Fluctuation of current in mesoscopic junctions

Transcription

1 Hlsinki Univrsity of Tchnology Dpartmnt of Enginring Physics and Mathmatics Spcial assignmnt Tfy Matrials physics 22nd April 24, rv. 3 Fluctuation of currnt in msoscopic junctions Pauli Virtann 5758F

2 Contnts Introduction 2 2 Formalism 3 2. Statistics of chargs and currnt Currnt corrlators Kldysh Grn s function tchniqu Full counting statistics Multipl trminals Statistics in a diffusiv wir by xpansion 3. Solving th cumulants First cumulant Scond cumulant Third cumulant Highr cumulants Rlaxation in th diffusiv wir Statistics of a gnric junction 9 5 Circuit thory for th first two cumulants Exampl: thr-trminal chaotic quantum dot Discussion 26 Introduction Th magnitud of currnt flowing through a rsistor is gnrally not constant in tim, but fluctuats around th avrag valu, givn for normal mtals usually by Ohm s law I = V/R. Many factors contribut to th fluctuation []: on part is du to a random thrmal nois, anothr part is inducd du to th discrtnss of charg and th rst is du to various sourcs, som of which ar not known. Of th two important sourcs considrd hr, th thrmal fluctuations occur also in quilibrium, but th part du to th discrtnss of charg shows up only in non-quilibrium transport procsss. Th fact that charg coms in discrt units implis that on can dfin a probability distribution for th numbr N of chargs transmittd in a crtain tim through a givn junction. Faturs of this distribution show up in fluctuations, which can b masurd to giv information about th lctron transport procss. Howvr, in macroscopic wirs th fluctuations ar gnrally Gaussian, and show lss information about th transport procsss. This ffct is usually du to th inlastic scattring of lctrons from phonons, which is, howvr, supprssd at low tmpraturs in suitably small systms. That is, at low tmpraturs msoscopic systms with thir lngth scals blow th inlastic scattring lngth, typically of th ordr of micromtrs ar wll suitd for obsrvation of phnomna of this kind. Th ffct of charg discrtnss on th currnt nois, i.., th scond momnt of th currnt probability distribution, was first masurd by Schottky in th 2

3 92s. Although som additional vn momnts hav bn masurd sinc thn, no masurmnts of highr odd momnts xistd until vry rcntly [2], sinc th high-ordr momnts ar xprimntally difficult to approach. Nonthlss, th highr momnts and th full probability distribution of currnt hav bn undr intnsiv thortical study undr th past fw yars. Morovr, many prdictions and thortical mthods hav bn put forth, spcially aftr a usful connction, th full counting statistics (FCS), btwn th statistics and th quantum-mchanical dscription was formulatd [3] for lctrons. Currntly thr ar many diffrnt tchniqus for calculating th probability distribution and its momnts, for xampl, th quasiclassical Kldyshformulation [4], th smiclassical Boltzmann-Langvin thory [5], a path intgral formulation [6], and th random scattring matrix thory [7]. In this work, w us th quasiclassical Kldysh formulation for obtaining th statistics of charg transmission. Th formalism is rathr gnral and abl to cop with many kinds of junctions, and w xplain in dtail how it can b applid. Much of this work concrns th diffusiv normal mtal wir, th momnts for which w calculat through two diffrnt routs. Morovr, w prsnt som obsrvations concrning th inlastic scattring in th diffusiv wir, driv rsults for othr kinds of junctions, and xamin many-prob systms. Howvr, w rstrict ourslvs to normal mtal structurs, as suprconducting parts mak th problm somwhat mor complicatd, vn in th static cas. 2 Formalism What w ar intrstd in is th statistical dscription for th procss of chargs passing through a givn junction. This kind of a procss may b dscribd by th probability distribution P t (N), corrsponding to N chargs bing transmittd during tim t. Th mthod w us to obtain information about this quantity is a vrsion of th quasiclassical Kldysh Grn s matrix circuit thory that also taks fluctuations of currnt into account. Howvr, bfor actually discussing this thory, w nd to considr first how probability distributions and th fluctuations of currnt may gnrally b charactrizd. 2. Statistics of chargs and currnt In practic, a discrt probability distribution is oftn charactrizd by a st of numbrs, th momnts or th cumulants. Gnrally, such a distribution say, th transmission probability P t (N) of N chargs to a crtain dirction can b xprssd in trms of its cumulant-gnrating function S(χ): S(χ) iχn = N= P t (N) iχn, P t (N) = π dχ S(χ) iχn. () 2π π That is, th probability distribution is xamind in th Fourir transformd form. Morovr, a probability distribution can b charactrizd by its cumulants [8], which this work mostly concrns. Thy ar dfind as C n n S(χ) (iχ) n, (2) χ= 3

4 and simply ar th cofficints of xpansion for S(χ) with rspct to iχ. Th raw momnts N n of transmittd chargs can now b xprssd as N n = n iχn (i χ) n = n χ= (i χ) n S(χ). (3) χ= Using this rsult, on can writ th first fw raw and cntral momnts, N n and (N N ) n, in trms of th cumulants: N = t C, N 2 = C 2 + C 2, (N N ) 2 = C 2, (4a) N 3 = C 3 + 3C C 2 + C 3, (N N ) 3 = C 3, (4b) N 4 = C 4 + 6C 2 C 2 + 3C C C 3 + C 4, (4c) (N N ) 4 3 (N N ) 2 2 = C4. (4d) W can asily s th rlation btwn th raw and th cntral momnts, n n N (N N ) n k N n k, (5) k k= which can b obtaind simply by xpanding th binomial. Th transmission of chargs can trivially b rlatd to th tim-avragd currnt I N/t = t t dt I(t), which immdiatly yilds th cumulants of th avrag currnt C I,n = ( t ) n C n. Howvr, th cumulants of charg transmission ar also connctd to th currnt corrlation functions, as discussd in th nxt sction. On thing to not is that th gnrating function () dos not contain much information about th tim-dpndnc of I(t). This could b rmdid by rplacing χn with t dt χ(t)i(t), but in th following w ar satisfid with ltting χ b indpndnt of th tim, which still yilds usful information about th fluctuations of th currnt. 2.. Currnt corrlators Oftn on is intrstd in th tim-dpndnt momnts of currnt fluctuations, in particular th currnt corrlation functions δi(t )δi(t 2 ), δi(t )δi(t 2 )δi(t 3 ),..., (6) and thir Fourir transforms. How do ths rlat to th momnts and cumulants of th tim-avragd currnt? As an xampl, w considr hr th cas of th third momnt along th lins prsntd in [9], but only for classical currnts not for quantum-mchanical currnt oprators. Th othr momnts can b handld in a similar fashion. First, lt us dfin th third corrlation function S 3 (t, t, t ) δi(t)δi(t )δi(t ) = δi(t t )δi(t t )δi() S 3 (t t, t t ), (7) whr δi(t) I(t) I(t). Th quality follows sinc w assum a stady-stat situation whr th fluctuations ar indpndnt of th absolut tim. Now, tak th Fourir transform Ŝ 3 (ω, ω ) = dt dt δi(t)δi(t )δi() iωt iω t, (8) 4

