A distributed device model for phonon-cooled HEB mixers predicting IV characteristics, gain, noise and IF bandwidth

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1 distributd dvi modl for honon-oold HEB mirs rditing V haratristis, gain, nois and F bandwidth. Khosroanah *, H. Mrkl *, S. Yngvsson #,. dam, S. hrdnihnko *, E. Kollbrg * * halmrs Univrsity of hnology, Dartmnt of Miroltronis, Mirowav ltronis aboratory, 4 96 Gothnburg, Swdn # Univrsity of Massahustts, Dartmnt of Eltrial and omutr Enginring, mhrst, M 3 Suél, Dartmnt of Eltron Dvi hysis, 99 Gif-sur-Yvtt, Fran bstrat distributd modl for honon-oold surondutor hot ltron bolomtr HEB mirs is givn, whih is basd on solving th on-dimnsional hat balan quation for th ltron tmratur rofil along th surondutor stri. n this modl it is assumd that th O owr is absorbd uniformly along th bridg but th D owr absortion dnds on th loal rsistivity and is thus not uniform. h ltron tmratur dndn of th rsistivity is assumd to b ontinuous and has a Frmi form. hs assumtions ar usd in stting u th non-linar hat balan quation, whih is solvd numrially for th ltron tmratur rofil along th bolomtr stri. Basd on this rofil th rsistan of th dvi and th V urvs ar alulatd. h V urvs ar in llnt agrmnt with masurmnt rsults. Using a small signal modl th onvrsion gain of th mir is obtaind. h rssions for Johnson nois

2 and thrmal flutuation nois ar drivd. h alulatd rsults ar in los agrmnt with masurmnts, rovidd that on of th aramtrs usd is adjustd.. ntrodution rviously rsntd HEB modls th "oint bolomtr " or "standard" modl [,] assum a uniform ltron and honon tmratur along th surondutor stri. lthough ths modls ar quit sussful to lain many rimntal rsults, som disranis hav bn rortd: t has bn shown that th modls ar not aabl of stimating th absorbd O owr orrtly whn orating at frqunis abov th quasiartil bandga [3]. lso aurat masurmnts and alulations hav shown suh th "oint bolomtr" modl annot lain th dndn of th masurd onvrsion loss and th ut nois tmratur on th bias voltag [4]. Nvrthlss in ordr to otimiz th dvi rforman for sa aliations, an aurat modl is ndd. On-dimnsional modls hav bn dvlod assuming that th ltron tmratur varis along th surondutor stri [5,6]. his assumtion lads to diffrnt hating ffiinis for th absorbd O and D owr imlying that th rsistan hang du to a small hang in absorbd O owr is not th sam as that for absorbd D owr. his is du to th fat that th D owr is only absorbd in th art of th stri whr th ltron tmratur is abov th ritial tmratur whras O owr absortion is uniform. n rvious on-dimnsional hot sot modls [5,7] th tmratur dndn of rsistivity is assumd to b a st funtion, whih gos from zro to normal rsistivity at th ritial tmratur. his and othr assumtions wr mad in ordr to mak it ossibl to obtain an analytial solution for th tmratur rofil. h main disadvantag of this modl is that whil th tmratur rofil dos not d th ritial tmratur no hot sot is formd, and th modl rdits zro rsistan. onsquntly it is not ossibl to alulat rasonabl V urvs at low bias voltags and low O owrs. n rati th rsistivity transition is smooth and masurmnts show that th transition width is ab. K [8]. n ordr to obtain omlt V urvs it is imortant to modl th transition in a mor ralisti way. h modl rsntd hr assums that th tmratur dndn of th rsistivity, although st lik, is ontinuous and has a Frmi form, and th transition width an b hosn as a variabl similar to a modl for Nb diffusion oold bolomtrs.f. [9].. his rsults in a non-linar hat balan quation, whih is solvd for th ltron tmratur rofil numrially. n stion th on-dimnsional hat balan quation and our larg signal modl ar disussd. Stion dsribs th small signal modl. h nois is disussd in stion V. Stion V lains a ossibl mthod for alulating th F imdan and th bandwidth of th bolomtr. Stion V summarizs th rsults.

