 2 August, ditd by S. Coopr h Max Plank Institut, Munich, Grmany, 997 p J. C. Mathr, Appl. Opt. 23, C. K. Stahl t al., Nucl. Instrum. Mthods Phys. Rs. A 436, E. Figuroa-Fliciano, Ph.D. thsis, Stanford Univrsity, M. Galazzi, E. Figuroa-Fliciano, D. Liu, D. McCammon, W.. Sandrs, C. K. Stahl, and P. an, AIP Conf. Proc. 605, Downloadd 25 Oct 2005 to Rdistribution subjct to AIP licns or copyright, s