5 and considr th zro-frquncy cas Ŝ 3 (ω =, ω = ) = dt dt δi(t)δi(t )δi() (9) W can connct this to th tim-indpndnt statistics as follows. First, I I = (N N ) = t dt (I(t) I(t) ) = t dt δi(t), () t t t which implis that t t t C 3 = t3 (I I ) 3 3 = 3 dt dt 2 dt 3 δi(t )δi(t 2 )δi(t 3 ) = 3 dt dt 2 dt 3 δi(t )δi(t 2 )δi(t 3 ) Ω(t,, t )Ω(t 2,, t )Ω(t 3,, t ), () whr w introducd a support function Ω: { for a t b Ω(t, a, b) =. (2) othrwis Now, bcaus δi(t)δi(t )δi(t ) is indpndnt of th absolut tim, w chang th variabls to th rlativ tim (th Jacobian of this transformation is ) t = 3 (t + t 2 + t 3 ), τ = t t 3, τ 2 = t 2 t 3. (3) Sinc Ω(t + c, a, b) = Ω(t, a c, b c), w obtain C 3 = 3 dτ dτ 2 δi(τ )δi(τ 2 )δi() dt A(t), A(t) Ω(t, 2τ τ 2 3, t 2τ τ 2 3 Ω(t, τ + τ 2, t + τ + τ 2 ) 3 3 Ω(t,, t ), for τ, τ 2 t, )Ω(t, 2τ 2 τ 3, t 2τ 2 τ ) 3 (4a) (4b) whr th approximation is valid in th sns that dt A(t) dt Ω(t,, t ) = t. W can us this approximation providd that th corrlation function δi(τ )δi(τ 2 )δi() is non-vanishing only for τ, τ 2 t, i.., th obsrvation tim t is longr than th corrlation tim. Thn, w s that for rlvant valus of τ and τ 2, th innr intgration yilds just t. Hnc w obtain C 3 t 3 dt dt δi(t)δi(t )δi() = t 3 Ŝ3(, ). (5) A similar drivation can b prformd also for othr momnts of currnt fluctuation, yilding for xampl for th scond cumulant C 2 t 2 dt δi(t)δi() = t 2 Ŝ2(, ) = t 2 2 S I, (6) 5

6 whr S I 2S 2 (, ) is calld th nois powr. Summarizing, th cntral momnts of charg transmission ar connctd to th Fourir transformations of th corrlation functions at zro frquncy, through rlations analogous to Eqs. (5,6). Now that w ar abl to connct th statistical dscription of currnt fluctuations to quantitis that ar masurabl at last in principl, th nxt task is to obtain th connction of statistics to th microscopical structur of th systm. 2.2 Kldysh Grn s function tchniqu Msoscopic lctrical circuits consisting of normal and suprconducting parts can b modld using th quasiclassical Kldysh Grn s function mthod, or mor spcifically, its matrix circuit thory [, ]. In its basic form, it offrs no information about th fluctuation of currnt, but it can b modifid to includ such ffcts. Howvr, lt us first discuss th unmodifid vrsion. Th main quantity which w attmpt to solv is th lctron distribution function f at vry point in th circuit, sinc knowing it is ndd for dtrmining th classical xpctation valus for th currnts in th structur. In th quasiclassical thory, this information is containd in th Grn s function Ǧ, along with th information about th ffcts du to possibl narby suprconductors. On nds thus to know how to dtrmin Ǧ from th structur, tmpraturs and potntials in qustion, and how to rlat it to th classical lctrical currnt. In th following, w summariz th formalism, but mak no attmpt to actually driv its rsults. First, w nd to discuss th proprtis of Grn s functions thmslvs. Th quasiclassical Kldysh Grn s functions discussd hr blong to th Kldysh Nambu spac, i.., ar 4 4 matrics with th structur G Ǧ = R G K G A, (7) whr ach lmnt G blongs to th Nambu spac and is a 2 2 matrix. Th Kldysh structur appars sinc th Kldysh formalism is usd to obtain nonquilibrium statistical proprtis, and th Nambu spac is ndd for considring both particls and hols. Morovr, th Grn s function hav th symmtry G A = τ 3 (G R ) τ 3, (8) and thy should also satisfy th normalization condition Ǧ2 = ˇ. This is achivd by rquiring that th rtardd, advancd and Kldysh componnts, G R, G A, and G K hav th following proprtis: G K = G R h hg A, h f L + f T τ 3, (9) whr f L = f( ε) f(ε) and f T = f( ε) f(ε) ar th antisymmtric and symmtric parts of th lctron distribution function f( r, ε). As th quasiclassical Grn s function dscrib th bhavior of lctrons and hols, it contains information about th lctrical currnts I in th systm. Not, howvr, that cumulants of highr ordr than 3 ar not th corrsponding cntral momnts. 6

7 A usful quantity is th spctral matrix currnt dnsity Ǐ, from which th xpctation valu I = Î for th lctrical currnt can b obtaind as I = 8 and for th nrgy currnt as dε Tr [ˇτ K Ǐ ], Ǐ σaǧ Ǧ, ˇτ K τ τ 3 (2a) I E = 8 dε ε Tr [ τ 3ˇτ K Ǐ ]. (2b) Hr, A and σ ar th cross-sctional ara and th normal-stat conductanc of th wir whr th currnt flows. Not that hr w mak th commonly usd assumption that th wirs ar quasi-d, i.., Grn s functions do not vary significantly across th cross sctions. Nxt, w nd to dscrib how th Grn s functions vary spatially in th circuit. Considr first th cas of diffusiv mtals, whr th lastic scattring of lctrons is strong, and th man fr path is much smallr than th lngth scals of intrst. According to th thory, in th diffusiv limit th Grn s function satisfis th Kldysh-Usadl diffusion quation D (Ǧ Ǧ) = [ iε τ 3 + ˇ + ˇΣ in, Ǧ]. (2) Hr, D is th diffusion constant of th mtal, is th pair potntial matrix and ˇΣ in contains th inlastic collision slf-nrgis. Th pair potntial matrix dscribs th attractiv lctron lctron intraction which inducs suprconductivity. W considr hr normal mtals, and thus ˇ =. Additionally, ˇΣ in = if inlastic lctron lctron and lctron phonon scattring is nglctd. Th Kldysh Usadl quation cannot handl lngth scals short compard to th Frmi wavlngth or th man fr path. Thus, th intrfacs btwn th rsrvoirs and th wirs nd to b considrd sparatly through boundary conditions. Th boundary condition at a clan rsrvoir contact is such that Ǧ obtains its bulk valu. For normal mtals, this implis Ǧ τ3 2h N = bulk τ 3 2f(ε), h τ bulk =, (22) 3 2f( ε) whr h bulk contains a Frmi function f(ε) ( + xp((ε + V )/T ) at tmpratur T in a rsrvoir biasd at potntial V. For suprconductors, th corrsponding bulk Grn s function is Ǧ ( ε G R S = S ( s)g R S f ) L ε2 2 sg R, S G R S = τ 3 + ε iτ3φ iτ 2, s sgn( ε ). (23) Hr, is th suprconducting nrgy gap and φ is th phas of th ordr paramtr in th suprconductor. Howvr, not that th tim-indpndnt formulation cannot in this form dscrib non-stationary situations or non-quilibrium Josphson ffcts, so on ncssarily nds to assum V = in th suprconductors which also implis f T = in thm. Considr thn a gnral intrfac btwn two rgions with diffrnt Grn s functions ǦL and ǦR,.g., a tunnl junction with rsrvoirs at both sids. 7