3 . arg signal modl Figur shows th shmati hat flow in a small sgmnt of a bolomtr. h ltrons ar hatd by th absorbd O owr O and D bias owr. O is absorbd uniformly along th bolomtr with lngth, but th absortion of D owr is not uniform and dnds on th bias urrnt and th loal rsistivity r. n stady stat this hat is artly transfrrd to th honons and artly to th ltrons in th nighboring sgmnts du to th ltron thrmal ondutivity. Hr it is assumd that owr that is arrid by th honons is dirtly transfrrd to th substrat substrat. hr th thrmal ondutivity of th honons in th dirtion of th bolomtr stri is ngltd. h ltron thrmal ondutivity, l, is a funtion of th ltron tmratur. Blow th ritial tmratur,, it is roortional to 3 and abov it is roortional to []. dnots th ross stion ara of th bolomtr stri. O λ quilibruim λ Eltrons in,, honons in quilibruim, substrat, substrat Figur. h hat flow in a small sgmnt of th bolomtr. h ltron-honon and honon-substrat thrmal ouling an b writtn as: n n σ substrat 4 4 σ substrat whr s, s ar th ltron-honon and honon-substrat ooling ffiiny rstivly [8]. For NbN n is found to b qual to 3.6 in [8]. h hat balan quations for a small sgmnt of th bolomtr an b writtn as: O λ σ σ σ 3. substrat

4 Solving 3. aroimatly for and substituting in 3. yilds: O λ σ ff substrat 4 8σ σ substrat whr σ ff. 8σ substrat 7σ Not that th dimnsion of all th trms in 4 is W/m. and s ff and l dnd on th ross stion ara of th bolomtr [7]. h orrsonding dimnsions of s ff and l ar givn in tabl. h ltron tmratur dndn of th rsistivity is assumd to b: N 5 δ whr th r N is th normal rsistivity, is th ritial tmratur and d is a masur for th transition width. Figur shows th rsistivity as a funtion of tmratur. f w dfin D as th width of th transition form % to 9% of th normal rsistivity, it is ossibl to show from 5 that: ln 9 δ 4. 4 δ. r rn Figur. sistivity as a funtion of tmratur Equation 4 is solvd numrially for with th boundary onditions: bath whr is th bolomtr lngth. Not that th -ais is in th dirtion of th bolomtr stri and th origin is at th ntr of th bolomtr. On th tmratur rofil is alulatd th rsistan of th bolomtr taks th following form: N d d 6 δ

5 . Small signal modl and onvrsion gain h bolomtr iruit is shown in figur 3. h miing trm at F auss rsistan flutuations in th bolomtr. h orrsonding small signal voltag drivs a urrnt through th amlifir at th F frquny. Not that this urrnt gos through th bolomtr as wll and auss additional hating and onsquntly additional hang in th rsistan, whih must b takn in to aount. his fft is rfrrd to as ltrothrmal fdbak. blok blok t os ω t O S O F F os ω t F F os ω F t Figur 3. h bolomtr iruit Bolomtr mlifir From th abov iruit th onvrsion gain of th mir is drivd as [7]: F O G 7 S D whr is th owr in th load rsistan and S is th absorbd signal owr. D and F ar th hating ffiinis of th absorbd D and F owr rstivly and dfind as: D and F. 8 D O O D n ordr to alulat D, th O owr is kt onstant and th rsistan hang du to th small hang in th bias urrnt is alulatd at ah orating oint. Sial ar has to b takn whn alulating F baus king th urrnt onstant and alying a small hang in O owr will hang th absorbd D owr as wll, whih in turn hangs th rsistan. his ontribution has to b ddutd from th total rsistan hang in ordr to alulat th rsistan hang du to th hang in th absorbd F owr only. ssuming qual valus for D and F, 7 is rdud to th orrsonding rlation known from th oint bolomtr modls [,4].