8 Thr is a gnral rlation conncting th Grn s functions and th transmission proprtis of th intrfac to th spctral matrix currnt passing through it []: ] Ǐ L R = 22 T n [ǦL, ǦR π 4 + T n ( { }. (24) Ǧ L, ǦR 2) n Hr, T n ar th transmission ignvalus of th intrfac, charactrizing is proprtis. In th scattring matrix thory, thy ar th ignvalus of th scattring probability matrix, and thir probability distribution is known for various typs of junctions (s Sct. 4 for xampls), also diffusiv wirs. Howvr, in th prsnc of suprconductivity, rlation (24) is valid only for zro-dimnsional junctions, hnc diffusiv wirs nd to b thn considrd using th Kldysh Usadl quation. If w considr a nod whr multipl wirs ar connctd with clan contacts, th boundary condition is th consrvation of th matrix currnts, Ǐ nod j =, (25) j whr th sum gos ovr th wirs connctd to th nod, and th spctral matrix currnts point towards th nod. Finally, w not that if thr ar only normal mtal componnts prsnt in th structur, all th sub-matrics of Ǧ commut with τ 3, sinc othr componnts than and τ 3 do not ntr. Thus, Eq. (2) rducs to D (Ǧ Ǧ) = [ˇΣin, Ǧ]. (26) If inlastic collisions ar additionally nglctd, this implis that th spctral matrix currnts ar consrvd, sparatly for ach nrgy ε. On additional simplification in th absnc of suprconductivity is that th Nambu spac is not strictly ncssary, as all lmnts in th Kldysh spac commut with ach othr and can hnc b tratd as scalars. Morovr, in th absnc of inlastic ffcts, th Grn s functions assum th structur g Ǧ(ε) = R (ε) g K (ε) g g O (ε) g A + R ( ε) g K ( ε) (ε) g O ( ε) g A, (27) ( ε) }{{} Ḡ(ε) sinc this is th form of th boundary conditions, and th quation (26) trats both parts sparatly: solving (Ḡ Ḡ) = is sufficint. Spctral matrix currnt has a form similar to (27), with th sign of th lattr part, th hol part, rvrsd. Using this, on nots that Eqs. (2) rduc to I = dε Tr[τ Ī], I E = dε ε Tr[τ Ī], Ī AσḠ Ḡ 4 4. (28) Howvr, in th following w do not xplicitly us this proprty Full counting statistics Th statistical proprtis of currnt can b modld using a mthod known as th full counting statistics [3] (FCS), originating from quantum optics, but 8

9 applicabl also to lctron systms. It conncts th quantum mchanical dscription of th lctrons in a conducting systm to th statistics of th currnt, through th cumulant-gnrating function S(χ), which in th stationary stat classical cas is writtn as in Eq. (). Th main ida is to valuat th quantum-mchanical cumulant-gnrating function, which can b writtn as S(χ) = ln T xp ( iχ 2 t dt Î(t) ) ( ˆT xp iχ 2 t ) dt Î(t), (29) whr T and ˆT ar th tim and anti-tim ordring oprators. Without th tim ordring, this function would not corrspond to th classical cumulant gnrating function. Morovr, th classical intrprtation is possibl only in th absnc of suprconductivity. A mor throughout discussion on this can b rad, for xampl, in Rfs. [2, 3]. It can b shown that th calculation of th gnrating function (29) can b don using a slightly modifid matrix circuit thory [9, 4, 4]. Th only chang rquird for calculation of th gnrating function S(χ) is th coupling of th counting fild χ to th systm through a modification of th boundary conditions. In th two-trminal cas, th modifid, Kldysh-rotatd, Grn s function in th lft rsrvoir is [4] Ǧ L (χ) = (i/2)χˇτk Ǧ bulk (i/2)χˇτk, ˇτ K τ τ 3, (3) and th usual boundary condition at th right rsrvoir may b rtaind. 2 Now, w may calculat th χ-dpndnt counting currnt I(χ) ithr by solving th Kldysh Usadl quation in th cas of diffusiv wirs, or by applying Eq. (24) for gnral junctions. (Th quantity I(χ) is again th xpctation valu of th corrsponding currnt oprator.) Nxt, th gnrating function and th cumulants can b obtaind from th χ-dpndncy of this currnt [4]: S(χ) (iχ) = t I(χ) C n = t n I(χ) (iχ) n. (3) χ= That is, all cumulants and momnts can asily b calculatd at last in principl by solving I(χ) using th quasiclassical Grn s function tchniqu and th matrix circuit thory. 3 Bfor w dlv into th actual calculations, lt us summariz som rlvant rsults from th prvious sctions: Ŝ (, ) = I = C = I(χ) t χ= (32) Ŝ 2 (, ) = dt δi(t)δi() = 2 C 2 = I(χ) t (iχ) (33) χ= Ŝ 3 (, ) = dt dt 2 δi(t )δi(t 2 )δi() = 3 C 3 = 2 2 I(χ) t (iχ) 2. (34) χ= Ths will b of us in conncting th rsults obtaind in th following sctions to quantitis that ar in principl masurabl. 2 Not that th rotation (3) prsrvs th structur (27). 3 W chos th dirction of currnts in Eq. (2) in a way that is usually usd in th circuit thory [, ], opposit to th on usd in Rf. [4]. Thus, th sign in Eq. (3) is diffrnt. 9

10 2.2.2 Multipl trminals Counting statistics and th circuit thory ar not rstrictd to two-trminal structurs. In fact, th gnralization is vry straightforward [5]: on nds only to introduc multipl counting filds χ = (χ,..., χ n ), on pr trminal. Thn, th gnrating function for th probability distribution P t (N,..., N n ) of N i chargs ntring th i =,..., n trminals is writtn as S(χ,...,χn) P t (N,..., N n ) i χ N, (35) N,...,N n whr N = (N,..., N n ). As in Sct. 2., this givs th cumulants C α α S( χ) C α, (36) i α α χ... αn χ n with α = (α,..., α N ) and α = α α N. Morovr, th cumulants can b connctd to th currnt corrlators as shown in Sct Using thm, on may also considr th cross-corrlations of currnts flowing to diffrnt trminals. Conncting th multipl counting filds to th quantum-mchanical dscription is don as in th two-trminal cas: on prforms th rotation (3) at ach trminal, using th corrsponding counting fild. (On of th counting filds may b chosn to b zro, as only th diffrncs mattr: th total currnt is consrvd so only n of th currnts to th trminals ar indpndnt.) Aftr modifying th boundary conditions, on can apply th circuit thory ruls, including currnt consrvation at nods (25), to obtain th counting currnts I ij ( χ) flowing from th nod or trminal i to th nod or trminal j. Aftr th counting currnts ar found, th gnrating function is found from ds( χ) = t N I ij ( χ) i dχ i, S( ) =, (37) i= j whr th summation with j xtnds ovr all th nods and trminals connctd to th trminal i, totaling th counting currnt flowing from th trminal. In th cas of two trminals, this naturally rducs to Eq. (3). 3 Statistics in a diffusiv wir by xpansion In this sction w considr th statistics of currnt in a diffusiv wir connctd to two trminals, rfrrd to as th lft and right rsrvoirs in th following. Additionally, w assum hr that inlastic collisions ar absnt. To calculat th cumulants of th currnt using th matrix circuit thory, w nd to impos th boundary conditions (3) on th lft rsrvoir, and solv th χ-dpndnt currnt from Eq. (26). From this rsult w obtain th cumulants of currnt. In th following, w calculat th first fw cumulants of th currnt by xpanding th boundary conditions and Ǧ in a sris of χ, and solv th similarly xpandd Kldysh Usadl quation sparatly for ach ordr, starting from th lowst on (in a way similar to th prturbation thory). For convninc, w scal th wir to b of lngth, thus th actual lngth L appars in th xprssion for th spctral matrix currnt. Morovr, w writ th sris using