6 V. Nois h alulation of th Johnson nois ontribution from th HEB nois modl in [4] ngltd th fat that th dissiatd owr in th nois sour is atually dissiatd within th HEB bridg, rsulting in additional hating. gnoring this trm lads to a disrany btwn th rsults whn th voltag nois sour is rlad by an quivalnt urrnt nois sour. omlt drivation for Johnson nois is givn in th andi, taking this trm in to aount. h ontribution of th Johnson nois to th total ut nois is: 4 Jn D 9 h rssion for th Johnson rivr nois tmratur is simly obtaind by dividing th ut nois 9 by two tims th gain 7: in Jn, F O n th on-dimnsional hot sot modl th ltron tmratur and th rsistivity vary satially, whih rquirs us to modify as: in Jn, F d O From what has bn drivd originally in th oint bolomtr modl [,] th ut thrmal flutuation F nois is writtn as: 4 Fn V bo D whr V is th bolomtr volum, is th ltron thrmal aaity, and t is th ltron rlaation tim. h doubl sid band rivr nois tmratur du to th F nois is obtaind by dividing th F ut nois by two tims th gain 7: in Fn,. F O 3 V

7 his rssion also has to b modifid for th on-dimnsional distributd modl. h modl rsntd hr assums that th ltron tmratur flutuations in th bolomtr sgmnts ar unorrlatd. h drivation is givn in th andi B in dtail and hr w only rall th rsult: in n Fn d F δ, O δ δ Not that and t dnd on th ltron tmratur, whih in turn dnds on. h disussion in andi B indiats that th ut flutuation nois is ssntially indndnt of th orrlation lngth assumd. h masurd nois at th ut of th mir has thr ontributions and an b writtn as: nois Fn Jn 95 G 5 h third trm is th ambint tmratur 95 K at th inut of th mir, whih ontributs to th ut nois during this masurmnt. h total rivr nois tmratur is alulatd using th usual rssion: in in F X, Fn, Jn, 6 G whr F is th nois ontribution from th low nois F K and G is th onvrsion gain 7. V. F mdan and Bandwidth W ar urrntly studying a ossibl mthod to alulat th F imdan and Bandwidth of th bolomtr. h mthod is basd on solving th tim varying small signal on-dimnsional hat balan quation at th orating oint. dding th tim varying trms to 4 yilds: λ σ ff substrat t 7 O S O jωt whr dnots th small signal owr du to th ltrothrmal fdbak r unit lngth. h ltron tmratur an b writtn as: ~ jwt 8

8 nsrting 8 in 7 and sarating th tim varying art, an quation is obtaind whih ~ an b solvd for. rliminary studis show a aaitiv bhavior of th bolomtr, as is wll-known from th oint bolomtr modl []. siml way to stimat th F bandwidth is to assign a tim onstant for th hononooling ross and anothr tim onstant for th diffusion-ooling ross. h diffusion-oold owr an b alulatd by valuating th gradint of ltron tmratur at th ads. h owr oold by honons is thn th diffrn btwn th total absorbd owr and th diffusion-oold owr. h fftiv mir tim onstant an b obtaind by wighting th tim onstants with thir ontributions to th total ooling owr. V. sults h rsults of our alulations and masurmnts rsntd hr aly to a.4 mm long, 4 mm wid and 5 nm thik NbN HEB dvi on MgO Dvi M-. h aramtr valus usd in 4 and 5 ar summarizd in tabl. aramtr l s ff d substrat N Valu 6œ -8 œ 3 œ Dimnsion Wm/K W/m.K 3.6 K K K W abl. h aramtr valus usd in 4 and 5. Equation 4 is solvd numrially. Figur 4 shows th tmratur rofil obtaind for 4 m urrnt and diffrnt absorbd O owrs. Eltron tmrtur HK nw nw 3 nw nw 6 nw ngth Hm Figur 4. h ltron tmratur rofil aross th bolomtr at 4 m bias urrnt and mntiond absorbd O owr.

9 Figur 5 shows th alulatd and masurd V urvs for dvi M-. Et at bias voltags blow.4 mv, th alulatd and masurd V urvs ar in llnt agrmnt. n gnral th alulatd V urvs tnd to bnd mor at lowr voltags than th masurd ons. On th othr hand blow.4 mv th nois is vry high and th onvrsion gain is vry low so this rgion is hardly of any intrst. h otimum orating oint was obsrvd around 4 m and.8 mv. hrfor w fous on th middl urv whih orrsonds to 5 nw absorbd O owr and alulat th onvrsion gain and th nois. urrnt Hm m O nw 5 nw O 3 nw VoltagHmV Figur 5. h masurd solid lins and alulatd dots V urs for dvi M-. h alulatd D and F 8 for th 5 nw absorbd O owr urv is shown in figur 6. d HW n W Voltag HmV rf HW n W Voltag HmV Figur 6. alulatd D lft and F right along th 5nW absorbd O owr urv for diffrnt bias voltags.