11 ω iχτ 3 instad of χ as th xpansion paramtr: Ǧ(x) j= j!ǧj+(x)τ 3 ω j, Ǐ Aσ Aσ Ǧ Ǧ L L j= j!ǐj+ω j. (38) Using th dfinitions abov, w obtain th cumulants by calculating C n = n S(χ) (iχ) n = t n I(χ) χ= (iχ) n = t χ= R I n, (39) whr I = Aσ }{{} L R j= j! I j+ω j, I j = 8 [ ] dεtr τ j 3 ˇτ K Ǐ j (4) as implid by Eq. (2). Hr, R is th rsistanc of th wir. First, w xpand th boundary conditions (3): Ǧ() ǦL = ω/2ˇτkτ3 (Ǧ N) ω/2ˇτkτ3, Ǧ() ǦR = Ǧ τ3 2h N = R τ 3. τ 3 cosh ω Ǧ() = hl sinh ω h L ( + cosh ω) sinh ω τ h L ( cosh ω) + sinh ω cosh ω + h L sinh ω 3. (4) Thus, ( ) h L hl h L Ǧ j () = δ h L j, + h L h L 2hR Ǧ j () = δ j,. for odd j, for vn j, (42) Nxt, w xpand th normalization condition Ǧ2 = ˇ: ( ) i i!ǧi+ω j j!ǧj+ω = i+j i! j!ǧi+ǧj+ω (43) i= = k k= l= j= (k l)! l!ǧl+ǧk l+ω k = i= j= k= k! ωk k l= k Ǧ l+ Ǧ k l+ = ˇ, l whr w changd variabls in th summation by introducing k = i+j and l = j. As th right-hand sid is indpndnt of ω, and ω k ar linarly indpndnt, w obtain th normalization conditions 4 k l= ( k l ) Ǧ l+ Ǧ k l+ = { Ǧ k+, Ǧ } ( 2 δ k) + 4 W us th usual convntion that b j=a = for a > b. k l= k Ǧ l+ Ǧ k l+ = δ kˇ. l (44)

12 Writing th first fw xplicitly, w gt Ǧ 2 = ˇ, { Ǧ, Ǧ2} = ˇ, { Ǧ, Ǧ3} + 2 Ǧ 2 Ǧ 2 = ˇ, (45) } } {Ǧ, Ǧ4 + 3 {Ǧ2, Ǧ3 = ˇ, { } } Ǧ, Ǧ5 + 4 {Ǧ2, Ǧ4 + 6 Ǧ 3 Ǧ 3 = ˇ. An xpandd xprssion for th spctral matrix currnt is obtaind in th sam mannr: Ǐ L/(Aσ) = }{{} Ǧ Ǧ = k= R Ǐ k+ k Th first fw ar xplicitly k! ωk Ǐ k+, Ǐ k+ k = Ǧ Ǧk+ + Ǧk+ Ǧ + l= ( k l k l= k Ǧ l+ Ǧk l+, l ) Ǧ l+ Ǧk l+. (46) Ǐ = Ǧ Ǧ, Ǐ 2 = Ǧ2 Ǧ + Ǧ Ǧ2, Ǐ 3 = Ǧ3 Ǧ + Ǧ Ǧ3 + 2Ǧ2 Ǧ2, (47) Ǐ 4 = Ǧ4 Ǧ + Ǧ Ǧ4 + 3Ǧ3 Ǧ2 + 3Ǧ2 Ǧ3. Now, Eq. (26) implis that in th absnc of inlastic collisions (with Σ in = ), Ǐk =, (48) that is, th partial spctral currnts spcifid abov ar consrvd. Thus, w hav formulatd th quations from which all cumulants of currnt can b solvd in a straightforward mannr, starting from th lowst cumulant. 3. Solving th cumulants Considr now what th normalization condition (44) suggsts about th structur of th matrics. In th following, w show that in gnral thy ar of th form gj g Ǧ j = j2. (49) g j3 g j W know from th usual Grn s function calculation that this is tru for j =, sinc Ǧ has g =, g 2 = 2h and g 3 =. Morovr, all matrics with th structur (49) hav th proprty } {Ǧi, Ǧj = (2gi g j + g i3 g j2 + g i2 g j3 )Ǐ m ijǐ. (5) Additionally, w s that for a gnral matrix, { } gj g Ǧ, j2 gj + hg = 2 j3 h(g j + g j4 ). (5) g j4 + hg j3 g j3 g j4 Not that all trms apparing in th normalization rlation (44) ar anticommutators btwn Ǧi-matrics, noting that ǦiǦi = 2 {Ǧi, Ǧi}. Thus, if th matrics i =,..., k ar of th form (49), thn by Eq. (5) th lattr sum in Eq. (44) givs no contribution to th off-diagonal lmnts. Thn also j = k + is of th form (49), sinc th only off-diagonal lmnt on th lft-hand sid in 2

13 Eq. (44) is 2h(g j + g j4 ) and it should vanish to guarant th normalization. Sinc this is asily sn for j = 2 (thr ar no trms bsids th on anticommutator with Ǧ), w dduc by induction that all Ǧ j -matrics ar of th form (49). W can now writ g j xplicitly, albit rcursivly, using Eq. (44): j> g j = { hgj3 2 (j 3)/2 ( j l= l (j 2)/2 ( j l= l ) m j+ ) ml+, j l 4( j j 2 2, j+ 2 ) ml+, j l, j 2N, j 2N + hg j3 2 (52) Thus, w hav shown that th normalization condition (44) is satisfid iff th matrics hav th form (49), with g j givn by Eq. (52). Now that w know th propr paramtrization for ths matrics, w may turn to solving th quations (48). First, w not that gj + 2h g Ǧ Ǧj + Ǧj Ǧ = j3 g j2 + 2g j h 2h g j g j3 g j + 2g j3 h (53) j 2 j 2 = Ǐj Ǧ l+ Ǧj l, l (54) l= whr Ǐj = const(x) sinc it is a consrvd quantity. W not that vrything on th right-hand sid is ithr constant or blongs to th alrady known ordrs of xpansion i < j. Now, g j3 can b solvd dirctly by intgrating th bottom lft cornr, and using th boundary conditions (42) to giv g j3 () and th bottom lft cornr of Ǐj. Aftr that, g j2 may also b solvd from th top right cornr in a similar fashion. W can now argu that th lmnts on th diagonal ar also consistnt (i.., sum to an x-indpndnt quantity), sinc th non-xpandd quation should hav a solution. Thus, w obtain an xplicit xprssion for th matrix Ǧj and th partial matrix currnt Ǐj, which can b usd for th calculation of th jth cumulant of th currnt. Thus th calculation, nglcting inlastic collisions, is a straightforward procss. Th only practical complication is th lngthning of xprssions with incrasing j. Howvr, using computr algbra systms, th first cumulants ar not too difficult to solv analytically, which is what w do in th following. 3.. First cumulant W first solv th first cumulant, i.., th quations of ordr ω. Th boundary conditions now imply a normal Ansatz 2h(x) G (x) =, (55) which also satisfis th normalization condition. For th spctral currnts, w obtain, using th spctral currnt consrvation Ǐ = (48) and th boundary conditions (42), Ǐ = 2 h(x) = 2(hR h L ) = const(x), h(x) = h L +(h R h L )x. (56) 3

14 Calculating now th first cumulant using (39), w obtain C = t 2 dε (f T,L f T,R ) = t R 2 2 dε (f L f R ), (57) R whr R is th rsistanc of th wir, and f T,L, f T,R th symmtric parts of th distribution function at th lft and right rsrvoirs. In th scond quality w xpandd f T,R/L and f L,R/L to th Frmi distribution functions f L and f R in th lft and right rsrvoirs, rspctivly. Assuming th whol systm to b at th sam tmpratur T and th potntials to b V L = and V R = V, w obtain C = t V R, (58) as xpctd, sinc th avrag currnt should b Ī = t C, according to Ohm s law Scond cumulant Th calculation of th scond cumulant procds similarly. First, w pick a paramtrization using Eqs. (49) and (52), agring with th boundary conditions and th normalization condition: g23 (x)h(x) g Ǧ 2 (x) = 22 (x) (59) g 23 (x) g 23 (x)h(x) Thus {G 2, G } =, as rquird. Now, th currnts bcom h g23 g Ǐ 2 = 23 h g h 2 g 23 hr j = 22 = const(x) (6) g 23 (h g 23 g 23 h) h R Hr, th last quality is obtaind as follows: First, w can intgrat th bottom lft cornr and us th boundary conditions, which yilds g 23 (x) = x. Similarly, th top right cornr yilds g 22 (x) = + j 22 x + 2 x dx h 2, whr j 22 = 2 3 (h2 L + h2 R + h Lh R ). Now, using (39) again, w obtain th scond cumulant: C 2 = t 2 dε ( fl,l 2 + ft,l 2 + fl,r 2 + ft,r 2 + f L,L f L,R + f T,L f T,R 3 ) R 6 (6) = t 2 dε ( 3f L ( f L ) + 3f R ( f R ) + (f L f R ) 2), (62) R 3 Intgrating (6) with T L = T R = T, V L =, V R = V, w obtain (dnoting p V/(2k B T )) C 2 = t = t 3R sinh (p) (V cosh(p) + 4k B T sinh(p)) (63) V 3 R, for k BT V = t 2 R 2k BT, for V k B T (64) 4