10 nsrting ths valus in 7, th onvrsion gain is alulatd. Figur 7a shows th alulatd and masurd onvrsion gain. lthough th sha of th urvs is similar, th alulatd valus ar ab 9 db highr that what w atually masurd. h alulatd ut nois 5 is a fator of 3 highr than what is masurd. Figur 8a shows th alulatd and masurd ut nois. On th othr hand baus of th high gain, th modl rdits vry low rivr nois tmratur 6, whih is shown in figur 9a. G ain H d B a G ain H d B b VoltagHmV VoltagHmV Figur 7. h masurd solid lin and alulatd dots onvrsion gain for 5 nw absorbd O owr; a for high F valus as shown in figur 4; b for F valus rdud by a fator of thr. his mans that th modl annot rdit th masurd urvs. Howvr a los study rvald that this is only du to th high F valus. f w rdu th F valus by a fator of thr, all th masurd and alulatd urvs fit togthr. h alulatd and masurd onvrsion gain, ut nois and rivr nois tmratur aftr this rdution is ditd in figurs 7, 8, 9 b rstivly. a 5 b Outut N ois HK 5 5 Outut N ois HK VoltagHmV VoltagHmV Figur 8 h masurd solid lin and alulatd dots ut nois for 5 nw absorbd O owr; a for high F valus as shown in figur 4 ; b for F valus rdud by a fator of thr.

11 riv rn ois HK 4 3 a riv rn ois HK b VoltagHmV VoltagHmV Figur 9. h masurd solid lin and alulatd dots rivr nois tmratur for 5 nw absorbd O owr; a for high F valus as shown in figur 4; b for F valus rdud by a fator of thr. ftr rduing F, th valu of F at a tyial otimum orating oint is.6. Sin / is th urrnt on th unumd V urv.f. [,4] is ab.4 at th otimum oint for rivr nois tmratur [4], F is ab.7. his quantity was usd as an adjustabl aramtr in [4], and for otimum rforman was found to b ab. his lains why th mirial "standard modl" in [4] yilds rsults in agrmnt with rimnt los to th otimum oint. V. onlusion h on-dimnsional distributd modl is fully aabl of rditing th V urvs, onvrsion gain, ut nois and rivr nois tmratur within atabl auray. Howvr this rquirs alying a tuning fator to th alulatd F. his tuning fator was found to b ab.34 to fit th masurd data for dvi M-. t simly mans that th rsistan osillation du to th miing trm in th F is ovr stimatd by alulating a small rsistan hang du to small hang in O owr. h hysis bhind this fator is urrntly undr invstigation. Major rogrss has bn mad in omarison with th oint-bolomtr modl in for aml [4], in that th variation of all masurabl quantitis onvrsion gain, rivr nois and ut nois with bias voltag and O owr is rditd vry wll. knowldgmnts his work was artly suortd by Euroan Sa gny, ES #738/95/nl/mv.