15 Th nois powr is connctd to this cumulant by th rlation S I = 22 t C 2 (th factor 2 coms only from th original dfinition of S I ). Th rsult is in agrmnt with [6], but w hav th additional 4k B T trm. Th low-tmpratur limit is a known rsult for th shot nois powr. Morovr, th high-tmpratur valu agrs with th fluctuation dissipation thorm Third cumulant Th calculation of th third cumulant procds in th sam mannr as abov. First, w choos a paramtrization satisfying th normalization condition: { g 3 }} { Ǧ 3 = hg 33 g 23 g 22 g23h 2 2 g 32. (65) g 33 g 3 Th boundary conditions rad g 32 () = g 33 () = h L, g 32 () = g 33 () =. (66) Nxt, w xpand th xprssion for th currnt Ǐ3 (46): ( ) j Ǐ 3 = 3 g 32 2h h(g23h 2 + g 33 ) + 2h((g23h hg 33 ) + 2g 33g 22 ) g 33 2g23h 2 j 3 = const(x), (67) whr j 3 = g 23g 22 g 22g 23 + hg 32 g 32 h. Intgrating th bottom lft cornr, w obtain g 33 (x) = h L j 32 x+ x dx ( 2g23 h), 2 whr j 32 = 3 (h L +2h R ) for g 33 to satisfy th boundary condition. Using a similar procdur, w solv g 32 (x) from th uppr lft cornr, and obtain a rathr lngthy xprssion: g 32 (x) = h L + 5 ( 55h L + 2h 3 L + h R + 4h 2 Lh R + 6h L h 2 R + 8h 3 R)x 2 3 (3h R + h L ( 6 + h 2 L + h L h R + h 2 R))x h L(h L h R ) 2 x (h L(2 + h 2 L) 2( + h 2 L)h R + h 3 R)x (h L h R ) 3 x 5. Howvr, aftr som algbra, w finally arriv at ( 3 h R(h L + 2h R ) Ǐ 3 = 5 (5h L 8h 3 L + h R 6h 2 L h R + 6h L h 2 R + 8h3 R ) 3 (h L + 2h R ) 3 h R(h L + 2h R ) (68) (69) Using this rsult, w may calculat th third cumulant of th currnt using Eq. (39) (th form with f L/T,R/L is omittd): ). C 3 = t 2 dε(f L f R )(5 + 6(f 2 R + f 2 R 5 L) 3(f R + f L ) + 28f L f R ) (7) Intgrating Eq. (7) with T L = T R = T, V L =, V R = V yilds C 3 = t = t = t 3R sinh 2 (p)(v ( 3 + cosh(2p)) + 2k B T sinh(2p)) (7) V 5 R, for k BT V V 3 R, for V k BT (72) 5

16 whr w dnotd p V/(2k B T ). Not that th high-tmpratur limit of th third cumulant has no dpndnc on th tmpratur, unlik th thrmal nois. Th rsults abov agr with thos obtaind via th non-linar σ-modl [7] Highr cumulants Also th highr cumulants ar not difficult to calculat, so w omit th intrmdiat stps (which ar rathr lngthy) and display only th rsults, in trms of Frmi functions (a f L, b f R ): C 4 = t 5 2 dε( 5a + 49a 2 672a a 4 5b + 7ab R 68a 2 b + 92a 3 b + 49b 2 68ab 2 2a 2 b 2 672b ab b 4 ) C 5 = t 5 2 dε( 5a + 5a 2 28a a 4 24a 5 R + 5b 42a 2 b + 96a 3 b 64a 4 b 5b ab a 3 b b 3 96ab 3 32a 2 b 3 288b ab b 5 ) C 6 = t 23 2 dε( 24a 6 644ba a b 2 a 4 R ba 4 484a 4 276b 3 a 3 22b 2 a ba a b 4 a 2 22b 3 a b 2 a ba a 2 644b 5 a b 4 a 8448b 3 a + 848b 2 a 54ba + 23a 24b b 5 484b b b b) (73) (74) (75) Finally, intgrating with T L = T R = T, V L =, V R = V and dnoting p V/(2k B T ), w obtain C 4 = t sinh 3 (p) 42 2 [ V (33 cosh(p) cosh(3p)) R + 4k B T ( 7 sinh(p) + 5 sinh(3p)) ] C 5 = t sinh 4 (p) [ 3V ( cosh(2p) + cosh(4p)) R + 2k B T (sinh(4p) 242 sinh(2p)) ] C 6 = t sinh 5 (p) 88 2 [ 3V (2594 cosh(p) cosh(3p) + cosh(5p)) R 4k B T (878 sinh(p) sinh(3p) + 7 sinh(5p)) ] (76) (77) (78) Ths yild th low-tmpratur limits, V/(k B T ), C 4 (T = ) = t V 5R, C 5(T = ) = t C 6 (T = ) = + t V 5R, (79) V 23R, (8) 6

17 which agr with prvious rsults [8]. Morovr, on obtains th high-tmpratur limits, V/(k B T ), C 4 (T ) = + t 2k B T 3R, C 5(T ) = t C 6 (T ) = t V 5R, (8) 2k B T 5R. (82) Not that th vn cumulants sm to b proportional to T in th limit of high tmpratur, whil th vn cumulants ar only proportional to V. This is in fact a quit gnral proprty of normal mtal junctions, as shown in Sct. 4, Eq. (93). 3.2 Rlaxation in th diffusiv wir Rlaxation of th lctron distribution function to th quilibrium form, causd by inlastic scattring, affcts th statistics of currnt fluctuations. Elctron phonon scattring tnds to supprss thm, but inlastic lctron lctron scattring dos ncssarily not. This has bn known for th nois [9] (i.. th scond cumulant) for som tim, and rcntly also th highr corrlators hav bn undr activ xamination. Prdictions hav bn drivd for th third cumulant [7, 5], but through approachs diffrnt from th matrix circuit thory usd in this work. In th following, w prsnt som obsrvations concrning th rlaxation and matrix circuit thory with counting statistics in diffusiv wirs. In diffusiv wirs, th rlaxation appars in th Kldysh Usadl diffusion quation (26) through th inlastic scattring slf-nrgy matrix ˇΣ in. In a normal mtal, it is a local functional of th Grn s function Ǧ intgratd ovr th nrgy. Howvr, although th xplicit xprssion is likly to b complicatd, on can considr th situation in th limit of high lctron lctron scattring in a way don in Rf. [9], sinc ˇΣ in has known proprtis in that limit. First, xpand Ǧ as in Eq. (38) and considr th first cumulant, th currnt. Taking th trac Tr[ ˇτ K ] of Eq. (26) w find th Boltzmann diffusion quation D 2 f = I in [f] for th lctron distribution function f. It contains th collision intgral I in [f] = Tr([ˇΣ in, Ǧ]ˇτ K ), dscribing th scattring. Morovr, th two first momnts dε and dε ε of th collision intgral ar known to vanish, if f is a Frmi function. In th limit of strong lctron lctron scattring, on can assum [9] that f tnds towards a Frmi function f (V (x), T (x)), rquir th first two momnts to vanish, solv f, and finally obtain th currnt using Eq. (2). On might now assum that th right-hand sid of Eq. (26) ffctivly vanishs if f has th quilibrium form, and tak inlastic ffcts into account only 7

18 in h. W find C 2 = 2t 2 dε RL C 3 = t 2 RL 2 dε L dx f(x)( f(x)), (83) L L dx ds [ 4 3 f()2 ( f()) + (4x 2)f(x)( f(x)) 32( x) + 4f()f(x)( f(x)) + f(x) 2 ( f(x)) 3 4( x) + (f(x) 2 f() 2 ) + 8(x θ(x s))f(s)( f(s)) (84) 3 + 6(θ(x s) )f(x)f(s)( f(s)) ].27 I, at T = with rlaxd f. This xprssion for C 3 is incorrct if inlastic ffcts ar prsnt, and th rsults with rlaxd f is invalid. (Compar th krnls with thos prsntd in Rfs. [7, 5]). Howvr, th xprssion for C 2 is valid also with inlastic scattring. Givn th ad-hoc assumption undrlying th calculation, it is surprising that th mthod happns to work for C 2. Th rason for this is, howvr, unclar. Morovr, on can us th normalization condition Ǧ2 = ˇ to driv consistncy rquirmnts on th right-hand sid of Eq. (26). Considr th quation for th scond cumulant hg 23 g Ǐ2 = 23 h g hh g h 2 g 23 α2 α g 23 (hg 23 g 23 h = 22 (85) ) α 23 α 2 Hr, α jl, l =,..., 3, ar th xpansion cofficints of [ ˇΣ in, Ǧ] corrsponding to ω j. Thr ar only thr indpndnt componnts, sinc th xpandd xprssion is a commutator. Thus, g j3 = α j3 and g 23 2 h + α 23 h + α 2 =, which prvnts th us of th simplst rlaxation tim approximation (Ǧ Ǧ) τ (Ǧ Ǧ)τ 3, Ǧ 2h = const(ω), (86) whr τ = const(ω) is th rlaxation tim and Ǧ = const(ω) th function Ǧ rlaxs into. This happns bcaus th normalization rquirs, by Eq. (52), that g 2 = hg 23. W would thn hav α 2 = hα 23 = hg 23 /τ, which in turn implis that 2 h and h = h, as g 23 du to boundary conditions (42). This implis that w would hav no rlaxation, which is not what w aimd at. Suppos thn w had Ǧ dpndnt on ω. Now w hav hα 23 + α 2 = g23(h h )/τ = g23 2 h, whr w usd 2 h = (h h)/τ obtaind from th zroth ordr of ω. Thus, g 23 = g23, and g 22 can b obtaind simply by intgrating twic. In fact, xamining Eqs. (52,53) w s similarly that g j3 can b obtaind dirctly for all ordrs j > and g j2 can always b intgratd: g j has th form hg j3 + known trms, so similarly as abov, w obtain g j3 = gj3 + known trms, providd 2 h. Aftr this solving g j2 is asy, and w can procd to solving th nxt cumulant. Howvr, th problm is that w do not know th ω-dpndnc of Ǧ, as w hav not obtaind rsults for th limit of high inlastic scattring. An actual calculation of rlaxation ffcts using th quasiclassical matrix circuit thory would likly rquir considring th structur of th inlastic scattring slf nrgy matrix ˇΣ in. This is bcaus th ω-dpndnc in ˇΣ in sms to b 8

19 ndd, as othrwis on can driv consistncy rlations btwn th lmnts of ˇΣ in, and ths rlations may b impossibl to satisfy with ω-indpndnt inlastic scattring slf-nrgis. 4 Statistics of a gnric junction In th prvious sctions w solvd a diffusion quation to obtain th cumulants of th currnt. Howvr, a gnral contact can also b charactrizd by its transmission cofficints T n, which can b usd, for xampl, to calculat th currnt through th junction using Eq. (24). Notably, this xprssion is also valid in th contxt of full counting statistics, so it can b usd for th calculation of th cumulants [9]. Morovr, th distribution of th transmission cofficints T n in a diffusiv wir is known [2], so w can obtain th rsults calculatd in th prvious sction in a much mor clan and gnral mannr. Substituting th boundary conditions (4) to Eq. (24), w obtain ] hr (h L ( c) + s) 2h R c+s h L (+c+2h R s) T n [ǦL, ǦR 4+T n ( { h L ( c) + s h R (h L ( c) + s) } =, Ǧ L, ǦR 2) 2/T n (h L h R )(c ) + (h L h R )s (87) whr s τ 3 sinh ω, c cosh ω and ω iχ. Now, w valuat th currnt using Eq. (2) and obtain I = 22 π 8 dε [ ] (h R h L )c + ( h L h R )s Tr 2 2/T n n + ( h L h R )(c ) + (h R h L )s τ 3 (88) Evaluating th trac furthr, w again not that th Nambu spac is not ndd, as both trms in th trac contribut th sam amount, totaling I = dε (f L f R ) cosh ω + (f L +f R 2f L f R ) sinh ω π /T n n + (f L f R ) sinh ω + (f L +f R 2f L f R )(cosh ω ). (89) Intgrating ovr χ, w obtain th gnrating function for a gnric N-N contact S = t dε ln [ R n + T n f L ( f R ) ω + T n f R ( f L ) ω], (9) π n with R n T n (f L ( f R )+f R ( f L )). This agrs with th rsult in Rf. [9]. Howvr, thr is a mor gnral rlation, xprssd in trms of Ǧ, valid also for othr than N-N junctions [9]. If w considr nrgy-indpndnt transmission cofficints T n, i.. linar rspons, and assum a voltag-biasd junction, f L = ( + xp(ε/t )), f R = (+xp((ε+v )/T )), th intgral ovr th nrgy in Eq. (89) may b valuatd: I = 2T ω π 2 arcosh [cosh p + T n(cosh(p + ω) cosh p)] 2, (9) n whr p V/(2T ). If th distribution of transmission cofficints in an N-N junction is known, all cumulants may b obtaind analytically using Eq. (9) and Eq. (3). 9

20 Equation (9) is a vry gnral rlation, and w can immdiatly s that thr ar also gnral diffrncs btwn th vn and odd cumulants. First, w not that I(ω, p) = ω f(ω, p), whr f(ω, p) f( ω, p). Thus, n ωf( ω, p) = ( ) n n ωf(ω, p), and C n (V ) = f (n,) (, p) = ( ) n f (n,) (, p) = ( ) n C n ( V ), (92) i.. odd cumulants ar odd functions of V and vn cumulants ar vn. In fact, this is obvious from Eq. (3), as ach drivativ changs th vnnss of ach trm on right-hand sid, and th trm C n appars without a prfactor. Morovr, Eq. (9) implis that in th high-tmpratur limit, i.. p, C 2n T, C 2n+ V. (93) This can b sn as follows, ltting f, f j b vn and g, g j odd functions of ω: x cosh p + T n (cosh(p+ω) cosh p) = + arcosh 2 x = S(ω) = T [f j + g j p] p 2j (94) j= [ c k f + kpg f + O(p 2 ) ] = f + pg + O(p 2 ) (95) k= d 2j ω 2j + V j= j= (V ) d 2j+ ω 2j+ 2 + O kb 2 T T, for V k B T. (96) Hr, c j and d j ar xpansion cofficints (with d = ) and T som tmpratur, all indpndnt of ω, p, V and T. Th rlation (93) thus applis for N-N junctions in th linar rspons rgim, indpndnt of thir xact structur, if th starting point, Eq. (24) is applicabl. Not, howvr, that an nrgy dpndnc of T n may brak this symmtry at last in som dgr. W can also calculat th zro-tmpratur limit, i.. p : x cosh p + T n (cosh(p+ω) cosh p) p + T n( ω ) 2 + O( p ), (97) ω arcosh 2 x = ω (ln 2+ln x) 2 + O( x ) = 2p ω ln(+t n ( ω )) + O(), (98) assuming V >. For V <, w nd to substitut p p and ω ω to Eq. (98). Givn this asymptotic bhavior, Eq. (9) yilds I(ω) = 2 V π ω ln[( T n ) + T n ω sgn(v ) ], for k B T V, (99) n in agrmnt with Rf. [9]. Hr it can b plainly sn that in th low-tmpratur limit, C n V sgn(v ) n for both vn and odd cumulants, i.. vn cumulants do not dpnd on th dirction of th currnt. Morovr, (V t /π) ln[( T n ) + T n iχ ] is th gnrating function for a binomial distribution Pt n. This can b sn using th dfinition (), which 2