12 frns [] H. Ekström, B. Karasik, E. Kollbrg, K. S. Yngvsson, EEE rans. M, Vol. 43 4, 995 [] B. Karasik,.. Elantv, ro. Sith ntrnational Symosium on Sa rahrtz hnology, 995. ags: [3] H. Mrkl,. Yagoubov,. Khosroanah, E. Kollbrg. ro. Sith EEE ntrnational onfrn on rahrtz Eltronis, 998, ags: [4] K. S. Yngvsson, E. Kollbrg, ro. nth EEE ntrnational Symosium on Sa rahrtz hnology, 999. ags: [5] H. Mrkl,. Khosroanah,. Yagoubov, E. Kollbrg, EEE ransations on lid Surondutivity Vol. 9 ssu: art: 3, Jun 999, ags: 4-44 [6] D. W. Flot, E. Midma, J.. Gao,.M. Klawijk, ro. Sith EEE ntrnational onfrn on rahrtz Eltronis, 998, ags: 9-. [7] H. Mrkl,. Khosroanah,. Yagoubov, E. Kollbrg, ro. nth ntrnational Symosium on Sa rahrtz hnology, 999. ags: [8] S. hrdnihnko,. Yagoubov, K. l'in, G. Gol'tsman, E. Grshnzon, ro. 8 th ntrnational Symosium on Sa rahrtz hnology, 997, ags: [9] D. W. Flot, J.. Gao,.M. Klawijk,.. J. d Kort, ro. nth ntrnational Symosium on Sa rahrtz hnology, 999. ags: [] harls. ool, Horaio. Farah, ihards J. rswik. Surondutivity. admi rss. 995 [] John. Mathr. lid Otis. Vol., No. 6. Marh 5, 98, ags 5-9. ndi : Johnson nois h thrmal nois from a rsistan at tmratur an b writtn as: vn 4k B,. whr k B is th Boltzmann onstant. n ordr to alulat th nois w an insrt this nois sour in th bolomtr iruit. Figur. shows this iruit. D and D ar small variations of rsistan and urrnt rstivly. ssuming a oint bolomtr at tmratur and rsistan, th nois quivalnt sour is:

13 vn 4k B. h total dissiatd owr in th bolomtr is: vn 3. Not that th dissiatd owr in th nois sour is atually dissiatd in th bolomtr as wll and it is of grat imortan to tak it into aount. gnoring this trm lads to a disrany btwn th rsults whn th voltag nois sour is rlad by an quivalnt urrnt nois sour. blok blok v n Bolomtr mlifir Figur.. Bolomtr iruit with quivalnt Johnson nois sour Eanding th right sid of 3. and ignoring th sond ordr small signal trms, 3. is simlifid to: vn 4. Sarating th small signal and larg signal arts yilds: vn 5. sin, w an rla D and solv 5. for D whih rsults in: D D Dvn 6. D Now it is ossibl to alulat th voltag aross th bolomtr: V V vn 7. Sarating th larg and small signal hr too givs: V vn 8. whr th small signal voltag DV is th F voltag aross th load. Solving for D and rlaing D from 6. yilds:

14 v D n 9. h ut nois owr is simly: v D n Jn. nsrting. in. rsults in: Jn B D B Jn B k B k 4. whr Jn is th ut Johnsson nois. 4 D Jn. h quivalnt nois tmratur at th inut of th mir is obtaind by dividing th ut nois by th onvrsion gain givn in 7. h doubl sid band quivalnt nois tmratur is a fator of two smallr than th SSB on. So th rssion for th Johnson nois tmratur at th inut boms: O F in Jn, 3. his rsult is in agrmnt with th rssion drivd by taking anothr aroah [], and also with th gnral rsults for bolomtrs drivd originally by Mathr []. ndi B: hrmal flutuation nois From th oint bolomtr modl [,] th ut thrmal flutuation F nois is writtn as: D Fn V 4.B whr V is th bolomtr volum, is th ltron thrmal aaity, and t th ltron rlaation tim. h doubl sid band rivr nois tmratur du to th F nois is obtaind by dividing th F ut nois.b by two tims th gain 7:

15 O F in Fn V,..B h total rsistan flutuation is th sum of all th rsistan flutuations of ah small sgmnt of th bolomtr du to tmratur flutuations. hus dr, th rsistan of suh a sgmnt with lngth d is writtn as: d dr 3.B Not that th rsistivity rofil is obtaind from th ltron tmratur rofil from 5. th total rsistan of th bolomtr is obtaind by: d dr. 5.B h rsistan flutuation of a small sgmnt of th bolomtr an b writtn as: d dr 6.B From 6.B and.b th thrmal flutuation nois tmratur from this small art is drivd as: d d d d dr d O F O F O F in Fn 3, B Now if w assum that all ths loal tmratur flutuations ar unorrlatd, th total nois tmratur is simly th intgral of all th small ontributions. d d n O F O F in Fn 3 4 3, δ δ δ 8.B w hav also rformd this alulation by assuming diffrnt valus for th orrlation lngth of th F nois. h rsult is ssntially unhangd, rovidd that varis smoothly. h abov assumtion of an infinitsimal orrlation lngth thus is justifid, unlss th orrlation lngth boms omarabl to th lngth of th bolomtr.

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