21 yilds Pt n (N) = π dχ iχn 2π π = M N= M N = M ( T n ) M N Tn N, N M N ( T n ) M N Tn N iχn () whr th numbr of attmpts is M = V t /h and th probability of succss is T n. Th total gnrating function is a sum of such functions, which implis that th total probability distribution is P t (N) = N +N 2+...=N n= Pt n (N n ). () Thus, in th physical intrprtation, particls ar transmittd indpndntly through ach ignchannl n, but only to on dirction, according to th corrsponding binomial distribution P n t (N). Thr is an uppr bound M for th numbr of attmpts for ach channl, sinc arbitrarily many frmions cannot b transmittd in a givn tim [9]. In fact, for ach nrgy in th finit tmpratur cas, Eq. (9) has a similar intrprtation: this tim w gt, howvr, a mor complicatd distribution, as particls may b transmittd to both dirctions. That is, transmission probability to th right is P R f L ( f R )T n and to th lft P L f R ( f L )T n for a givn channl at a givn nrgy. Considr now four diffrnt systms, for which th distributions of transmission cofficints ar known. Symmtric chaotic cavity: a small island coupld to both trminals with prfctly transmitting similar contacts. 2. Diffusiv wir. 3. Dirty intrfac. 4. Tunnl junction: a contact whos transmission cofficints satisfy T n. Th distributions of transmission cofficints for th first thr systms ar th following [2, 2, 22]: ρ chao (t) = G 2 t /2 t, ρ diff(t) = G 2 At a vanishing tmpratur w obtain t t, ρ dirty(t) = G π t 3/2 t. (2) I chao (ω) = 4I sgn(v ) ω ln( + ω sgn(v )/2 ), (3) I diff (ω) = I sgn(v ) ω arcosh 2 ( ω sgn(v )/2 ), (4) I dirty (ω) = 2I sgn(v ) ω ( ω sgn(v )/2 ). (5) Using ths, w obtain th Fano factors F C 2 /(I): F chao = 4, F diff = 3, F dirty = 2. (6) 2

22 At a finit tmpratur, Eq. (9) yilds cumulants, which can b mchanically calculatd (hr up to th fourth on): C 2 /C = coth(p) + 6(/2p) 4 Chaotic cavity:, C 3 /C = 3 (/2p) sinh(2p) 8 sinh 2, (7) (p) (3 cosh(p) cosh(3p)) + (/2p)( 27 sinh(p) + sinh(3p)) C 4 /C = 28 sinh 3, (8) (p) C 2 /C = coth(p) + 2(/2p) 2 Dirty intrfac: (cosh(2p) 4) + 3(/2p) sinh(2p), C 3 /C = 8 sinh 2, (p) (9) (7 cosh(p) + cosh(3p)) + 3(/2p)( 5 sinh(p) + sinh(3p)) C 4 /C = 32 sinh 3, () (p) whr again p V/(2k B T ). In practic, calculating th cumulants using this mthod rducs to straightforward valuation of intgrals and drivativs. Morovr, obtaining th analytic low-tmpratur form of th gnrating function also succdd. Finally, as a somwhat diffrnt xampl, lt us considr th tunnl junction without using Eq. (9), as th tunnling limit T n is asir to tak arlir, in Eq. (24): Ǐ = G T 2 [ǦL, ǦR], G T 2 π T n. () W could calculat this by insrting th full form (3) of Ǧ L and thn taking th currnt trac (2). Howvr, it is slightly asir to us th xpansion (3), valuat th commutator for ach powr and tak th currnt trac to find I 2n+ = G T I 2n = G T dε (f L f R ) = G T V = t C 2n+, (2) dε (f L + f R 2f L f R ) = G T V coth V 2T = t C 2n. (3) Th zro- and high-tmpratur limits for th vn cumulants ar C 2n = G T V t / and C 2n = 2G T T t /. At zro tmpratur w thus obtain th cumulantgnrating function S(χ) = G T V t / (xp(iχ) ), which corrsponds to a Poisson distribution [8] with th xpctation valu C n as notd also in Rf. [9]. This is a known proprty for charg transmission in tunnl junctions, and this paragraph simply contains a quantum mchanical drivation for it. Summarizing, if th transmission ignvalus for a junction ar known, thn th calculation of th cumulants is not difficult, at last in th linar rspons rgim. 5 Circuit thory for th first two cumulants Th mthod of xpansion usd in Sct. 3 can also b usd for studying multitrminal structurs. Expanding th Grn s functions and counting currnts in th n 22

23 counting filds, w obtain from th first ordr th Kirchhoff circuit ruls, and th scond ordr yilds ruls for th calculation of th nois corrlations. W considr hr only th first two cumulants, as thy ar not difficult to valuat. Lt us first dfin th structur of th circuit: w hav N t trminals and N n nods, th trminals indxd as i {, 2,..., N t }, and th nods as j {N t +, N t +2,..., N t +N n }. In th trminal or nod j, th Grn s function is dnotd by Ǧj. Morovr, w hav btwn th nods or trminals i and j th connctor (i, j) charactrizd by th transmission ignvalus Tn ij. For convninc of notation, dfin Tn ii =, and lt Tn ij = for non-xistnt connctors (i, j). Th counting currnt flowing in th connctor (i, j) is dnotd by I ij ( χ), and its magnitud is dtrmind using th rlation (24). As in th two-trminal cas, xpand first th Grn s functions and counting currnts as follows: N t Ǧ j = τ 3 Ǧ j + iχ k Ǧ j 2k (4) k= To th first dgr in χ this xpansion dos not diffr from that mad in Sct. 3, in particular th boundary conditions and th xpandd normalization conditions ar not changd from Eqs. (42,44). That is, th paramtrization (49) is valid for Ǧj 2k, and hnc w may writ Ǧ j 2h j, Ǧ j 2k ( hb j k b j k a j k hb j k ), (5) maintaining th normalization. In trminals, k =,..., N t, h j = h j bulk, bj k = δ jk, and a j k = δ jk. Th paramtr h is again th lctron-hol distribution matrix, and a and b ar rlatd to th rspons a fluctuation in trminal k causs in nod j. Considr nxt th xpansion of Eq. (24). First, writ Ǐ ij = [Ǧi, Ǧj ]Z({Ǧi, Ǧj }), Z(x) (2 2 /π) n T ij n (4 + T ij n (x 2)), (6) and xpand th xprssion in χ, noting that {Ǧi, Ǧj } = 2, N t Ǐ ij = [Ǧi, Ǧj ]Z(2) + (([Ǧi iχ k 2k, Ǧj ] + [Ǧi, Ǧj 2k ])Z(2) (7) k= ) Nt + [Ǧi, Ǧj ]Z (2)({Ǧi 2k, Ǧj } + {Ǧi, Ǧj 2k }) +... = Ǐij + iχ k τ 3 Ǐ ij 2k Applying now th paramtrization, w obtain Ǐ ij h = j h i 2σij, (8a) ( Ǐ ij h 2k = i b j σij k hj b i k a j k ai k + 2hi h j (b j k bi k ) ) b i k bj k (h i b j k hj b i k ) (8b) ( + σ ij (h j h i ) 2 (b j k bi k ) ) 2, k= 23

24 2 π n (T ij n ) k. Now, sinc iχ k ar linarly indpndnt, consrva- with σ ij k tion of spctral currnts applis for ach ordr sparatly, yilding th following consrvation ruls at th nods (i = N t +..., N t + N n ) for all k =,..., N t : j σ ij (hj h i ) =, j σ ij (bj k bi k) =, j Î ij 2k =, (9a) Î ij 2k σij (aj k ai k) + (σ ij (2hi h j ) + σ ij 2 (hj h i ) 2 )(b j k bi k). (9b) Hr, th sums ovr j xtnd ovr all nods and trminals (σ ij = for nonxistnt connctors), and Tr[Îij 2k ]/8 = iχ k I ij ( χ) χ= is th first-ordr xpansion cofficint in iχ k for th counting currnt from point i to j. Th quations abov may as wll b writtn in trms of th lctron distribution functions f j : j σ ij (f j f i ) =, j σ ij (bj k bi k) =, j Tr[Îij 2k ] =, 8 Tr[Îij 2k ] = (cj k ci k) (b j k bi k) [ σ ij (f i( f i ) + f j ( f j )) + (σ ij σij 2 )(f i f j ) 2], (2a) (2b) c j k 4σij (aj k + bj k ), cj k = at rsrvoirs. (2c) Th boundary conditions ar now indd simplr: at th trminals (for j =,..., N t ) c j k = and f j ar th corrsponding lctron distribution functions. Solving th linar Kirchhoff-lik quations (2) yilds dirctly th first two cumulants. From Eqs. (37) and (2), C,i = t 2 j σ ij dε (f i f j ), C 2,ij = t k 8 dε Tr[Îik 2j], (2) which satisfy t C i = I i and 2 t C 2,ij = dt δi i (t)δi j () S ij, with I i bing th currnt flowing from trminal i, and S th nois corrlation matrix for th circuit. Equations (2) ar asy to intrprt. W rcogniz th first of Eqs. (2) as th consrvation of th avrag currnts, and th third on dscribs th consrvation of th counting currnt in th first ordr in iχ. Comparing th scond condition to th first, w not that b i k indd dscribs th rspons of th distribution function f i to a chang in th distribution function f k in th trminal k. Furthrmor, th trm apparing insid th brackts in Eq. (2b) is simply th two-trminal spctral nois [, Eq. 6] btwn th points i and j in th circuit. It is wighd according to th cofficints b, and propagats to th trminals via th variabls c. That is, th nois obsrvd in a trminal is a suprposition of th noiss gnratd in individual junctions, wighd according to th proprtis of th ntwork. 5. Exampl: thr-trminal chaotic quantum dot As an xampl, lt us considr th quantum-dot structur in Fig. : a zrodimnsional island of normal mtal, connctd to rsrvoirs through thr idntical connctors. This structur is also considrd in Rf. [5] using th sam 24

25 2 4 3 Figur : Lft: A quantum dot (4) connctd to thr rsrvoirs (-3) through connctors, ij 4, 24, 34, which ar charactrizd by thir transmission ignvalus Tn ij. matrix circuit thory as considrd hr. In th following w dmonstrat xplicitly that th gnral xpansion drivd abov indd producs th sam rsults. Morovr, a similar calculation has bn don in [23]. Lt us assum that ach contact has th rsistanc R and th Fano factor F C 2 /(I), which implis that σ = /R and σ 2 = ( F )/R. Apart from th rquirmnt of symmtry, this puts no othr constraints on th typs of junctions (as long as thy may b dscribd with transmission ignvalus): th scond cumulant dpnds only on th first and scond powrs of Tn ij. Th Fano factors for various connctors ar discussd in Sct. 4. For th stup in qustion, th first of Eqs. (9) taks th form 3 R h4 = h + h 2 + h 3 R h 4 = 3 (h +h 2 +h 3 ) f 4 = 3 (f +f 2 +f 3 ), (22a) whr f j is th lctron distribution function in th rsrvoir or nod j. Th scond quation of consrvation yilds a similar rsult, 3 R b4 k = b k + b2 k + b3 k R b 4 k = 3, (22b) du to th boundary condition of b j k. For a4 5 k, w obtain 3 R a4 k = R 3 [ δ kj + ( 2h j h 4 + ( F )(h j h 4 ) 2) (δ jk 3 ] ), j= (22c) Tr[a 4 ] = (f 3( f 3 ) + f 2 ( f 2 ) 2f ( f )) + 8F 27 (f2 2 + f 2 3 2f 2 + 2f (f 2 + f 3 ) 4f 2 f 3 ), (22d) whr th boundary conditions for a and b ar writtn out, and b 4 k and h4 ar substitutd. Th quantitis Tr[a 4 2] and Tr[a 4 3] may b obtaind by prmuting th subscript indxs in Eq. (22d), du to th symmtry of th structur. Insrting Tr[a 4 k ] to th xprssion for Tr[Î4j 2k ] now yilds, aftr som simpl 5 By virtu of symmtry (27), both lmnts of th Nambu spac contribut th sam amount, hnc taking th trac may b don by substituting h j ( 2f j ) and multiplying th rsulting xprssion by 2. 25

26 but long-windd algbra, 2(3F 7)f 2 + (3F 5)f (3F 5)f f + 9(f 2 + f 3 ) 4(f f 2 + f f 3 + f 2 f 3 ) for j, k = Tr[Î4j 2k ] = 6F f (f 2 + f 3 ) 27R (7 3F )(f 2 + f 2 2) 2f (f f 2 + f f 3 + f 2 f 3 ) for j =, k = 2. 9(f + f 2 ) + 6F f f 2 (23) Othr lmnts may again b obtaind by prmuting th indxs. For a structur at a constant tmpratur T, k B T = dε f i ( f i ), (V i V j ) = dε (f i f j ), and (24a) w ij (Vi V j ) dε (f i + f j 2f i f j ) = (V i V j ) coth, (24b) 2k B T whr f j = /( ε Vj + ) and V j ar th voltags at th rsrvoirs. Intgrating Eq. (23) ovr th nrgy thus yilds lmnts of th nois corrlation matrix: 27R S ii = 2(2 F )k B T + (2 + 3F )(w ij + w ik ) + 2w jk, 27R S ij = 6(2 + F )k B T ( + 3F )w ij w ik w jk. (25a) (25b) whr i, j, k {, 2, 3} and i j, i k, j k. As a sid not, at constant tmpratur, th formalism always yilds linar combinations of th trms apparing in (24), hnc it is asy to obtain rsults at a finit-tmpratur. Th limit of vanishing tmpratur is asy to tak: w ij V i V j as T, and th trms proportional to T vanish. For xampl, considr a masurmnt whr V = V 2 =, V 3 = V and th tmpratur is vanishing. Th nois corrlations in such a situation ar asily found from Eq. (25): S = V 27R 4 + 3F 2 (2 + 3F ) F (2 + 3F ). (26) (2 + 3F ) (2 + 3F ) 4 + 6F Th rsult abov fully agrs with th on obtaind in Rf. [5]. Morovr, not that th zro-tmpratur shot nois hr is not proportional to F as it would b in two-trminal junctions. Th xtra nois is du to th fact that th lctrons at th nods ar not in (quasi-)quilibrium and f j ar not Frmi functions at th nods. This may b sn by intgrating Eqs. (2) ovr th nrgy assuming that f j ar Frmi functions, which yilds shot nois that is proportional to F in ach junction. 6 Discussion In this work, w applid th quasiclassical Kldysh-formulation of full counting statistics to th calculation of th currnt probability distributions in various msoscopic lctrical circuits. Most of th rsults prsntd hr hav also bn obtaind arlir, although th systmatical xpansion of th Grn s functions 26